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Matroid.uniqueBaseOn_dual_eq | Mathlib/Data/Matroid/Constructions.lean | theorem uniqueBaseOn_dual_eq (I E : Set α) :
(uniqueBaseOn I E)✶ = uniqueBaseOn (E \ I) E | α : Type u_1
I E B : Set α
hB : B ⊆ E
h : B = E \ I
⊢ E \ B = I ∩ E | rw [h, inter_comm I] | α : Type u_1
I E B : Set α
hB : B ⊆ E
h : B = E \ I
⊢ E \ (E \ I) = E ∩ I | 60744eced319574b |
Finsupp.linearCombinationOn_range | Mathlib/LinearAlgebra/Finsupp/LinearCombination.lean | theorem linearCombinationOn_range (s : Set α) :
LinearMap.range (linearCombinationOn α M R v s) = ⊤ | α : Type u_1
M : Type u_2
R : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
v : α → M
s : Set α
⊢ span R (v '' s) ≤ map (linearCombination R v) (supported R R s) | exact (span_image_eq_map_linearCombination _ _).le | no goals | 5d96391416327877 |
CoxeterSystem.prod_alternatingWord_eq_prod_alternatingWord_sub | Mathlib/GroupTheory/Coxeter/Basic.lean | theorem prod_alternatingWord_eq_prod_alternatingWord_sub (i i' : B) (m : ℕ) (hm : m ≤ M i i' * 2) :
π (alternatingWord i i' m) = π (alternatingWord i' i (M i i' * 2 - m)) | B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
hm : m ≤ M.M i i' * 2
⊢ (if Even ↑m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (↑m / ↑2) =
(if Even (↑(M.M i i' * 2) - ↑m) then 1 else cs.simple i) *
(cs.simple i' * cs.simple i) ^ ((↑(M.M i i' * 2) - ↑m) / ↑2) | generalize (m : ℤ) = m' | B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
hm : m ≤ M.M i i' * 2
m' : ℤ
⊢ (if Even m' then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m' / ↑2) =
(if Even (↑(M.M i i' * 2) - m') then 1 else cs.simple i) *
(cs.simple i' * cs.simple i) ^ ((↑(M.M i i' * 2) - m') / ↑2) | 8b276f2d43ad82bd |
IsCompactlyGenerated.BooleanGenerators.sSup_inter | Mathlib/Order/BooleanGenerators.lean | lemma sSup_inter (hS : BooleanGenerators S) {T₁ T₂ : Set α} (hT₁ : T₁ ⊆ S) (hT₂ : T₂ ⊆ S) :
sSup (T₁ ∩ T₂) = (sSup T₁) ⊓ (sSup T₂) | case a.a
α : Type u_1
inst✝¹ : CompleteLattice α
S : Set α
inst✝ : IsCompactlyGenerated α
hS : BooleanGenerators S
T₁ T₂ : Set α
hT₁ : T₁ ⊆ S
hT₂ : T₂ ⊆ S
⊢ sSup (T₁ ∩ T₂) ≤ sSup T₁ | apply sSup_le_sSup Set.inter_subset_left | no goals | 7804e1cb5854118b |
CategoryTheory.ShortComplex.exact_iff_mono | Mathlib/Algebra/Homology/ShortComplex/Exact.lean | lemma exact_iff_mono [HasZeroObject C] (hf : S.f = 0) :
S.Exact ↔ Mono S.g | case mp
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex C
inst✝ : HasZeroObject C
hf : S.f = 0
this✝ : S.HasHomology
h : IsZero S.homology
this : IsIso S.pOpcycles
⊢ Mono S.g | have := mono_of_isZero_kernel' _ S.homologyIsKernel h | case mp
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex C
inst✝ : HasZeroObject C
hf : S.f = 0
this✝¹ : S.HasHomology
h : IsZero S.homology
this✝ : IsIso S.pOpcycles
this : Mono S.fromOpcycles
⊢ Mono S.g | f4e5eedef442a362 |
Real.sign_eq_zero_iff | Mathlib/Data/Real/Sign.lean | theorem sign_eq_zero_iff {r : ℝ} : sign r = 0 ↔ r = 0 | case inl
r : ℝ
h : r.sign = 0
hn : r < 0
⊢ r = 0 | rw [sign_of_neg hn, neg_eq_zero] at h | case inl
r : ℝ
h : 1 = 0
hn : r < 0
⊢ r = 0 | 2e6f33eeda35157a |
WeierstrassCurve.exists_variableChange_of_char_two | Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean | private lemma exists_variableChange_of_char_two (heq : E.j = E'.j) :
∃ C : VariableChange F, E.variableChange C = E' | case intro.of_j_ne_zero.intro.of_j_ne_zero
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 2
heq : E.j = E'.j
C : VariableChange F
inst✝¹ : (E.variableChange C).IsCharTwoJNeZeroNF
C' : VariableChange F
inst✝ : (E'.variableChange C').IsCharTwoJNeZeroNF
⊢ ∃ C, E.variableChange C = E' | simp_rw [← variableChange_j E C, ← variableChange_j E' C',
j_of_isCharTwoJNeZeroNF_of_char_two, one_div, inv_inj] at heq | case intro.of_j_ne_zero.intro.of_j_ne_zero
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 2
C : VariableChange F
inst✝¹ : (E.variableChange C).IsCharTwoJNeZeroNF
C' : VariableChange F
inst✝ : (E'.variableChange C').IsCharTwoJNeZeroNF
heq : (E.variableChange C).a₆ = (E'.variableChange C').a₆
⊢ ∃ C, E.variableChange C = E' | 777fc2b6b6b2f928 |
Std.Sat.CNF.nonempty_or_impossible | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/CNF/Relabel.lean | theorem nonempty_or_impossible (f : CNF α) : Nonempty α ∨ ∃ n, f = List.replicate n [] | case nil
α : Type u_1
⊢ Nonempty α ∨ ∃ n, [] = List.replicate n [] | exact Or.inr ⟨0, rfl⟩ | no goals | eb153cbe9df80b19 |
MeasureTheory.ae_nonneg_of_forall_setIntegral_nonneg | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | theorem ae_nonneg_of_forall_setIntegral_nonneg (hf : Integrable f μ)
(hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
b : ℝ
hb_neg : b < 0
s : Set α := {x | f x ≤ b}
hs : NullMeasurableSet s μ
mus : μ s < ⊤
⊢ ∀ᵐ (x : α) ∂μ, x ∈ s → f x ≤ b | exact Eventually.of_forall fun x hxs => hxs | no goals | d52a385303ec1354 |
MvPowerSeries.WithPiTopology.tendsto_pow_of_constantCoeff_nilpotent_iff | Mathlib/RingTheory/MvPowerSeries/PiTopology.lean | theorem tendsto_pow_of_constantCoeff_nilpotent_iff [CommRing R] [DiscreteTopology R] (f) :
Tendsto (fun n : ℕ => f ^ n) atTop (nhds 0) ↔
IsNilpotent (constantCoeff σ R f) | σ : Type u_1
R : Type u_2
inst✝² : TopologicalSpace R
inst✝¹ : CommRing R
inst✝ : DiscreteTopology R
f : MvPowerSeries σ R
h : Tendsto (fun n => f ^ n) atTop (nhds 0)
this : Tendsto (fun n => (constantCoeff σ R) (f ^ n)) atTop (nhds 0)
⊢ IsNilpotent ((constantCoeff σ R) f) | simp only [tendsto_def] at this | σ : Type u_1
R : Type u_2
inst✝² : TopologicalSpace R
inst✝¹ : CommRing R
inst✝ : DiscreteTopology R
f : MvPowerSeries σ R
h : Tendsto (fun n => f ^ n) atTop (nhds 0)
this : ∀ s ∈ nhds 0, (fun n => (constantCoeff σ R) (f ^ n)) ⁻¹' s ∈ atTop
⊢ IsNilpotent ((constantCoeff σ R) f) | 75b216f0c0004b82 |
RCLike.add_conj | Mathlib/Analysis/RCLike/Basic.lean | theorem add_conj (z : K) : z + conj z = 2 * re z :=
calc
z + conj z = re z + im z * I + (re z - im z * I) | K : Type u_1
inst✝ : RCLike K
z : K
⊢ ↑(re z) + ↑(im z) * I + (↑(re z) - ↑(im z) * I) = 2 * ↑(re z) | rw [add_add_sub_cancel, two_mul] | no goals | 935131c31d9ee0b1 |
Array.exists_of_eraseP | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Erase.lean | theorem exists_of_eraseP {l : Array α} {a} (hm : a ∈ l) (hp : p a) :
∃ a l₁ l₂, (∀ b ∈ l₁, ¬p b) ∧ p a ∧ l = l₁.push a ++ l₂ ∧ l.eraseP p = l₁ ++ l₂ | α : Type u_1
p : α → Bool
l : Array α
a : α
hm : a ∈ l
hp : p a = true
⊢ ∃ a l₁ l₂, (∀ (b : α), b ∈ l₁ → ¬p b = true) ∧ p a = true ∧ l = l₁.push a ++ l₂ ∧ l.eraseP p = l₁ ++ l₂ | rcases l with ⟨l⟩ | case mk
α : Type u_1
p : α → Bool
a : α
hp : p a = true
l : List α
hm : a ∈ { toList := l }
⊢ ∃ a l₁ l₂,
(∀ (b : α), b ∈ l₁ → ¬p b = true) ∧
p a = true ∧ { toList := l } = l₁.push a ++ l₂ ∧ { toList := l }.eraseP p = l₁ ++ l₂ | 3081e6f43f7730fa |
Matrix.zpow_sub_one | Mathlib/LinearAlgebra/Matrix/ZPow.lean | theorem zpow_sub_one {A : M} (h : IsUnit A.det) (n : ℤ) : A ^ (n - 1) = A ^ n * A⁻¹ :=
calc
A ^ (n - 1) = A ^ (n - 1) * A * A⁻¹ | n' : Type u_1
inst✝² : DecidableEq n'
inst✝¹ : Fintype n'
R : Type u_2
inst✝ : CommRing R
A : M
h : IsUnit A.det
n : ℤ
⊢ A ^ (n - 1) = A ^ (n - 1) * A * A⁻¹ | rw [mul_assoc, mul_nonsing_inv _ h, mul_one] | no goals | 21343db4aa6e3915 |
Complex.norm_sub_mem_Icc_angle | Mathlib/Analysis/Complex/Angle.lean | /-- Chord-length is a multiple of arc-length up to constants. -/
lemma norm_sub_mem_Icc_angle (hx : ‖x‖ = 1) (hy : ‖y‖ = 1) :
‖x - y‖ ∈ Icc (2 / π * angle x y) (angle x y) | x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ 2 * (1 - Real.cos θ) = (Real.cos θ - 1) ^ 2 + Real.sin θ ^ 2 | linear_combination -θ.cos_sq_add_sin_sq | no goals | 5f1a2d709d27832f |
UniformSpace.secondCountable_of_separable | Mathlib/Topology/UniformSpace/Cauchy.lean | theorem secondCountable_of_separable [SeparableSpace α] : SecondCountableTopology α | case intro.intro.intro.intro.refine_2.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u
uniformSpace : UniformSpace α
inst✝¹ : (𝓤 α).IsCountablyGenerated
inst✝ : SeparableSpace α
s : Set α
hsc : s.Countable
hsd : Dense s
t : ℕ → Set (α × α)
h_basis : (𝓤 α).HasAntitoneBasis t
ht_mem : ∀ (i : ℕ), t i ∈ (𝓤 α).sets
hto : ∀ (i : ℕ), IsOpen (t i)
hts : ∀ (i : ℕ), SymmetricRel (t i)
x : α
V : Set α
hxV : x ∈ V
hVo : IsOpen V
U : Set (α × α)
hU : U ∈ 𝓤 α
hUV : ball x U ⊆ V
U' : Set (α × α)
hU' : U' ∈ 𝓤 α
left✝ : ∀ {a b : α}, (a, b) ∈ U' → (b, a) ∈ U'
hUU' : U' ○ U' ⊆ U
k : ℕ
hk : t k ⊆ U'
y : α
hxy : y ∈ ball x (t k)
hys : y ∈ s
⊢ ∃ v, (∃ i ∈ s, ∃ y, ball i (t y) = v) ∧ x ∈ v ∧ v ⊆ V | refine ⟨_, ⟨y, hys, k, rfl⟩, (hts k).subset hxy, fun z hz => ?_⟩ | case intro.intro.intro.intro.refine_2.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u
uniformSpace : UniformSpace α
inst✝¹ : (𝓤 α).IsCountablyGenerated
inst✝ : SeparableSpace α
s : Set α
hsc : s.Countable
hsd : Dense s
t : ℕ → Set (α × α)
h_basis : (𝓤 α).HasAntitoneBasis t
ht_mem : ∀ (i : ℕ), t i ∈ (𝓤 α).sets
hto : ∀ (i : ℕ), IsOpen (t i)
hts : ∀ (i : ℕ), SymmetricRel (t i)
x : α
V : Set α
hxV : x ∈ V
hVo : IsOpen V
U : Set (α × α)
hU : U ∈ 𝓤 α
hUV : ball x U ⊆ V
U' : Set (α × α)
hU' : U' ∈ 𝓤 α
left✝ : ∀ {a b : α}, (a, b) ∈ U' → (b, a) ∈ U'
hUU' : U' ○ U' ⊆ U
k : ℕ
hk : t k ⊆ U'
y : α
hxy : y ∈ ball x (t k)
hys : y ∈ s
z : α
hz : z ∈ ball y (t k)
⊢ z ∈ V | 4ad08f66c78319f3 |
Std.Tactic.BVDecide.BVExpr.bitblast.go_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Impl/Expr.lean | theorem bitblast.go_decl_eq (aig : AIG BVBit) (expr : BVExpr w) :
∀ (idx : Nat) (h1) (h2), (go aig expr).val.aig.decls[idx]'h2 = aig.decls[idx]'h1 | case const
w idx w✝ : Nat
val✝ : BitVec w✝
aig : AIG BVBit
h1 : idx < aig.decls.size
h2 : idx < (go aig (const val✝)).val.aig.decls.size
⊢ (go aig (const val✝)).val.aig.decls[idx] = aig.decls[idx] | dsimp only [go] | case const
w idx w✝ : Nat
val✝ : BitVec w✝
aig : AIG BVBit
h1 : idx < aig.decls.size
h2 : idx < (go aig (const val✝)).val.aig.decls.size
⊢ (blastConst aig val✝).aig.decls[idx] = aig.decls[idx] | f92988d5a1595b39 |
ContinuousMultilinearMap.changeOrigin_toFormalMultilinearSeries | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | theorem changeOrigin_toFormalMultilinearSeries [DecidableEq ι] :
continuousMultilinearCurryFin1 𝕜 (∀ i, E i) F (f.toFormalMultilinearSeries.changeOrigin x 1) =
f.linearDeriv x | case h.inr
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
x : (i : ι) → E i
inst✝ : DecidableEq ι
y : (i : ι) → E i
h✝ : Nonempty ι
⊢ (∑ a : { s // #s = Fintype.card ι - 1 },
((f.toFormalMultilinearSeries.changeOriginSeriesTerm 1 (Fintype.card ι - 1) ↑a ⋯) fun x_1 => x) fun x => y) =
∑ i : ι, f (Function.update x i (y i)) | simp_rw [changeOriginSeriesTerm_apply] | case h.inr
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
x : (i : ι) → E i
inst✝ : DecidableEq ι
y : (i : ι) → E i
h✝ : Nonempty ι
⊢ ∑ x_1 : { s // #s = Fintype.card ι - 1 },
(f.toFormalMultilinearSeries (1 + (Fintype.card ι - 1))) ((↑x_1).piecewise (fun x_2 => x) fun x => y) =
∑ i : ι, f (Function.update x i (y i)) | d7dded01cf2a4e1b |
QuaternionAlgebra.Basis.lift_mul | Mathlib/Algebra/QuaternionBasis.lean | theorem lift_mul (x y : ℍ[R,c₁,c₂,c₃]) : q.lift (x * y) = q.lift x * q.lift y | R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
c₁ c₂ c₃ : R
q : Basis A c₁ c₂ c₃
x y : ℍ[R,c₁,c₂,c₃]
⊢ (x * y).re • 1 + (x * y).imI • q.i + (x * y).imJ • q.j + (x * y).imK • q.k =
(x.re * y.re) • 1 + (x.re * y.imI) • q.i + (x.re * y.imJ) • q.j + (x.re * y.imK) • q.k + (x.imI * y.re) • q.i +
(x.imI * y.imI) • (c₁ • 1 + c₂ • q.i) +
(x.imI * y.imJ) • q.k +
(x.imI * y.imK) • (c₁ • q.j + c₂ • q.k) +
(x.imJ * y.re) • q.j +
(x.imJ * y.imI) • (c₂ • q.j + -q.k) +
(x.imJ * c₃ * y.imJ) • 1 +
(x.imJ * y.imK) • ((c₂ * c₃) • 1 + -(c₃ • q.i)) +
(x.imK * y.re) • q.k +
-((c₁ * x.imK * y.imI) • q.j) +
(x.imK * c₃ * y.imJ) • q.i +
-((c₁ * c₃ * x.imK * y.imK) • 1) | simp only [mul_re, sub_eq_add_neg, add_smul, neg_smul, mul_imI, ← add_assoc, mul_imJ, mul_imK] | R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
c₁ c₂ c₃ : R
q : Basis A c₁ c₂ c₃
x y : ℍ[R,c₁,c₂,c₃]
⊢ (x.re * y.re) • 1 + (c₁ * x.imI * y.imI) • 1 + (c₃ * x.imJ * y.imJ) • 1 + (c₂ * c₃ * x.imJ * y.imK) • 1 +
-((c₁ * c₃ * x.imK * y.imK) • 1) +
(x.re * y.imI) • q.i +
(x.imI * y.re) • q.i +
(c₂ * x.imI * y.imI) • q.i +
-((c₃ * x.imJ * y.imK) • q.i) +
(c₃ * x.imK * y.imJ) • q.i +
(x.re * y.imJ) • q.j +
(c₁ * x.imI * y.imK) • q.j +
(x.imJ * y.re) • q.j +
(c₂ * x.imJ * y.imI) • q.j +
-((c₁ * x.imK * y.imI) • q.j) +
(x.re * y.imK) • q.k +
(x.imI * y.imJ) • q.k +
(c₂ * x.imI * y.imK) • q.k +
-((x.imJ * y.imI) • q.k) +
(x.imK * y.re) • q.k =
(x.re * y.re) • 1 + (x.re * y.imI) • q.i + (x.re * y.imJ) • q.j + (x.re * y.imK) • q.k + (x.imI * y.re) • q.i +
(x.imI * y.imI) • (c₁ • 1 + c₂ • q.i) +
(x.imI * y.imJ) • q.k +
(x.imI * y.imK) • (c₁ • q.j + c₂ • q.k) +
(x.imJ * y.re) • q.j +
(x.imJ * y.imI) • (c₂ • q.j + -q.k) +
(x.imJ * c₃ * y.imJ) • 1 +
(x.imJ * y.imK) • ((c₂ * c₃) • 1 + -(c₃ • q.i)) +
(x.imK * y.re) • q.k +
-((c₁ * x.imK * y.imI) • q.j) +
(x.imK * c₃ * y.imJ) • q.i +
-((c₁ * c₃ * x.imK * y.imK) • 1) | 09bfbbd836e86c34 |
CategoryTheory.Dial.tensor_comp | Mathlib/CategoryTheory/Dialectica/Monoidal.lean | theorem tensor_comp {X₁ Y₁ Z₁ X₂ Y₂ Z₂ : Dial C}
(f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : Y₁ ⟶ Z₁) (g₂ : Y₂ ⟶ Z₂) :
tensorHom (f₁ ≫ g₁) (f₂ ≫ g₂) = tensorHom f₁ f₂ ≫ tensorHom g₁ g₂ | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasFiniteProducts C
inst✝ : HasPullbacks C
X₁ Y₁ Z₁ X₂ Y₂ Z₂ : Dial C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
g₁ : Y₁ ⟶ Z₁
g₂ : Y₂ ⟶ Z₂
⊢ tensorHom (f₁ ≫ g₁) (f₂ ≫ g₂) = tensorHom f₁ f₂ ≫ tensorHom g₁ g₂ | ext <;> simp | case hF
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasFiniteProducts C
inst✝ : HasPullbacks C
X₁ Y₁ Z₁ X₂ Y₂ Z₂ : Dial C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
g₁ : Y₁ ⟶ Z₁
g₂ : Y₂ ⟶ Z₂
⊢ π(prod.map π₁ π₁ ≫ π(π₁, prod.map f₁.f (𝟙 Z₁.tgt) ≫ g₁.F) ≫ f₁.F,
prod.map π₂ π₂ ≫ π(π₁, prod.map f₂.f (𝟙 Z₂.tgt) ≫ g₂.F) ≫ f₂.F) =
π(π(π₁ ≫ π₁, prod.map (π₁ ≫ f₁.f) π₁ ≫ g₁.F) ≫ f₁.F, π(π₁ ≫ π₂, prod.map (π₂ ≫ f₂.f) π₂ ≫ g₂.F) ≫ f₂.F) | 1e9289a58304a4a7 |
Module.support_of_noZeroSMulDivisors | Mathlib/RingTheory/Support.lean | lemma Module.support_of_noZeroSMulDivisors [NoZeroSMulDivisors R M] [Nontrivial M] :
Module.support R M = Set.univ | R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : NoZeroSMulDivisors R M
inst✝ : Nontrivial M
⊢ support R M = Set.univ | simp only [Set.eq_univ_iff_forall, mem_support_iff', ne_eq, smul_eq_zero, not_or] | R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : NoZeroSMulDivisors R M
inst✝ : Nontrivial M
⊢ ∀ (x : PrimeSpectrum R), ∃ m, ∀ r ∉ x.asIdeal, ¬r = 0 ∧ ¬m = 0 | 40f3ce29daf90d9a |
Ordinal.CNF_sorted | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·) | b o : Ordinal.{u_1}
⊢ Sorted (fun x1 x2 => x1 > x2) (map Prod.fst (CNF b o)) | refine CNFRec b ?_ (fun o ho IH ↦ ?_) o | case refine_1
b o : Ordinal.{u_1}
⊢ Sorted (fun x1 x2 => x1 > x2) (map Prod.fst (CNF b 0))
case refine_2
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : Sorted (fun x1 x2 => x1 > x2) (map Prod.fst (CNF b (o % b ^ log b o)))
⊢ Sorted (fun x1 x2 => x1 > x2) (map Prod.fst (CNF b o)) | d982c7966c162266 |
interior_subset_gauge_lt_one | Mathlib/Analysis/Convex/Gauge.lean | theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ { x | gauge s x < 1 } | E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
s : Set E
⊢ interior s ⊆ {x | gauge s x < 1} | intro x hx | E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
s : Set E
x : E
hx : x ∈ interior s
⊢ x ∈ {x | gauge s x < 1} | 3cd737264597e0ba |
PMF.support_normalize | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | theorem support_normalize : (normalize f hf0 hf).support = Function.support f :=
Set.ext fun a => by simp [hf, mem_support_iff]
| α : Type u_1
f : α → ℝ≥0∞
hf0 : tsum f ≠ 0
hf : tsum f ≠ ⊤
a : α
⊢ a ∈ (normalize f hf0 hf).support ↔ a ∈ Function.support f | simp [hf, mem_support_iff] | no goals | 6e34b3d5b5061659 |
continuousAt_gauge_zero | Mathlib/Analysis/Convex/Gauge.lean | theorem continuousAt_gauge_zero (hs : s ∈ 𝓝 0) : ContinuousAt (gauge s) 0 | E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
hs : s ∈ 𝓝 0
⊢ ContinuousAt (gauge s) 0 | rw [ContinuousAt, gauge_zero] | E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
hs : s ∈ 𝓝 0
⊢ Tendsto (gauge s) (𝓝 0) (𝓝 0) | 5dd1f9e9b5f3bede |
t2_separation_compact_nhds | Mathlib/Topology/Separation/Hausdorff.lean | theorem t2_separation_compact_nhds [LocallyCompactSpace X] [T2Space X] {x y : X} (h : x ≠ y) :
∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ IsCompact u ∧ IsCompact v ∧ Disjoint u v | X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : LocallyCompactSpace X
inst✝ : T2Space X
x y : X
h : x ≠ y
⊢ ∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ IsCompact u ∧ IsCompact v ∧ Disjoint u v | simpa only [exists_prop, ← exists_and_left, and_comm, and_assoc, and_left_comm] using
((compact_basis_nhds x).disjoint_iff (compact_basis_nhds y)).1 (disjoint_nhds_nhds.2 h) | no goals | ee342c49880ccde1 |
sum_mul_eq_sub_integral_mul₀ | Mathlib/NumberTheory/AbelSummation.lean | theorem sum_mul_eq_sub_integral_mul₀ (hc : c 0 = 0) (b : ℝ)
(hf_diff : ∀ t ∈ Set.Icc 1 b, DifferentiableAt ℝ f t)
(hf_int : IntegrableOn (deriv f) (Set.Icc 1 b)) :
∑ k ∈ Icc 0 ⌊b⌋₊, f k * c k =
f b * (∑ k ∈ Icc 0 ⌊b⌋₊, c k) - ∫ t in Set.Ioc 1 b, deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k | 𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
hc : c 0 = 0
b : ℝ
hf_diff : ∀ t ∈ Set.Icc 1 b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc 1 b) volume
⊢ ∑ k ∈ Icc 0 ⌊b⌋₊, f ↑k * c k =
f b * ∑ k ∈ Icc 0 ⌊b⌋₊, c k - ∫ (t : ℝ) in Set.Ioc 1 b, deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k | obtain hb | hb := le_or_gt 1 b | case inl
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
hc : c 0 = 0
b : ℝ
hf_diff : ∀ t ∈ Set.Icc 1 b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc 1 b) volume
hb : 1 ≤ b
⊢ ∑ k ∈ Icc 0 ⌊b⌋₊, f ↑k * c k =
f b * ∑ k ∈ Icc 0 ⌊b⌋₊, c k - ∫ (t : ℝ) in Set.Ioc 1 b, deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k
case inr
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
hc : c 0 = 0
b : ℝ
hf_diff : ∀ t ∈ Set.Icc 1 b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc 1 b) volume
hb : 1 > b
⊢ ∑ k ∈ Icc 0 ⌊b⌋₊, f ↑k * c k =
f b * ∑ k ∈ Icc 0 ⌊b⌋₊, c k - ∫ (t : ℝ) in Set.Ioc 1 b, deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k | 715329581e1c96e7 |
Nat.minSqFac_has_prop | Mathlib/Data/Nat/Squarefree.lean | theorem minSqFac_has_prop (n : ℕ) : MinSqFacProp n (minSqFac n) | case neg.inr
n : ℕ
d2 : ¬2 ∣ n
n0 : n > 0
⊢ n.MinSqFacProp (n.minSqFacAux 3) | refine minSqFacAux_has_prop _ n0 0 rfl ?_ | case neg.inr
n : ℕ
d2 : ¬2 ∣ n
n0 : n > 0
⊢ ∀ (m : ℕ), Prime m → m ∣ n → 3 ≤ m | 60eba7de40a3a105 |
UniformContinuousOn.continuousOn | Mathlib/Topology/UniformSpace/Basic.lean | theorem UniformContinuousOn.continuousOn [UniformSpace α] [UniformSpace β] {f : α → β} {s : Set α}
(h : UniformContinuousOn f s) : ContinuousOn f s | α : Type ua
β : Type ub
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
s : Set α
h : UniformContinuousOn f s
⊢ ContinuousOn f s | rw [uniformContinuousOn_iff_restrict] at h | α : Type ua
β : Type ub
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
s : Set α
h : UniformContinuous (s.restrict f)
⊢ ContinuousOn f s | 5802946f1c9cbf58 |
irreducible_pow_sup_of_le | Mathlib/RingTheory/DedekindDomain/Ideal.lean | theorem irreducible_pow_sup_of_le (hJ : Irreducible J) (n : ℕ) (hn : n ≤ emultiplicity J I) :
J ^ n ⊔ I = J ^ n | case neg
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hJ : Irreducible J
n : ℕ
hn : ↑n ≤ emultiplicity J I
hI : ¬I = ⊥
⊢ n ≤ count J (normalizedFactors I) | rw [emultiplicity_eq_count_normalizedFactors hJ hI, normalize_eq J] at hn | case neg
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hJ : Irreducible J
n : ℕ
hn : ↑n ≤ ↑(count J (normalizedFactors I))
hI : ¬I = ⊥
⊢ n ≤ count J (normalizedFactors I) | c29a75c2640d6fa6 |
PadicInt.bojanic_mahler_step2 | Mathlib/NumberTheory/Padics/MahlerBasis.lean | /--
Second step in Bojanić's proof of Mahler's theorem (equation (11) of [bojanic74]): show that values
`Δ_[1]^[n + p ^ t] f 0` for large enough `n` are bounded by the max of `(‖f‖ / p ^ s)` and `1 / p`
times a sup over values for smaller `n`.
We use `nnnorm`s on the RHS since `Finset.sup` requires an order with a bottom element.
-/
private lemma bojanic_mahler_step2 {f : C(ℤ_[p], E)} {s t : ℕ}
(hst : ∀ x y : ℤ_[p], ‖x - y‖ ≤ p ^ (-t : ℤ) → ‖f x - f y‖ ≤ ‖f‖ / p ^ s) (n : ℕ) :
‖Δ_[1]^[n + p ^ t] f 0‖ ≤ max ↑((Finset.range (p ^ t - 1)).sup
fun j ↦ ‖Δ_[1]^[n + (j + 1)] f 0‖₊ / p) (‖f‖ / p ^ s) | p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℚ_[p] E
inst✝ : IsUltrametricDist E
f : C(ℤ_[p], E)
s t : ℕ
hst : ∀ (x y : ℤ_[p]), ‖x - y‖ ≤ ↑p ^ (-↑t) → ‖f x - f y‖ ≤ ‖f‖ / ↑p ^ s
n : ℕ
⊢ ‖Δ_[1]^[n + p ^ t] (⇑f) 0‖ ≤ ↑((range (p ^ t - 1)).sup fun j => ‖Δ_[1]^[n + (j + 1)] (⇑f) 0‖₊ / ↑p) ⊔ ‖f‖ / ↑p ^ s | rw [bojanic_mahler_step1 _ _ (one_le_pow₀ hp.out.one_le)] | p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℚ_[p] E
inst✝ : IsUltrametricDist E
f : C(ℤ_[p], E)
s t : ℕ
hst : ∀ (x y : ℤ_[p]), ‖x - y‖ ≤ ↑p ^ (-↑t) → ‖f x - f y‖ ≤ ‖f‖ / ↑p ^ s
n : ℕ
⊢ ‖-∑ j ∈ range (p ^ t - 1), (p ^ t).choose (j + 1) • Δ_[1]^[n + (j + 1)] (⇑f) 0 +
∑ k ∈ range (n + 1), ((-1) ^ (n - k) * ↑(n.choose k)) • (f (↑k + ↑(p ^ t)) - f ↑k)‖ ≤
↑((range (p ^ t - 1)).sup fun j => ‖Δ_[1]^[n + (j + 1)] (⇑f) 0‖₊ / ↑p) ⊔ ‖f‖ / ↑p ^ s | 2db3408803885b9e |
Real.strictConcaveOn_rpow | Mathlib/Analysis/Convex/SpecificFunctions/Pow.lean | lemma strictConcaveOn_rpow {p : ℝ} (hp₀ : 0 < p) (hp₁ : p < 1) :
StrictConcaveOn ℝ (Set.Ici 0) fun x : ℝ ↦ x ^ p | case a
p : ℝ
hp₀ : 0 < p
hp₁ : p < 1
x : ℝ
hx : x ∈ Ici 0
y : ℝ
hy : y ∈ Ici 0
hxy : x ≠ y
a b : ℝ
ha : 0 < a
hb : 0 < b
hab : a + b = 1
x' : ℝ≥0 := ⟨x, hx⟩
y' : ℝ≥0 := ⟨y, hy⟩
a' : ℝ≥0 := ⟨a, ⋯⟩
b' : ℝ≥0 := ⟨b, ⋯⟩
hxy' : x' ≠ y'
⊢ ↑(a' + b') = ↑1 | simp [a', b', hab] | no goals | e355702cbbe8ec4b |
IncidenceAlgebra.mu_toDual | Mathlib/Combinatorics/Enumerative/IncidenceAlgebra.lean | @[simp]
lemma mu_toDual (a b : α) : mu 𝕜 (toDual a) (toDual b) = mu 𝕜 b a | 𝕜 : Type u_2
α : Type u_5
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x1 x2 => x1 ≤ x2 := Classical.decRel fun x1 x2 => x1 ≤ x2
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
⊢ mud * zeta 𝕜 = 1 | ext a b | case h
𝕜 : Type u_2
α : Type u_5
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x1 x2 => x1 ≤ x2 := Classical.decRel fun x1 x2 => x1 ≤ x2
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : αᵒᵈ
a✝ : a ≤ b
⊢ (mud * zeta 𝕜) a b = 1 a b | 95f4eed14862eb7a |
BoundedContinuousFunction.isBounded_range_integral | Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean | lemma isBounded_range_integral
{ι : Type*} (μs : ι → Measure X) [∀ i, IsProbabilityMeasure (μs i)] (f : X →ᵇ E) :
Bornology.IsBounded (Set.range (fun i ↦ ∫ x, f x ∂ (μs i))) | case intro
X : Type u_1
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : OpensMeasurableSpace X
inst✝⁴ : SecondCountableTopology E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℝ E
ι : Type u_3
μs : ι → Measure X
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
f : X →ᵇ E
v : E
i : ι
hi : (fun i => ∫ (x : X), f x ∂μs i) i = v
⊢ ‖v‖ ≤ ‖f‖ | rw [← hi] | case intro
X : Type u_1
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : OpensMeasurableSpace X
inst✝⁴ : SecondCountableTopology E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℝ E
ι : Type u_3
μs : ι → Measure X
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
f : X →ᵇ E
v : E
i : ι
hi : (fun i => ∫ (x : X), f x ∂μs i) i = v
⊢ ‖(fun i => ∫ (x : X), f x ∂μs i) i‖ ≤ ‖f‖ | ab2ff73039861d6a |
Set.range_ite_subset' | Mathlib/Data/Set/Image.lean | theorem range_ite_subset' {p : Prop} [Decidable p] {f g : α → β} :
range (if p then f else g) ⊆ range f ∪ range g | case neg
α : Type u_1
β : Type u_2
p : Prop
inst✝ : Decidable p
f g : α → β
h : ¬p
⊢ range g ⊆ range f ∪ range g | exact subset_union_right | no goals | 62472e83fd94611c |
Basis.maximal | Mathlib/LinearAlgebra/Basis/Basic.lean | theorem maximal [Nontrivial R] (b : Basis ι R M) : b.linearIndependent.Maximal := fun w hi h => by
-- If `w` is strictly bigger than `range b`,
apply le_antisymm h
-- then choose some `x ∈ w \ range b`,
intro x p
by_contra q
-- and write it in terms of the basis.
have e := b.linearCombination_repr x
-- This then expresses `x` as a linear combination
-- of elements of `w` which are in the range of `b`,
let u : ι ↪ w :=
⟨fun i => ⟨b i, h ⟨i, rfl⟩⟩, fun i i' r =>
b.injective (by simpa only [Subtype.mk_eq_mk] using r)⟩
simp_rw [Finsupp.linearCombination_apply] at e
change ((b.repr x).sum fun (i : ι) (a : R) ↦ a • (u i : M)) = ((⟨x, p⟩ : w) : M) at e
rw [← Finsupp.sum_embDomain (f := u) (g := fun x r ↦ r • (x : M)),
← Finsupp.linearCombination_apply] at e
-- Now we can contradict the linear independence of `hi`
refine hi.linearCombination_ne_of_not_mem_support _ ?_ e
simp only [Finset.mem_map, Finsupp.support_embDomain]
rintro ⟨j, -, W⟩
simp only [u, Embedding.coeFn_mk, Subtype.mk_eq_mk] at W
apply q ⟨j, W⟩
| ι : Type u_1
R : Type u_3
M : Type u_5
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
b : Basis ι R M
w : Set M
hi : LinearIndependent R Subtype.val
h : range ⇑b ≤ w
⊢ w ≤ range ⇑b | intro x p | ι : Type u_1
R : Type u_3
M : Type u_5
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
b : Basis ι R M
w : Set M
hi : LinearIndependent R Subtype.val
h : range ⇑b ≤ w
x : M
p : x ∈ w
⊢ x ∈ range ⇑b | 26f21cbf99980896 |
PowerSeries.trunc_X | Mathlib/RingTheory/PowerSeries/Trunc.lean | theorem trunc_X (n) : trunc (n + 2) X = (Polynomial.X : R[X]) | case a
R : Type u_1
inst✝ : Semiring R
n d : ℕ
⊢ (if d < n + 2 then if d = 1 then 1 else 0 else 0) = Polynomial.X.coeff d | split_ifs with h₁ h₂ | case pos
R : Type u_1
inst✝ : Semiring R
n d : ℕ
h₁ : d < n + 2
h₂ : d = 1
⊢ 1 = Polynomial.X.coeff d
case neg
R : Type u_1
inst✝ : Semiring R
n d : ℕ
h₁ : d < n + 2
h₂ : ¬d = 1
⊢ 0 = Polynomial.X.coeff d
case neg
R : Type u_1
inst✝ : Semiring R
n d : ℕ
h₁ : ¬d < n + 2
⊢ 0 = Polynomial.X.coeff d | b873ec511fef51ee |
MeasureTheory.Measure.pi_pi_aux | Mathlib/MeasureTheory/Constructions/Pi.lean | theorem pi_pi_aux [∀ i, SigmaFinite (μ i)] (s : ∀ i, Set (α i)) (hs : ∀ i, MeasurableSet (s i)) :
Measure.pi μ (pi univ s) = ∏ i, μ i (s i) | case refine_2
ι : Type u_1
α : ι → Type u_3
inst✝² : Fintype ι
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (α i)
hs : ∀ (i : ι), MeasurableSet (s i)
this : Encodable ι
⊢ (pi' μ) (univ.pi s) ≤ (OuterMeasure.pi fun i => (μ i).toOuterMeasure) (univ.pi s) | suffices (pi' μ).toOuterMeasure ≤ OuterMeasure.pi fun i => (μ i).toOuterMeasure by exact this _ | case refine_2
ι : Type u_1
α : ι → Type u_3
inst✝² : Fintype ι
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (α i)
hs : ∀ (i : ι), MeasurableSet (s i)
this : Encodable ι
⊢ (pi' μ).toOuterMeasure ≤ OuterMeasure.pi fun i => (μ i).toOuterMeasure | d6eb1f2f929e1b76 |
IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic | Mathlib/NumberTheory/Cyclotomic/Basic.lean | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} | case refine_2
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
n : ℕ+
inst✝ : IsDomain B
ζ : B
hζ : IsPrimitiveRoot ζ ↑n
x : B
hx : x = ζ
⊢ x ∈ (cyclotomic (↑n) A).rootSet B | simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos) | no goals | 47345c8f34a7fd33 |
factorsThrough_of_pullbackCondition | Mathlib/Condensed/TopComparison.lean | theorem factorsThrough_of_pullbackCondition {Z B : C} {π : Z ⟶ B} [HasPullback π π]
[PreservesLimit (cospan π π) G]
{a : C(G.obj Z, X)}
(ha : a ∘ (G.map (pullback.fst _ _)) = a ∘ (G.map (pullback.snd π π))) :
Function.FactorsThrough a (G.map π) | C : Type u
inst✝³ : Category.{v, u} C
G : C ⥤ TopCat
X : Type w'
inst✝² : TopologicalSpace X
Z B : C
π : Z ⟶ B
inst✝¹ : HasPullback π π
inst✝ : PreservesLimit (cospan π π) G
a : C(↑(G.obj Z), X)
ha : ⇑a ∘ ⇑(ConcreteCategory.hom (G.map (pullback.fst π π))) = ⇑a ∘ ⇑(ConcreteCategory.hom (G.map (pullback.snd π π)))
x y : ↑(G.obj Z)
hxy : (ConcreteCategory.hom (G.map π)) x = (ConcreteCategory.hom (G.map π)) y
⊢ a x = a y | let xy : G.obj (pullback π π) := (PreservesPullback.iso G π π).inv <|
(TopCat.pullbackIsoProdSubtype (G.map π) (G.map π)).inv ⟨(x, y), hxy⟩ | C : Type u
inst✝³ : Category.{v, u} C
G : C ⥤ TopCat
X : Type w'
inst✝² : TopologicalSpace X
Z B : C
π : Z ⟶ B
inst✝¹ : HasPullback π π
inst✝ : PreservesLimit (cospan π π) G
a : C(↑(G.obj Z), X)
ha : ⇑a ∘ ⇑(ConcreteCategory.hom (G.map (pullback.fst π π))) = ⇑a ∘ ⇑(ConcreteCategory.hom (G.map (pullback.snd π π)))
x y : ↑(G.obj Z)
hxy : (ConcreteCategory.hom (G.map π)) x = (ConcreteCategory.hom (G.map π)) y
xy : ↑(G.obj (pullback π π)) :=
(ConcreteCategory.hom (PreservesPullback.iso G π π).inv)
((ConcreteCategory.hom (TopCat.pullbackIsoProdSubtype (G.map π) (G.map π)).inv) ⟨(x, y), hxy⟩)
⊢ a x = a y | 579a4db837d3077a |
Module.Finite.exists_fin' | Mathlib/RingTheory/Finiteness/Cardinality.lean | /-- A finite module admits a surjective linear map from a finite free module. -/
lemma exists_fin' [Module.Finite R M] : ∃ (n : ℕ) (f : (Fin n → R) →ₗ[R] M), Surjective f | R : Type u
M : Type u_3
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Module.Finite R M
⊢ ∃ n f, Surjective ⇑f | have ⟨n, s, hs⟩ := exists_fin (R := R) (M := M) | R : Type u
M : Type u_3
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Module.Finite R M
n : ℕ
s : Fin n → M
hs : span R (Set.range s) = ⊤
⊢ ∃ n f, Surjective ⇑f | a44152e4175c7cd5 |
intervalIntegral.integral_comp_neg | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | theorem integral_comp_neg : (∫ x in a..b, f (-x)) = ∫ x in -b..-a, f x | E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
⊢ ∫ (x : ℝ) in a..b, f (-x) = ∫ (x : ℝ) in -b..-a, f x | simpa only [zero_sub] using integral_comp_sub_left f 0 | no goals | 3ac1cbee3bf8f34a |
VectorFourier.integral_bilin_fourierIntegral_eq_flip | Mathlib/Analysis/Fourier/FourierTransform.lean | theorem integral_bilin_fourierIntegral_eq_flip
{f : V → E} {g : W → F} (M : E →L[ℂ] F →L[ℂ] G) (he : Continuous e)
(hL : Continuous fun p : V × W ↦ L p.1 p.2) (hf : Integrable f μ) (hg : Integrable g ν) :
∫ ξ, M (fourierIntegral e μ L f ξ) (g ξ) ∂ν =
∫ x, M (f x) (fourierIntegral e ν L.flip g x) ∂μ | case hf.h.h
𝕜 : Type u_1
inst✝²³ : CommRing 𝕜
V : Type u_2
inst✝²² : AddCommGroup V
inst✝²¹ : Module 𝕜 V
inst✝²⁰ : MeasurableSpace V
W : Type u_3
inst✝¹⁹ : AddCommGroup W
inst✝¹⁸ : Module 𝕜 W
E : Type u_4
F : Type u_5
G : Type u_6
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace ℂ E
inst✝¹⁵ : NormedAddCommGroup F
inst✝¹⁴ : NormedSpace ℂ F
inst✝¹³ : NormedAddCommGroup G
inst✝¹² : NormedSpace ℂ G
inst✝¹¹ : TopologicalSpace 𝕜
inst✝¹⁰ : IsTopologicalRing 𝕜
inst✝⁹ : TopologicalSpace V
inst✝⁸ : BorelSpace V
inst✝⁷ : TopologicalSpace W
inst✝⁶ : MeasurableSpace W
inst✝⁵ : BorelSpace W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜
ν : Measure W
inst✝⁴ : SigmaFinite μ
inst✝³ : SigmaFinite ν
inst✝² : SecondCountableTopology V
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
f : V → E
g : W → F
M : E →L[ℂ] F →L[ℂ] G
he : Continuous ⇑e
hL : Continuous fun p => (L p.1) p.2
hf : Integrable f μ
hg : Integrable g ν
hG : CompleteSpace G
this : Integrable (fun p => ‖M‖ * (‖g p.1‖ * ‖f p.2‖)) (ν.prod μ)
ξ : W
x : V
⊢ ‖(M (f x)) (g ξ)‖ ≤ ‖M‖ * ‖f (ξ, x).2‖ * ‖g (ξ, x).1‖ | exact M.le_opNorm₂ (f x) (g ξ) | no goals | 81aa292dd9850619 |
List.dedup_eq_cons | Mathlib/Data/List/Dedup.lean | theorem dedup_eq_cons (l : List α) (a : α) (l' : List α) :
l.dedup = a :: l' ↔ a ∈ l ∧ a ∉ l' ∧ l.dedup.tail = l' | case refine_1
α : Type u_1
inst✝ : DecidableEq α
l : List α
a : α
l' : List α
h : l.dedup = a :: l'
⊢ a ∈ l ∧ a ∉ l' ∧ l.dedup.tail = l' | refine ⟨mem_dedup.1 (h.symm ▸ mem_cons_self _ _), fun ha => ?_, by rw [h, tail_cons]⟩ | case refine_1
α : Type u_1
inst✝ : DecidableEq α
l : List α
a : α
l' : List α
h : l.dedup = a :: l'
ha : a ∈ l'
⊢ False | 31be0178b993dde6 |
iInter_halfSpaces_eq | Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean | theorem iInter_halfSpaces_eq (hs₁ : Convex ℝ s) (hs₂ : IsClosed s) :
⋂ l : E →L[ℝ] ℝ, { x | ∃ y ∈ s, l x ≤ l y } = s | E : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s : Set E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s
hs₂ : IsClosed s
⊢ {x | ∀ (i : E →L[ℝ] ℝ), ∃ y ∈ s, i x ≤ i y} = s | refine Set.Subset.antisymm (fun x hx => ?_) fun x hx l => ⟨x, hx, le_rfl⟩ | E : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s : Set E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s
hs₂ : IsClosed s
x : E
hx : x ∈ {x | ∀ (i : E →L[ℝ] ℝ), ∃ y ∈ s, i x ≤ i y}
⊢ x ∈ s | 62fc6b991b51d567 |
AlgebraicGeometry.Scheme.Opens.toSpecΓ_top | Mathlib/AlgebraicGeometry/AffineScheme.lean | @[simp]
lemma Scheme.Opens.toSpecΓ_top {X : Scheme} :
(⊤ : X.Opens).toSpecΓ = (⊤ : X.Opens).ι ≫ X.toSpecΓ | X : Scheme
⊢ (↑⊤).toSpecΓ ≫ Spec.map (X.presheaf.map (eqToHom ⋯)) = (↑⊤).toSpecΓ ≫ Spec.map (X.presheaf.map (homOfLE ⋯).op) | rfl | no goals | 30e215b07924d3af |
polynomialFunctions.comap_compRightAlgHom_iccHomeoI | Mathlib/Topology/ContinuousMap/Polynomial.lean | theorem polynomialFunctions.comap_compRightAlgHom_iccHomeoI (a b : ℝ) (h : a < b) :
(polynomialFunctions I).comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm) =
polynomialFunctions (Set.Icc a b) | case h.mpr.intro.intro.refine_2.h
a b : ℝ
h : a < b
p : ℝ[X]
q : ℝ[X] := p.comp ((b - a) • X + Polynomial.C a)
x : ↑I
⊢ (↑(toContinuousMapOnAlgHom I) q) x =
(↑(compRightAlgHom ℝ ℝ ↑(iccHomeoI a b h).symm) (↑(toContinuousMapOnAlgHom (Set.Icc a b)) p)) x | simp [q, mul_comm] | no goals | c86eff4051268cff |
Submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul | Mathlib/RingTheory/Finiteness/Nakayama.lean | theorem exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
[AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
∃ r : R, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = (0 : M) | case intro.intro.refine_2
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
s✝ : Set M
hfs : s✝.Finite
i : M
s : Set M
x✝¹ : i ∉ s
x✝ : s.Finite
ih :
(∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N) → ∃ r, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
H : ∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R (insert i s)) ∧ insert i s ⊆ ↑N
⊢ ∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N | rcases H with ⟨r, hr1, hrn, hs⟩ | case intro.intro.refine_2.intro.intro.intro
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
s✝ : Set M
hfs : s✝.Finite
i : M
s : Set M
x✝¹ : i ∉ s
x✝ : s.Finite
ih :
(∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N) → ∃ r, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hrn : N ≤ comap ((LinearMap.lsmul R M) r) (I • span R (insert i s))
hs : insert i s ⊆ ↑N
⊢ ∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N | ddde3c87abdd1812 |
LieSubmodule.isCompactElement_lieSpan_singleton | Mathlib/Algebra/Lie/Submodule.lean | lemma isCompactElement_lieSpan_singleton (m : M) :
CompleteLattice.IsCompactElement (lieSpan R L {m}) | R : Type u
L : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : LieRingModule L M
m : M
s : Set (LieSubmodule R L M)
hne : s.Nonempty
hdir : DirectedOn (fun x1 x2 => x1 ≤ x2) s
hsup : m ∈ ↑(sSup s)
this : ↑(sSup s) = ⋃ N ∈ s, ↑N
N : LieSubmodule R L M
hN : N ∈ s
hN' : m ∈ N
⊢ lieSpan R L {m} ≤ N | simpa | no goals | 016ba20f16d8a5de |
exists_continuous_add_one_of_isCompact_nnreal | Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/Basic.lean | lemma exists_continuous_add_one_of_isCompact_nnreal
{s₀ s₁ : Set X} {t : Set X} (s₀_compact : IsCompact s₀) (s₁_compact : IsCompact s₁)
(t_compact : IsCompact t) (disj : Disjoint s₀ s₁) (hst : s₀ ∪ s₁ ⊆ t) :
∃ (f₀ f₁ : C_c(X, ℝ≥0)), EqOn f₀ 1 s₀ ∧ EqOn f₁ 1 s₁ ∧ EqOn (f₀ + f₁) 1 t | case h.refine_3
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j => if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j => if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i : Fin 2, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
x : X
hx : x ∈ t
⊢ ((f 1) x).toNNReal + ((f 0) x).toNNReal = 1 | rw [Real.toNNReal_add_toNNReal (f_in_icc 1 x).1 (f_in_icc 0 x).1, add_comm] | case h.refine_3
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j => if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j => if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i : Fin 2, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
x : X
hx : x ∈ t
⊢ ((f 0) x + (f 1) x).toNNReal = 1 | c2d96a23593bed78 |
AddCircle.closedBall_ae_eq_ball | Mathlib/MeasureTheory/Group/AddCircle.lean | theorem closedBall_ae_eq_ball {x : AddCircle T} {ε : ℝ} : closedBall x ε =ᵐ[volume] ball x ε | T : ℝ
hT : Fact (0 < T)
x : AddCircle T
ε : ℝ
⊢ closedBall x ε =ᶠ[ae volume] ball x ε | rcases le_or_lt ε 0 with hε | hε | case inl
T : ℝ
hT : Fact (0 < T)
x : AddCircle T
ε : ℝ
hε : ε ≤ 0
⊢ closedBall x ε =ᶠ[ae volume] ball x ε
case inr
T : ℝ
hT : Fact (0 < T)
x : AddCircle T
ε : ℝ
hε : 0 < ε
⊢ closedBall x ε =ᶠ[ae volume] ball x ε | d4d4a4368219224a |
DerivedCategory.mem_distTriang_iff | Mathlib/Algebra/Homology/DerivedCategory/Basic.lean | lemma mem_distTriang_iff (T : Triangle (DerivedCategory C)) :
(T ∈ distTriang (DerivedCategory C)) ↔ ∃ (X Y : CochainComplex C ℤ) (f : X ⟶ Y),
Nonempty (T ≅ Q.mapTriangle.obj (CochainComplex.mappingCone.triangle f)) | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
T : Triangle (DerivedCategory C)
⊢ T ∈ distinguishedTriangles ↔ ∃ X Y f, Nonempty (T ≅ Q.mapTriangle.obj (CochainComplex.mappingCone.triangle f)) | constructor | case mp
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
T : Triangle (DerivedCategory C)
⊢ T ∈ distinguishedTriangles → ∃ X Y f, Nonempty (T ≅ Q.mapTriangle.obj (CochainComplex.mappingCone.triangle f))
case mpr
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
T : Triangle (DerivedCategory C)
⊢ (∃ X Y f, Nonempty (T ≅ Q.mapTriangle.obj (CochainComplex.mappingCone.triangle f))) → T ∈ distinguishedTriangles | 03b436a0c0486942 |
MeasureTheory.L1.setToL1_mono | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem setToL1_mono {T : Set α → G' →L[ℝ] G''} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C)
(hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f g : α →₁[μ] G'}
(hfg : f ≤ g) : setToL1 hT f ≤ setToL1 hT g | α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁴ : NormedLatticeAddCommGroup G''
inst✝³ : NormedSpace ℝ G''
inst✝² : CompleteSpace G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : ↥(Lp G' 1 μ)
hfg : 0 ≤ g - f
⊢ 0 ≤ (setToL1 hT) g - (setToL1 hT) f | rw [← (setToL1 hT).map_sub] | α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁴ : NormedLatticeAddCommGroup G''
inst✝³ : NormedSpace ℝ G''
inst✝² : CompleteSpace G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : ↥(Lp G' 1 μ)
hfg : 0 ≤ g - f
⊢ 0 ≤ (setToL1 hT) (g - f) | d553f717872d4afe |
separableClosure_le_iff | Mathlib/FieldTheory/PurelyInseparable/Basic.lean | theorem separableClosure_le_iff [Algebra.IsAlgebraic F E] (L : IntermediateField F E) :
separableClosure F E ≤ L ↔ IsPurelyInseparable L E | F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsAlgebraic F E
L : IntermediateField F E
⊢ separableClosure F E ≤ L ↔ IsPurelyInseparable (↥L) E | refine ⟨fun h ↦ ?_, fun _ ↦ separableClosure_le F E L⟩ | F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsAlgebraic F E
L : IntermediateField F E
h : separableClosure F E ≤ L
⊢ IsPurelyInseparable (↥L) E | 3ca97f86bd963620 |
strictConvexOn_rpow | Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean | theorem strictConvexOn_rpow {p : ℝ} (hp : 1 < p) : StrictConvexOn ℝ (Ici 0) fun x : ℝ ↦ x ^ p | p : ℝ
hp : 1 < p
x y z : ℝ
hx : 0 ≤ x
hz : 0 ≤ z
hxy : x < y
hyz : y < z
hy : 0 < y
hy' : 0 < y ^ p
q : 0 < y - x
⊢ (y ^ p - x ^ p) / (y - x) < p * y ^ (p - 1) | rw [div_lt_iff₀ q, ← div_lt_div_iff_of_pos_right hy', _root_.sub_div, div_self hy'.ne',
← div_rpow hx hy.le, sub_lt_comm, ← add_sub_cancel_right (x / y) 1, add_comm, add_sub_assoc,
← div_mul_eq_mul_div, mul_div_assoc, ← rpow_sub hy, sub_sub_cancel_left, rpow_neg_one,
mul_assoc, ← div_eq_inv_mul, sub_eq_add_neg, ← mul_neg, ← neg_div, neg_sub, _root_.sub_div,
div_self hy.ne'] | p : ℝ
hp : 1 < p
x y z : ℝ
hx : 0 ≤ x
hz : 0 ≤ z
hxy : x < y
hyz : y < z
hy : 0 < y
hy' : 0 < y ^ p
q : 0 < y - x
⊢ 1 + p * (x / y - 1) < (1 + (x / y - 1)) ^ p | 003004303f55f1b7 |
NumberField.InfinitePlace.mk_eq_iff | Mathlib/NumberTheory/NumberField/Embeddings.lean | theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ | case mp.intro.inl.h
K : Type u_2
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥φ.range := RingEquiv.ofLeftInverse hiφ
hlip : LipschitzWith 1 ⇑(ψ.comp ι.symm.toRingHom)
h : (↑↑(ψ.comp ι.symm.toRingHom)).toFun = ⇑φ.fieldRange.subtype
⊢ φ = ψ | ext1 x | case mp.intro.inl.h.a
K : Type u_2
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
j : ℂ → K
hiφ : Function.LeftInverse j ⇑φ
ι : K ≃+* ↥φ.range := RingEquiv.ofLeftInverse hiφ
hlip : LipschitzWith 1 ⇑(ψ.comp ι.symm.toRingHom)
h : (↑↑(ψ.comp ι.symm.toRingHom)).toFun = ⇑φ.fieldRange.subtype
x : K
⊢ φ x = ψ x | 1d2ef4e3545a7979 |
CliffordAlgebra.involute_eq_of_mem_odd | Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | theorem involute_eq_of_mem_odd {x : CliffordAlgebra Q} (h : x ∈ evenOdd Q 1) : involute x = -x | R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x : CliffordAlgebra Q
h : x ∈ evenOdd Q 1
⊢ involute x = -x | induction x, h using odd_induction with
| ι m => exact involute_ι _
| add x y _hx _hy ihx ihy =>
rw [map_add, ihx, ihy, neg_add]
| ι_mul_ι_mul m₁ m₂ x _hx ihx =>
rw [map_mul, map_mul, involute_ι, involute_ι, ihx, neg_mul_neg, mul_neg] | no goals | de256de08fdc980b |
AlgebraicGeometry.spread_out_of_isGermInjective | Mathlib/AlgebraicGeometry/SpreadingOut.lean | /--
Given `S`-schemes `X Y` and points `x : X` `y : Y` over `s : S`.
Suppose we have the following diagram of `S`-schemes
```
Spec 𝒪_{X, x} ⟶ X
|
Spec(φ)
↓
Spec 𝒪_{Y, y} ⟶ Y
```
Then the map `Spec(φ)` spreads out to an `S`-morphism on an open subscheme `U ⊆ X`,
```
Spec 𝒪_{X, x} ⟶ U ⊆ X
| |
Spec(φ) |
↓ ↓
Spec 𝒪_{Y, y} ⟶ Y
```
provided that `Y` is locally of finite type over `S` and
`X` is "germ-injective" at `x` (e.g. when it's integral or locally noetherian).
TODO: The condition on `X` is unnecessary when `Y` is locally of finite presentation.
-/
@[stacks 0BX6]
lemma spread_out_of_isGermInjective [LocallyOfFiniteType sY] {x : X} [X.IsGermInjectiveAt x] {y : Y}
(e : sX.base x = sY.base y) (φ : Y.presheaf.stalk y ⟶ X.presheaf.stalk x)
(h : sY.stalkMap y ≫ φ =
S.presheaf.stalkSpecializes (Inseparable.of_eq e).specializes ≫ sX.stalkMap x) :
∃ (U : X.Opens) (hxU : x ∈ U) (f : U.toScheme ⟶ Y),
Spec.map φ ≫ Y.fromSpecStalk y = U.fromSpecStalkOfMem x hxU ≫ f ∧
f ≫ sY = U.ι ≫ sX | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
X Y S : Scheme
sX : X ⟶ S
sY : Y ⟶ S
inst✝¹ : LocallyOfFiniteType sY
x : ↑↑X.toPresheafedSpace
inst✝ : X.IsGermInjectiveAt x
y : ↑↑Y.toPresheafedSpace
e : (ConcreteCategory.hom sX.base) x = (ConcreteCategory.hom sY.base) y
φ : Y.presheaf.stalk y ⟶ X.presheaf.stalk x
h : Scheme.Hom.stalkMap sY y ≫ φ = S.presheaf.stalkSpecializes ⋯ ≫ Scheme.Hom.stalkMap sX x
U : TopologicalSpace.Opens ↑↑S.toPresheafedSpace
hU : U ∈ S.affineOpens
hxU : (ConcreteCategory.hom sX.base) x ∈ ↑U
hyU : (ConcreteCategory.hom sY.base) y ∈ U
V : Y.Opens
hV : V ∈ Y.affineOpens
hyV : y ∈ ↑V
iVU : ↑V ⊆ (sY ⁻¹ᵁ U).carrier
this :
Scheme.Hom.appLE sY U V iVU ≫ Y.presheaf.germ V y hyV ≫ φ = Scheme.Hom.app sX U ≫ X.presheaf.germ (sX ⁻¹ᵁ U) x hxU
⊢ ∃ U, ∃ (hxU : x ∈ U), ∃ f, Spec.map φ ≫ Y.fromSpecStalk y = U.fromSpecStalkOfMem x hxU ≫ f ∧ f ≫ sY = U.ι ≫ sX | obtain ⟨W, hxW, φ', i, hW, h₁, h₂⟩ :=
exists_lift_of_germInjective (R := Γ(S, U)) (A := Γ(Y, V)) (U := sX ⁻¹ᵁ U) (x := x) hxU
(Y.presheaf.germ _ y hyV ≫ φ) (sY.appLE U V iVU) (sX.app U)
(LocallyOfFiniteType.finiteType_of_affine_subset ⟨_, hU⟩ ⟨_, hV⟩ _) this | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
X Y S : Scheme
sX : X ⟶ S
sY : Y ⟶ S
inst✝¹ : LocallyOfFiniteType sY
x : ↑↑X.toPresheafedSpace
inst✝ : X.IsGermInjectiveAt x
y : ↑↑Y.toPresheafedSpace
e : (ConcreteCategory.hom sX.base) x = (ConcreteCategory.hom sY.base) y
φ : Y.presheaf.stalk y ⟶ X.presheaf.stalk x
h : Scheme.Hom.stalkMap sY y ≫ φ = S.presheaf.stalkSpecializes ⋯ ≫ Scheme.Hom.stalkMap sX x
U : TopologicalSpace.Opens ↑↑S.toPresheafedSpace
hU : U ∈ S.affineOpens
hxU : (ConcreteCategory.hom sX.base) x ∈ ↑U
hyU : (ConcreteCategory.hom sY.base) y ∈ U
V : Y.Opens
hV : V ∈ Y.affineOpens
hyV : y ∈ ↑V
iVU : ↑V ⊆ (sY ⁻¹ᵁ U).carrier
this :
Scheme.Hom.appLE sY U V iVU ≫ Y.presheaf.germ V y hyV ≫ φ = Scheme.Hom.app sX U ≫ X.presheaf.germ (sX ⁻¹ᵁ U) x hxU
W : X.Opens
hxW : x ∈ W
φ' : Γ(Y, V) ⟶ Γ(X, W)
i : W ≤ sX ⁻¹ᵁ U
hW : IsAffineOpen W
h₁ : Y.presheaf.germ V y hyV ≫ φ = φ' ≫ X.presheaf.germ W x hxW
h₂ : Scheme.Hom.app sX U ≫ X.presheaf.map i.hom.op = Scheme.Hom.appLE sY U V iVU ≫ φ'
⊢ ∃ U, ∃ (hxU : x ∈ U), ∃ f, Spec.map φ ≫ Y.fromSpecStalk y = U.fromSpecStalkOfMem x hxU ≫ f ∧ f ≫ sY = U.ι ≫ sX | 6cace8db5fa36278 |
List.Sublist.of_sublist_append_right | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sublist.lean | theorem Sublist.of_sublist_append_right (w : ∀ a, a ∈ l → a ∉ l₁) (h : l <+ l₁ ++ l₂) : l <+ l₂ | case intro.intro.intro.intro
α✝ : Type u_1
l₁ l₂ l₁' l₂' : List α✝
w : ∀ (a : α✝), a ∈ l₁' ++ l₂' → ¬a ∈ l₁
h₁ : l₁' <+ l₁
h₂ : l₂' <+ l₂
⊢ l₁' ++ l₂' <+ l₂ | have : l₁' = [] := by
rw [eq_nil_iff_forall_not_mem]
exact fun x m => w x (mem_append_left l₂' m) (h₁.mem m) | case intro.intro.intro.intro
α✝ : Type u_1
l₁ l₂ l₁' l₂' : List α✝
w : ∀ (a : α✝), a ∈ l₁' ++ l₂' → ¬a ∈ l₁
h₁ : l₁' <+ l₁
h₂ : l₂' <+ l₂
this : l₁' = []
⊢ l₁' ++ l₂' <+ l₂ | a561ed959cac614f |
Vector.append_eq_append_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem append_eq_append_iff {a : Vector α n} {b : Vector α m} {c : Vector α k} {d : Vector α l}
(w : k + l = n + m) :
a ++ b = (c ++ d).cast w ↔
if h : n ≤ k then
∃ a' : Vector α (k - n), c = (a ++ a').cast (by omega) ∧ b = (a' ++ d).cast (by omega)
else
∃ c' : Vector α (n - k), a = (c ++ c').cast (by omega) ∧ d = (c' ++ b).cast (by omega) | case mk.mk.mk.mk.mpr.isFalse.intro.intro
α : Type u_1
a b c : Array α
h : ¬a.size ≤ c.size
c' : Vector α (a.size - c.size)
ha : a = ({ toArray := c, size_toArray := ⋯ } ++ c').toArray
w : c.size + (c' ++ { toArray := b, size_toArray := ⋯ }).size = a.size + b.size
⊢ (∃ a', c = a ++ a' ∧ b = a' ++ (c' ++ { toArray := b, size_toArray := ⋯ }).toArray) ∨
∃ c'_1, a = c ++ c'_1 ∧ (c' ++ { toArray := b, size_toArray := ⋯ }).toArray = c'_1 ++ b | right | case mk.mk.mk.mk.mpr.isFalse.intro.intro.h
α : Type u_1
a b c : Array α
h : ¬a.size ≤ c.size
c' : Vector α (a.size - c.size)
ha : a = ({ toArray := c, size_toArray := ⋯ } ++ c').toArray
w : c.size + (c' ++ { toArray := b, size_toArray := ⋯ }).size = a.size + b.size
⊢ ∃ c'_1, a = c ++ c'_1 ∧ (c' ++ { toArray := b, size_toArray := ⋯ }).toArray = c'_1 ++ b | 6aa03c7084d30f0e |
Complex.HadamardThreeLines.norm_le_interp_of_mem_verticalClosedStrip₀₁' | Mathlib/Analysis/Complex/Hadamard.lean | /-- **Hadamard three-line theorem** on `re ⁻¹' [0, 1]` (Variant in simpler terms): Let `f` be a
bounded function, continuous on the closed strip `re ⁻¹' [0, 1]` and differentiable on open strip
`re ⁻¹' (0, 1)`. If, for all `z.re = 0`, `‖f z‖ ≤ a` for some `a ∈ ℝ` and, similarly, for all
`z.re = 1`, `‖f z‖ ≤ b` for some `b ∈ ℝ` then for all `z` in the closed strip
`re ⁻¹' [0, 1]` the inequality `‖f(z)‖ ≤ a ^ (1 - z.re) * b ^ z.re` holds. -/
lemma norm_le_interp_of_mem_verticalClosedStrip₀₁' (f : ℂ → E) {z : ℂ} {a b : ℝ}
(hz : z ∈ verticalClosedStrip 0 1) (hd : DiffContOnCl ℂ f (verticalStrip 0 1))
(hB : BddAbove ((norm ∘ f) '' verticalClosedStrip 0 1))
(ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a) (hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b) :
‖f z‖ ≤ a ^ (1 - z.re) * b ^ z.re | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSupNormIm f 0 ^ (1 - z.re) ≤ a ^ (1 - z.re) | apply Real.rpow_le_rpow (sSupNormIm_nonneg f _) _ (sub_nonneg.mpr hz.2) | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSupNormIm f 0 ≤ a | 0fab8ad239caab3f |
BitVec.iunfoldr_getLsbD' | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Folds.lean | theorem iunfoldr_getLsbD' {f : Fin w → α → α × Bool} (state : Nat → α)
(ind : ∀(i : Fin w), (f i (state i.val)).fst = state (i.val+1)) :
(∀ i : Fin w, getLsbD (iunfoldr f (state 0)).snd i.val = (f i (state i.val)).snd)
∧ (iunfoldr f (state 0)).fst = state w | case right
w : Nat
α : Type u_1
f : Fin w → α → α × Bool
state : Nat → α
ind : ∀ (i : Fin w), (f i (state ↑i)).fst = state (↑i + 1)
j : Fin w
s : α
v : BitVec ↑j
ih : ∀ (hj : ↑j ≤ w), (∀ (i : Fin ↑j), (s, v).snd.getLsbD ↑i = (f ⟨↑i, ⋯⟩ (state ↑i)).snd) ∧ (s, v).fst = state ↑j
hj : ↑j + 1 ≤ w
⊢ ((f j (s, v).fst).fst, cons (f j (s, v).fst).snd (s, v).snd).fst = state (↑j + 1) | case right =>
simp
have hj2 : j.val ≤ w := by simp
rw [← ind j, ← (ih hj2).2] | no goals | 2142a073bff43190 |
IntermediateField.mem_adjoin_iff | Mathlib/FieldTheory/IntermediateField/Adjoin/Basic.lean | theorem mem_adjoin_iff (x : E) :
x ∈ adjoin F S ↔ ∃ r s : MvPolynomial S F,
x = MvPolynomial.aeval Subtype.val r / MvPolynomial.aeval Subtype.val s | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set E
x : E
⊢ x ∈ adjoin F S ↔ ∃ r s, x = (MvPolynomial.aeval Subtype.val) r / (MvPolynomial.aeval Subtype.val) s | rw [← mem_adjoin_range_iff, Subtype.range_coe] | no goals | 366eafc3485b2f7a |
Topology.IsUpperSet.monotone_to_upperTopology_continuous | Mathlib/Topology/Order/UpperLowerSetTopology.lean | lemma monotone_to_upperTopology_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [IsUpper β] {f : α → β} (hf : Monotone f) : Continuous f | α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : IsUpper β
f : α → β
hf : Monotone f
s : Set β
hs : IsOpen s
⊢ IsUpperSet (f ⁻¹' s) | exact (IsUpper.isUpperSet_of_isOpen hs).preimage hf | no goals | ee33d0fa84466562 |
recursion' | Mathlib/Data/Real/Pi/Irrational.lean | /--
Auxiliary for the proof that `π` is irrational.
While it is most natural to give the recursive formula for `I (n + 2) θ`, as well as give the second
base case of `I 1 θ`, it is in fact more convenient to give the recursive formula for `I (n + 1) θ`
in terms of `I n θ` and `I (n - 1) θ` (note the natural subtraction!).
Despite the usually inconvenient subtraction, this in fact allows deducing both of the above facts
with significantly fewer analysis computations.
In addition, note the `0 ^ n` on the right hand side - this is intentional, and again allows
combining the proof of the "usual" recursion formula and the base case `I 1 θ`.
-/
private lemma recursion' (n : ℕ) :
I (n + 1) θ * θ ^ 2 = - (2 * 2 * ((n + 1) * (0 ^ n * cos θ))) +
2 * (n + 1) * (2 * n + 1) * I n θ - 4 * (n + 1) * n * I (n - 1) θ | case convert_2.convert_2.hg
θ : ℝ
n : ℕ
f : ℝ → ℝ := fun x => 1 - x ^ 2
u₁ : ℝ → ℝ := fun x => f x ^ (n + 1)
u₁' : ℝ → ℝ := fun x => -(2 * (↑n + 1) * x * f x ^ n)
v₁ : ℝ → ℝ := fun x => sin (x * θ)
v₁' : ℝ → ℝ := fun x => cos (x * θ) * θ
u₂ : ℝ → ℝ := fun x => x * f x ^ n
u₂' : ℝ → ℝ := fun x => f x ^ n - 2 * ↑n * x ^ 2 * f x ^ (n - 1)
v₂ : ℝ → ℝ := fun x => cos (x * θ)
v₂' : ℝ → ℝ := fun x => -sin (x * θ) * θ
hfd : Continuous f
hu₁d : Continuous u₁'
hv₁d : Continuous v₁'
hu₂d : Continuous u₂'
hv₂d : Continuous v₂'
hu₁_eval_one : u₁ 1 = 0
hu₁_eval_neg_one : u₁ (-1) = 0
t : u₂ 1 * v₂ 1 - u₂ (-1) * v₂ (-1) = 2 * (0 ^ n * cos θ)
hf : ∀ (x : ℝ), HasDerivAt f (-2 * x) x
hu₁ : ∀ (x : ℝ), HasDerivAt u₁ (u₁' x) x
hv₁ : ∀ (x : ℝ), HasDerivAt v₁ (v₁' x) x
hu₂ : ∀ (x : ℝ), HasDerivAt u₂ (u₂' x) x
hv₂ : ∀ (x : ℝ), HasDerivAt v₂ (v₂' x) x
this✝ : ∀ (x : ℝ), u₂' x = (2 * ↑n + 1) * f x ^ n - 2 * ↑n * f x ^ (n - 1)
this : Continuous v₂
⊢ IntervalIntegrable (fun x => 2 * ↑n * (f x ^ (n - 1) * v₂ x)) volume (-1) 1 | exact Continuous.intervalIntegrable (by fun_prop) _ _ | no goals | 834ac76d1ec60c71 |
AlgebraicGeometry.Scheme.ofRestrict_appIso | Mathlib/AlgebraicGeometry/OpenImmersion.lean | @[simp]
lemma Scheme.ofRestrict_appIso (U) :
(X.ofRestrict h).appIso U = Iso.refl _ | case w
U✝ : TopCat
X : Scheme
f : U✝ ⟶ TopCat.of ↑↑X.toPresheafedSpace
h : IsOpenEmbedding ⇑(ConcreteCategory.hom f)
U : (X.restrict h).Opens
⊢ (Hom.appIso (X.ofRestrict h) U).hom = (Iso.refl Γ(X, X.ofRestrict h ''ᵁ U)).hom | simp only [restrict_presheaf_obj, Hom.appIso_hom', ofRestrict_appLE, homOfLE_refl, op_id,
CategoryTheory.Functor.map_id, Iso.refl_hom] | no goals | 7473df1570c2c905 |
Array.size_eraseIdxIfInBounds | Mathlib/.lake/packages/batteries/Batteries/Data/Array/Lemmas.lean | theorem size_eraseIdxIfInBounds (a : Array α) (i : Nat) :
(a.eraseIdxIfInBounds i).size = if i < a.size then a.size-1 else a.size | case isTrue
α : Type u_1
a : Array α
i : Nat
h✝ : i < a.size
⊢ (a.eraseIdx i h✝).size = a.size - 1
case isFalse
α : Type u_1
a : Array α
i : Nat
h✝ : ¬i < a.size
⊢ a.size = a.size | simp | case isFalse
α : Type u_1
a : Array α
i : Nat
h✝ : ¬i < a.size
⊢ a.size = a.size | c70fb068dccb7279 |
DFinsupp.neLocus_self_sub_left | Mathlib/Data/DFinsupp/NeLocus.lean | theorem neLocus_self_sub_left : neLocus (f - g) f = g.support | α : Type u_1
N : α → Type u_2
inst✝² : DecidableEq α
inst✝¹ : (a : α) → DecidableEq (N a)
inst✝ : (a : α) → AddGroup (N a)
f g : Π₀ (a : α), N a
⊢ (f - g).neLocus f = g.support | rw [neLocus_comm, neLocus_self_sub_right] | no goals | 552fe3903be4c087 |
MeasureTheory.AEStronglyMeasurable.piecewise | Mathlib/MeasureTheory/Function/StronglyMeasurable/AEStronglyMeasurable.lean | theorem piecewise {s : Set α} [DecidablePred (· ∈ s)]
(hs : MeasurableSet s) (hf : AEStronglyMeasurable f (μ.restrict s))
(hg : AEStronglyMeasurable g (μ.restrict sᶜ)) :
AEStronglyMeasurable (s.piecewise f g) μ | case h
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f g : α → β
s : Set α
inst✝ : DecidablePred fun x => x ∈ s
hs : MeasurableSet s
hf : AEStronglyMeasurable f (μ.restrict s)
hg : AEStronglyMeasurable g (μ.restrict sᶜ)
h : ∀ᵐ (x : α) ∂μ, x ∈ sᶜ → g x = AEStronglyMeasurable.mk g hg x
x : α
hx : x ∈ sᶜ → g x = AEStronglyMeasurable.mk g hg x
⊢ x ∈ sᶜ → s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) x | intro hx_mem | case h
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f g : α → β
s : Set α
inst✝ : DecidablePred fun x => x ∈ s
hs : MeasurableSet s
hf : AEStronglyMeasurable f (μ.restrict s)
hg : AEStronglyMeasurable g (μ.restrict sᶜ)
h : ∀ᵐ (x : α) ∂μ, x ∈ sᶜ → g x = AEStronglyMeasurable.mk g hg x
x : α
hx : x ∈ sᶜ → g x = AEStronglyMeasurable.mk g hg x
hx_mem : x ∈ sᶜ
⊢ s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) x | 3fd4cff7b1ce1750 |
ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib' | Mathlib/Probability/Kernel/CondDistrib.lean | theorem condExp_prod_ae_eq_integral_condDistrib' [NormedSpace ℝ F] [CompleteSpace F]
(hX : Measurable X) (hY : AEMeasurable Y μ)
(hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
μ[fun a => f (X a, Y a)|mβ.comap X] =ᵐ[μ] fun a => ∫ y, f (X a,y) ∂condDistrib Y X μ (X a) | case refine_2.intro.intro
α : Type u_1
β : Type u_2
Ω : Type u_3
F : Type u_4
inst✝⁶ : MeasurableSpace Ω
inst✝⁵ : StandardBorelSpace Ω
inst✝⁴ : Nonempty Ω
inst✝³ : NormedAddCommGroup F
mα : MeasurableSpace α
μ : Measure α
inst✝² : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
f : β × Ω → F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hX : Measurable X
hY : AEMeasurable Y μ
hf_int : Integrable f (Measure.map (fun a => (X a, Y a)) μ)
hf_int' : Integrable (fun a => f (X a, Y a)) μ
t : Set β
ht : MeasurableSet t
a✝ : μ (X ⁻¹' t) < ⊤
⊢ ∫ (y : β), ∫ (y_1 : Ω), f (y, y_1) ∂(condDistrib Y X μ) y ∂Measure.map X (μ.restrict (X ⁻¹' t)) =
∫ (a : α) in X ⁻¹' t, f (X a, Y a) ∂μ | rw [← Measure.restrict_map hX ht, ← Measure.fst_map_prod_mk₀ hY, condDistrib,
Measure.setIntegral_condKernel_univ_right ht hf_int.integrableOn,
setIntegral_map (ht.prod MeasurableSet.univ) hf_int.1 (hX.aemeasurable.prod_mk hY),
mk_preimage_prod, preimage_univ, inter_univ] | no goals | 824ee395dfccb749 |
CategoryTheory.isUniversalColimit_extendCofan | Mathlib/CategoryTheory/Limits/VanKampen.lean | theorem isUniversalColimit_extendCofan {n : ℕ} (f : Fin (n + 1) → C)
{c₁ : Cofan fun i : Fin n ↦ f i.succ} {c₂ : BinaryCofan (f 0) c₁.pt}
(t₁ : IsUniversalColimit c₁) (t₂ : IsUniversalColimit c₂)
[∀ {Z} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i] :
IsUniversalColimit (extendCofan c₁ c₂) | case refine_3
C : Type u
inst✝¹ : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Discrete (Fin (n + 1)) ⥤ C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
j : Discrete (Fin n)
⊢ IsPullback
((Cofan.mk (pullback c₂.inr i) fun j =>
pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯).ι.app
j)
((Discrete.natTrans fun i => α.app { as := i.as.succ }).app j) (pullback.fst c₂.inr i) (c₁.ι.app j) | simp only [pair_obj_right, Functor.const_obj_obj, Discrete.functor_obj, id_eq,
extendCofan_pt, eq_mpr_eq_cast, Cofan.mk_pt, Cofan.mk_ι_app, Discrete.natTrans_app] | case refine_3
C : Type u
inst✝¹ : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Discrete (Fin (n + 1)) ⥤ C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
j : Discrete (Fin n)
⊢ IsPullback (pullback.lift (α.app { as := j.as.succ } ≫ c₁.inj j.as) (c.ι.app { as := j.as.succ }) ⋯)
(α.app { as := j.as.succ }) (pullback.fst c₂.inr i) (c₁.ι.app j) | 26415e6516008f86 |
Ordinal.preOmega_natCast | Mathlib/SetTheory/Cardinal/Aleph.lean | theorem preOmega_natCast (n : ℕ) : preOmega n = n | case succ
n : ℕ
IH : preOmega ↑n = ↑n
⊢ preOmega ↑(n + 1) = ↑(n + 1) | apply (le_preOmega_self _).antisymm' | case succ
n : ℕ
IH : preOmega ↑n = ↑n
⊢ preOmega ↑(n + 1) ≤ ↑(n + 1) | 5ffb6949c81c19a7 |
Std.DHashMap.Internal.List.getValueCast?_insertList_of_mem | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem getValueCast?_insertList_of_mem [BEq α] [LawfulBEq α]
{l toInsert : List ((a : α) × β a)}
{k k' : α} (k_beq : k == k') {v : β k}
(distinct_l : DistinctKeys l)
(distinct_toInsert : toInsert.Pairwise (fun a b => (a.1 == b.1) = false))
(mem : ⟨k, v⟩ ∈ toInsert) :
getValueCast? k' (insertList l toInsert) =
some (cast (by congr; exact LawfulBEq.eq_of_beq k_beq) v) | case e_a
α : Type u
β : α → Type v
γ : α → Type w
inst✝¹ : BEq α
inst✝ : LawfulBEq α
l toInsert : List ((a : α) × β a)
k k' : α
k_beq : (k == k') = true
v : β k
distinct_l : DistinctKeys l
distinct_toInsert : List.Pairwise (fun a b => (a.fst == b.fst) = false) toInsert
mem : ⟨k, v⟩ ∈ toInsert
⊢ k = k' | exact LawfulBEq.eq_of_beq k_beq | no goals | 82c18001be4198ba |
Traversable.toList_spec | Mathlib/Control/Fold.lean | theorem toList_spec (xs : t α) : toList xs = FreeMonoid.toList (foldMap FreeMonoid.of xs) :=
Eq.symm <|
calc
FreeMonoid.toList (foldMap FreeMonoid.of xs) =
FreeMonoid.toList (foldMap FreeMonoid.of xs).reverse.reverse | α : Type u
t : Type u → Type u
inst✝¹ : Traversable t
inst✝ : LawfulTraversable t
xs : t α
⊢ (unop ((Foldl.ofFreeMonoid (flip cons)) (foldMap FreeMonoid.of xs)) []).reverse = toList xs | rw [foldMap_hom_free (Foldl.ofFreeMonoid (flip <| @cons α))] | α : Type u
t : Type u → Type u
inst✝¹ : Traversable t
inst✝ : LawfulTraversable t
xs : t α
⊢ (unop (foldMap (⇑(Foldl.ofFreeMonoid (flip cons)) ∘ FreeMonoid.of) xs) []).reverse = toList xs | 9cc7a5054b1c54eb |
ProbabilityTheory.IndepFun.variance_add | Mathlib/Probability/Variance.lean | theorem IndepFun.variance_add [IsProbabilityMeasure μ] {X Y : Ω → ℝ} (hX : MemLp X 2 μ)
(hY : MemLp Y 2 μ) (h : IndepFun X Y μ) : variance (X + Y) μ = variance X μ + variance Y μ :=
calc
variance (X + Y) μ = μ[fun a => X a ^ 2 + Y a ^ 2 + 2 * X a * Y a] - μ[X + Y] ^ 2 | case hg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X Y : Ω → ℝ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
h : IndepFun X Y μ
⊢ Integrable (fun a => Y a ^ 2) μ | exact hY.integrable_sq | no goals | 6996a4817b022b25 |
Finset.mulEnergy_univ_left | Mathlib/Combinatorics/Additive/Energy.lean | @[to_additive (attr := simp)]
lemma mulEnergy_univ_left : Eₘ[univ, t] = Fintype.card α * t.card ^ 2 | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : CommGroup α
inst✝ : Fintype α
t : Finset α
f : α × α × α → (α × α) × α × α := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)
⊢ Set.InjOn f ↑(univ ×ˢ t ×ˢ t) | rintro ⟨a₁, b₁, c₁⟩ _ ⟨a₂, b₂, c₂⟩ h₂ h | case mk.mk.mk.mk
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : CommGroup α
inst✝ : Fintype α
t : Finset α
f : α × α × α → (α × α) × α × α := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)
a₁ b₁ c₁ : α
a✝ : (a₁, b₁, c₁) ∈ ↑(univ ×ˢ t ×ˢ t)
a₂ b₂ c₂ : α
h₂ : (a₂, b₂, c₂) ∈ ↑(univ ×ˢ t ×ˢ t)
h : f (a₁, b₁, c₁) = f (a₂, b₂, c₂)
⊢ (a₁, b₁, c₁) = (a₂, b₂, c₂) | 744bbda61960c041 |
Nat.succ_dvd_centralBinom | Mathlib/Data/Nat/Choose/Central.lean | theorem succ_dvd_centralBinom (n : ℕ) : n + 1 ∣ n.centralBinom | n : ℕ
h_s : (n + 1).Coprime (2 * n + 1)
⊢ n + 1 ∣ (2 * n + 1) * n.centralBinom | apply Nat.dvd_of_mul_dvd_mul_left zero_lt_two | n : ℕ
h_s : (n + 1).Coprime (2 * n + 1)
⊢ 2 * (n + 1) ∣ 2 * ((2 * n + 1) * n.centralBinom) | fb1b90bbcc5f1bf9 |
LinearMap.trace_one | Mathlib/LinearAlgebra/Trace.lean | theorem trace_one : trace R M 1 = (finrank R M : R) | R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Free R M
inst✝ : Module.Finite R M
⊢ (trace R M) 1 = ↑(finrank R M) | cases subsingleton_or_nontrivial R | case inl
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Free R M
inst✝ : Module.Finite R M
h✝ : Subsingleton R
⊢ (trace R M) 1 = ↑(finrank R M)
case inr
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Free R M
inst✝ : Module.Finite R M
h✝ : Nontrivial R
⊢ (trace R M) 1 = ↑(finrank R M) | 4959d10d80e34ef9 |
Equiv.Perm.signAux_swap_zero_one' | Mathlib/GroupTheory/Perm/Sign.lean | theorem signAux_swap_zero_one' (n : ℕ) : signAux (swap (0 : Fin (n + 2)) 1) = -1 :=
show _ = ∏ x ∈ {(⟨1, 0⟩ : Σ _ : Fin (n + 2), Fin (n + 2))},
if (Equiv.swap 0 1) x.1 ≤ swap 0 1 x.2 then (-1 : ℤˣ) else 1 by
refine Eq.symm (prod_subset (fun ⟨x₁, x₂⟩ => by
simp +contextual [mem_finPairsLT, Fin.one_pos]) fun a ha₁ ha₂ => ?_)
rcases a with ⟨a₁, a₂⟩
replace ha₁ : a₂ < a₁ := mem_finPairsLT.1 ha₁
dsimp only
rcases a₁.zero_le.eq_or_lt with (rfl | H)
· exact absurd a₂.zero_le ha₁.not_le
rcases a₂.zero_le.eq_or_lt with (rfl | H')
· simp only [and_true, eq_self_iff_true, heq_iff_eq, mem_singleton, Sigma.mk.inj_iff] at ha₂
have : 1 < a₁ := lt_of_le_of_ne (Nat.succ_le_of_lt ha₁)
(Ne.symm (by intro h; apply ha₂; simp [h]))
have h01 : Equiv.swap (0 : Fin (n + 2)) 1 0 = 1 | case mk
n : ℕ
a₁ a₂ : Fin (n + 2)
ha₂ : ⟨a₁, a₂⟩ ∉ {⟨1, 0⟩}
ha₁ : a₂ < a₁
⊢ (if (swap 0 1) a₁ ≤ (swap 0 1) a₂ then -1 else 1) = 1 | rcases a₁.zero_le.eq_or_lt with (rfl | H) | case mk.inl
n : ℕ
a₂ : Fin (n + 2)
ha₂ : ⟨0, a₂⟩ ∉ {⟨1, 0⟩}
ha₁ : a₂ < 0
⊢ (if (swap 0 1) 0 ≤ (swap 0 1) a₂ then -1 else 1) = 1
case mk.inr
n : ℕ
a₁ a₂ : Fin (n + 2)
ha₂ : ⟨a₁, a₂⟩ ∉ {⟨1, 0⟩}
ha₁ : a₂ < a₁
H : 0 < a₁
⊢ (if (swap 0 1) a₁ ≤ (swap 0 1) a₂ then -1 else 1) = 1 | 31014fc4c4be8869 |
abs_sub_round_eq_min | Mathlib/Algebra/Order/Round.lean | theorem abs_sub_round_eq_min (x : α) : |x - round x| = min (fract x) (1 - fract x) | α : Type u_2
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
x : α
hx : fract x ≥ 1 - fract x
⊢ 0 < fract x | replace hx : 0 < fract x + fract x := lt_of_lt_of_le zero_lt_one (tsub_le_iff_left.mp hx) | α : Type u_2
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
x : α
hx : 0 < fract x + fract x
⊢ 0 < fract x | 04106663884a2f2b |
Module.End.eigenspace_restrict_le_eigenspace | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | theorem eigenspace_restrict_le_eigenspace (f : End R M) {p : Submodule R M} (hfp : ∀ x ∈ p, f x ∈ p)
(μ : R) : (eigenspace (f.restrict hfp) μ).map p.subtype ≤ f.eigenspace μ | case intro.intro
R : Type v
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : End R M
p : Submodule R M
hfp : ∀ x ∈ p, f x ∈ p
μ : R
x : ↥p
hx : x ∈ ↑(eigenspace (LinearMap.restrict f hfp) μ)
⊢ p.subtype x ∈ f.eigenspace μ | simp only [SetLike.mem_coe, mem_eigenspace_iff, LinearMap.restrict_apply] at hx ⊢ | case intro.intro
R : Type v
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : End R M
p : Submodule R M
hfp : ∀ x ∈ p, f x ∈ p
μ : R
x : ↥p
hx : ⟨f ↑x, ⋯⟩ = μ • x
⊢ f (p.subtype x) = μ • p.subtype x | b2a6d59fdcc08e4e |
IsFractional.mul | Mathlib/RingTheory/FractionalIdeal/Basic.lean | theorem _root_.IsFractional.mul {I J : Submodule R P} :
IsFractional S I → IsFractional S J → IsFractional S (I * J : Submodule R P)
| ⟨aI, haI, hI⟩, ⟨aJ, haJ, hJ⟩ =>
⟨aI * aJ, S.mul_mem haI haJ, fun b hb => by
refine Submodule.mul_induction_on hb ?_ ?_
· intro m hm n hn
obtain ⟨n', hn'⟩ := hJ n hn
rw [mul_smul, mul_comm m, ← smul_mul_assoc, ← hn', ← Algebra.smul_def]
apply hI
exact Submodule.smul_mem _ _ hm
· intro x y hx hy
rw [smul_add]
apply isInteger_add hx hy⟩
| case refine_1
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
⊢ ∀ m ∈ I, ∀ n ∈ J, IsInteger R ((aI * aJ) • (m * n)) | intro m hm n hn | case refine_1
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
m : P
hm : m ∈ I
n : P
hn : n ∈ J
⊢ IsInteger R ((aI * aJ) • (m * n)) | 003413a5af38d292 |
AlgebraicGeometry.affineAnd_isLocal | Mathlib/AlgebraicGeometry/Morphisms/AffineAnd.lean | /-- `affineAnd P` is local if `P` is local on the (algebraic) source. -/
lemma affineAnd_isLocal (hPi : RingHom.RespectsIso Q) (hQl : RingHom.LocalizationAwayPreserves Q)
(hQs : RingHom.OfLocalizationSpan Q) : (affineAnd Q).IsLocal where
respectsIso := affineAnd_respectsIso hPi
to_basicOpen {X Y _} f r := fun ⟨hX, hf⟩ ↦ by
simp only [Opens.map_top] at hf
constructor
· simp only [Scheme.preimage_basicOpen, Opens.map_top]
exact (isAffineOpen_top X).basicOpen _
· dsimp only
rw [morphismRestrict_appTop, CommRingCat.hom_comp, hPi.cancel_right_isIso]
-- Not sure why the `show` fixes the following `rw` complaining about "motive is incorrect"
show Q (Scheme.Hom.app f ((Y.basicOpen r).ι ''ᵁ ⊤)).hom
rw [Scheme.Opens.ι_image_top]
rw [(isAffineOpen_top Y).app_basicOpen_eq_away_map f (isAffineOpen_top X),
CommRingCat.hom_comp, hPi.cancel_right_isIso, ← Scheme.Hom.appTop]
dsimp only [Opens.map_top]
haveI := (isAffineOpen_top X).isLocalization_basicOpen (f.appTop r)
apply hQl
exact hf
of_basicOpenCover {X Y _} f s hs hf | Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => Q
hQl : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => Q
hQs : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] => Q
X Y : Scheme
x✝ : IsAffine Y
f : X ⟶ Y
s : Finset ↑Γ(Y, ⊤)
hs : Ideal.span ↑s = ⊤
hf :
∀ (r : { x // x ∈ s }),
IsAffine ↑(f ⁻¹ᵁ Y.basicOpen ↑r) ∧ Q (CommRingCat.Hom.hom (Scheme.Hom.appTop (f ∣_ Y.basicOpen ↑r)))
this : IsAffine X
⊢ affineAnd (fun {R S} [CommRing R] [CommRing S] => Q) f | refine ⟨inferInstance, hQs.ofIsLocalization' hPi (f.appTop).hom s hs fun a ↦ ?_⟩ | Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => Q
hQl : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => Q
hQs : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] => Q
X Y : Scheme
x✝ : IsAffine Y
f : X ⟶ Y
s : Finset ↑Γ(Y, ⊤)
hs : Ideal.span ↑s = ⊤
hf :
∀ (r : { x // x ∈ s }),
IsAffine ↑(f ⁻¹ᵁ Y.basicOpen ↑r) ∧ Q (CommRingCat.Hom.hom (Scheme.Hom.appTop (f ∣_ Y.basicOpen ↑r)))
this : IsAffine X
a : ↑↑s
⊢ ∃ Rᵣ Sᵣ x x_1 x_2 x_3,
∃ (x_4 : IsLocalization.Away (↑a) Rᵣ) (x_5 :
IsLocalization.Away ((CommRingCat.Hom.hom (Scheme.Hom.appTop f)) ↑a) Sᵣ),
Q (IsLocalization.Away.map Rᵣ Sᵣ (CommRingCat.Hom.hom (Scheme.Hom.appTop f)) ↑a) | 2b5f5ef2a8aa609c |
List.pairwise_filter | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Pairwise.lean | theorem pairwise_filter {p : α → Prop} [DecidablePred p] {l : List α} :
Pairwise R (filter p l) ↔ Pairwise (fun x y => p x → p y → R x y) l | α : Type u_1
R : α → α → Prop
p : α → Prop
inst✝ : DecidablePred p
l : List α
⊢ Pairwise R (filter (fun b => decide (p b)) l) ↔ Pairwise (fun x y => p x → p y → R x y) l | rw [← filterMap_eq_filter, pairwise_filterMap] | α : Type u_1
R : α → α → Prop
p : α → Prop
inst✝ : DecidablePred p
l : List α
⊢ Pairwise
(fun a a' =>
∀ (b : α),
b ∈ Option.guard (fun x => decide (p x) = true) a →
∀ (b' : α), b' ∈ Option.guard (fun x => decide (p x) = true) a' → R b b')
l ↔
Pairwise (fun x y => p x → p y → R x y) l | 7ca88e024fd3a25b |
SetTheory.PGame.insertRight_numeric | Mathlib/SetTheory/Surreal/Basic.lean | theorem insertRight_numeric {x x' : PGame} (x_num : x.Numeric) (x'_num : x'.Numeric)
(h : x ≤ x') : (insertRight x x').Numeric | x x' : PGame
x_num : x.Numeric
x'_num : x'.Numeric
h : x ≤ x'
⊢ (-(-x).insertLeft (-x')).Numeric | apply Numeric.neg | case x
x x' : PGame
x_num : x.Numeric
x'_num : x'.Numeric
h : x ≤ x'
⊢ ((-x).insertLeft (-x')).Numeric | 3fe52f8d55a21b54 |
SimpleGraph.Walk.concat_inj | Mathlib/Combinatorics/SimpleGraph/Walk.lean | theorem concat_inj {u v v' w : V} {p : G.Walk u v} {h : G.Adj v w} {p' : G.Walk u v'}
{h' : G.Adj v' w} (he : p.concat h = p'.concat h') : ∃ hv : v = v', p.copy rfl hv = p' | case nil.cons.intro
V : Type u
G : SimpleGraph V
u v v' w : V
h' : G.Adj v' w
u✝ : V
h h✝ : G.Adj u✝ w
p✝ : G.Walk w v'
he : nil = p✝.concat h'
⊢ False | exact concat_ne_nil _ _ he.symm | no goals | 6aa77e2092977f18 |
contDiffWithinAt_localInvariantProp_of_le | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | theorem contDiffWithinAt_localInvariantProp_of_le (n m : WithTop ℕ∞) (hmn : m ≤ n) :
(contDiffGroupoid n I).LocalInvariantProp (contDiffGroupoid n I')
(ContDiffWithinAtProp I I' m) where
is_local {s x u f} u_open xu | case convert_2
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
n m : WithTop ℕ∞
hmn : m ≤ n
s : Set H
x : H
f : H → H'
e : PartialHomeomorph H H
he : e ∈ contDiffGroupoid n I
hx : x ∈ e.source
h : ContDiffWithinAt 𝕜 m (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) ((↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x)))
this✝¹ : ↑I x = (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x))
this✝ : ↑I (↑e x) ∈ ↑I.symm ⁻¹' e.target ∩ range ↑I
this : ContDiffWithinAt 𝕜 n (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ range ↑I) (↑I (↑e x))
⊢ ↑I.symm ⁻¹' e.target ∩ (↑I.symm ⁻¹' (↑e.symm ⁻¹' s) ∩ range ↑I) ⊆
↑I.symm ⁻¹' e.target ∩ range ↑I ∩ ↑I ∘ ↑e.symm ∘ ↑I.symm ⁻¹' (↑I.symm ⁻¹' s ∩ range ↑I) | mfld_set_tac | no goals | 841928255da4a913 |
MeasureTheory.Measure.eq_withDensity_rnDeriv | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | theorem eq_withDensity_rnDeriv {s : Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (hs : s ⟂ₘ ν)
(hadd : μ = s + ν.withDensity f) : ν.withDensity f = ν.withDensity (μ.rnDeriv ν) | α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hs : s ⟂ₘ ν
hadd : μ = s + ν.withDensity f
⊢ ν.withDensity f = ν.withDensity (μ.rnDeriv ν) | have : HaveLebesgueDecomposition μ ν := ⟨⟨⟨s, f⟩, hf, hs, hadd⟩⟩ | α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hs : s ⟂ₘ ν
hadd : μ = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
⊢ ν.withDensity f = ν.withDensity (μ.rnDeriv ν) | fc3f06ec8038d87c |
MeasureTheory.continuous_of_dominated | Mathlib/MeasureTheory/Integral/Bochner.lean | theorem continuous_of_dominated {F : X → α → G} {bound : α → ℝ}
(hF_meas : ∀ x, AEStronglyMeasurable (F x) μ) (h_bound : ∀ x, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a)
(bound_integrable : Integrable bound μ) (h_cont : ∀ᵐ a ∂μ, Continuous fun x => F x a) :
Continuous fun x => ∫ a, F x a ∂μ | case pos
α : Type u_1
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
F : X → α → G
bound : α → ℝ
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) μ
h_bound : ∀ (x : X), ∀ᵐ (a : α) ∂μ, ‖F x a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_cont : ∀ᵐ (a : α) ∂μ, Continuous fun x => F x a
hG : CompleteSpace G
⊢ Continuous fun x => ∫ (a : α), F x a ∂μ | simp only [integral, hG, L1.integral] | case pos
α : Type u_1
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
F : X → α → G
bound : α → ℝ
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) μ
h_bound : ∀ (x : X), ∀ᵐ (a : α) ∂μ, ‖F x a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_cont : ∀ᵐ (a : α) ∂μ, Continuous fun x => F x a
hG : CompleteSpace G
⊢ Continuous fun x =>
if h : True then
if hf : Integrable (fun a => F x a) μ then L1.integralCLM (Integrable.toL1 (fun a => F x a) hf) else 0
else 0 | 4a2462339b98c2fb |
Localization.smul_mk | Mathlib/GroupTheory/MonoidLocalization/Basic.lean | theorem smul_mk [SMul R M] [IsScalarTower R M M] (c : R) (a b) :
c • (mk a b : Localization S) = mk (c • a) b | M : Type u_1
inst✝² : CommMonoid M
S : Submonoid M
R : Type u_4
inst✝¹ : SMul R M
inst✝ : IsScalarTower R M M
c : R
a : M
b : ↥S
⊢ (c • 1) • a /ₒ (b * 1) = c • a /ₒ b | rw [smul_assoc, one_smul, mul_one] | no goals | f28c11ec5885a209 |
isZero_Ext_succ_of_projective | Mathlib/CategoryTheory/Abelian/Ext.lean | /-- If `X : C` is projective and `n : ℕ`, then `Ext^(n + 1) X Y ≅ 0` for any `Y`. -/
lemma isZero_Ext_succ_of_projective (X Y : C) [Projective X] (n : ℕ) :
IsZero (((Ext R C (n + 1)).obj (Opposite.op X)).obj Y) | case hf.h
R : Type u_1
inst✝⁵ : Ring R
C : Type u_2
inst✝⁴ : Category.{u_3, u_2} C
inst✝³ : Abelian C
inst✝² : Linear R C
inst✝¹ : EnoughProjectives C
X Y : C
inst✝ : Projective X
n : ℕ
⊢ (ModuleCat.Hom.hom (𝟙 (((linearYoneda R C).obj Y).obj (Opposite.op (((ChainComplex.single₀ C).obj X).X (n + 1))))))
0 =
(ModuleCat.Hom.hom 0) 0 | rfl | no goals | 40abdca1318b58b3 |
eVariationOn.add_point | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | theorem add_point (f : α → E) {s : Set α} {x : α} (hx : x ∈ s) (u : ℕ → α) (hu : Monotone u)
(us : ∀ i, u i ∈ s) (n : ℕ) :
∃ (v : ℕ → α) (m : ℕ), Monotone v ∧ (∀ i, v i ∈ s) ∧ x ∈ v '' Iio m ∧
(∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) ≤
∑ j ∈ Finset.range m, edist (f (v (j + 1))) (f (v j)) | case inr.refine_2
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i => if i < N then u i else if i = N then x else u (i - 1)
ws : ∀ (i : ℕ), w i ∈ s
hw : Monotone w
⊢ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ ∑ j ∈ Finset.range (n + 1), edist (f (w (j + 1))) (f (w j)) | rcases eq_or_lt_of_le (zero_le N) with (Npos | Npos) | case inr.refine_2.inl
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i => if i < N then u i else if i = N then x else u (i - 1)
ws : ∀ (i : ℕ), w i ∈ s
hw : Monotone w
Npos : 0 = N
⊢ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ ∑ j ∈ Finset.range (n + 1), edist (f (w (j + 1))) (f (w j))
case inr.refine_2.inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i => if i < N then u i else if i = N then x else u (i - 1)
ws : ∀ (i : ℕ), w i ∈ s
hw : Monotone w
Npos : 0 < N
⊢ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ ∑ j ∈ Finset.range (n + 1), edist (f (w (j + 1))) (f (w j)) | 760e94573f7a043d |
Homotopy.nullHomotopicMap_comp | Mathlib/Algebra/Homology/Homotopy.lean | theorem nullHomotopicMap_comp (hom : ∀ i j, C.X i ⟶ D.X j) (g : D ⟶ E) :
nullHomotopicMap hom ≫ g = nullHomotopicMap fun i j => hom i j ≫ g.f j | case h
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
hom : (i j : ι) → C.X i ⟶ D.X j
g : D ⟶ E
n : ι
⊢ (C.dFrom n ≫ hom (c.next n) n + hom n (c.prev n) ≫ D.dTo n) ≫ g.f n =
C.dFrom n ≫ hom (c.next n) n ≫ g.f n + (hom n (c.prev n) ≫ g.f (c.prev n)) ≫ E.dTo n | simp only [Preadditive.add_comp, assoc, g.comm] | no goals | 08696897e099d0bd |
ContinuousMultilinearMap.hasFTaylorSeriesUpTo_iteratedFDeriv | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | theorem hasFTaylorSeriesUpTo_iteratedFDeriv :
HasFTaylorSeriesUpTo ⊤ f (fun v n ↦ f.iteratedFDeriv n v) | 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
⊢ HasFTaylorSeriesUpTo ⊤ ⇑f fun v n => f.iteratedFDeriv n v | constructor | case zero_eq
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
⊢ ∀ (x : (i : ι) → E i), (f.iteratedFDeriv 0 x).curry0 = f x
case fderiv
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
⊢ ∀ (m : ℕ),
↑m < ⊤ → ∀ (x : (i : ι) → E i), HasFDerivAt (fun y => f.iteratedFDeriv m y) (f.iteratedFDeriv m.succ x).curryLeft x
case cont
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
⊢ ∀ (m : ℕ), ↑m ≤ ⊤ → Continuous fun x => f.iteratedFDeriv m x | ae8227810170d45c |
IsPGroup.smul_mul_inv_trivial_or_surjective | Mathlib/GroupTheory/SpecificGroups/ZGroup.lean | theorem smul_mul_inv_trivial_or_surjective [IsCyclic G] (hG : IsPGroup p G)
{K : Type*} [Group K] [MulDistribMulAction K G] (hGK : (Nat.card G).Coprime (Nat.card K)) :
(∀ g : G, ∀ k : K, k • g * g⁻¹ = 1) ∨ (∀ g : G, ∃ k : K, ∃ q : G, k • q * q⁻¹ = g) | case neg.inr.intro.intro.mk
G : Type u_1
inst✝⁴ : Group G
p✝ : ℕ
inst✝³ : Fact (Nat.Prime p✝)
inst✝² : IsCyclic G
K : Type u_4
inst✝¹ : Group K
inst✝ : MulDistribMulAction K G
hGK : (Nat.card G).Coprime (Nat.card K)
hc : ¬Nat.card G = 0
this : Finite G
ϕ : K →* ZMod (Nat.card G) := MulDistribMulAction.toMonoidHomZModOfIsCyclic G K ⋯
h : ∀ (g : G) (k : K) (n : ℤ), ϕ k - 1 = ↑n → k • g * g⁻¹ = g ^ n
hG : ∀ (k : K), ϕ k = 1 ∨ IsUnit (ϕ k - 1)
k : K
hk : ¬ϕ k = 1
u v : ZMod (Nat.card G)
hvu : v.cast * u.cast ≡ 1 [ZMOD ↑(Nat.card G)]
hu : ↑u.cast = ϕ k - 1
p : G
⊢ k • p ^ v.cast * (p ^ v.cast)⁻¹ = p ^ 1 | rw [h (p ^ v.cast) k u.cast hu.symm, ← zpow_mul, zpow_eq_zpow_iff_modEq] | case neg.inr.intro.intro.mk
G : Type u_1
inst✝⁴ : Group G
p✝ : ℕ
inst✝³ : Fact (Nat.Prime p✝)
inst✝² : IsCyclic G
K : Type u_4
inst✝¹ : Group K
inst✝ : MulDistribMulAction K G
hGK : (Nat.card G).Coprime (Nat.card K)
hc : ¬Nat.card G = 0
this : Finite G
ϕ : K →* ZMod (Nat.card G) := MulDistribMulAction.toMonoidHomZModOfIsCyclic G K ⋯
h : ∀ (g : G) (k : K) (n : ℤ), ϕ k - 1 = ↑n → k • g * g⁻¹ = g ^ n
hG : ∀ (k : K), ϕ k = 1 ∨ IsUnit (ϕ k - 1)
k : K
hk : ¬ϕ k = 1
u v : ZMod (Nat.card G)
hvu : v.cast * u.cast ≡ 1 [ZMOD ↑(Nat.card G)]
hu : ↑u.cast = ϕ k - 1
p : G
⊢ v.cast * u.cast ≡ 1 [ZMOD ↑(orderOf p)] | b33bc739e8adc37b |
abs_one_div | Mathlib/Algebra/Order/Field/Basic.lean | theorem abs_one_div (a : α) : |1 / a| = 1 / |a| | α : Type u_2
inst✝ : LinearOrderedField α
a : α
⊢ |1 / a| = 1 / |a| | rw [abs_div, abs_one] | no goals | ed2a7b8e8201fe19 |
Nat.card_pair_lcm_eq.f_img | Mathlib/Algebra/Order/Antidiag/Nat.lean | theorem f_img {n : ℕ} (hn : Squarefree n) (a : Fin 3 → ℕ)
(ha : a ∈ finMulAntidiag 3 n) :
f a ha ∈ Finset.filter (fun ⟨x, y⟩ => x.lcm y = n) (n.divisors ×ˢ n.divisors) | case h
n : ℕ
hn : Squarefree n
a : Fin 3 → ℕ
ha : a ∈ finMulAntidiag 3 n
⊢ a 0 * a 1 * a 2 = a 1 * a 2 * a 0 | ring | no goals | 266d4fcf66130dfe |
Batteries.RBNode.Balanced.zoom | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Alter.lean | theorem _root_.Batteries.RBNode.Balanced.zoom : t.Balanced c n → path.Balanced c₀ n₀ c n →
zoom cut t path = (t', path') → ∃ c n, t'.Balanced c n ∧ path'.Balanced c₀ n₀ c n
| .nil, hp => fun e => by cases e; exact ⟨_, _, .nil, hp⟩
| .red ha hb, hp => by
unfold zoom; split
· exact ha.zoom (.redL hb hp)
· exact hb.zoom (.redR ha hp)
· intro e; cases e; exact ⟨_, _, .red ha hb, hp⟩
| .black ha hb, hp => by
unfold zoom; split
· exact ha.zoom (.blackL hb hp)
· exact hb.zoom (.blackR ha hp)
· intro e; cases e; exact ⟨_, _, .black ha hb, hp⟩
| case h_3
α✝ : Type u_1
cut : α✝ → Ordering
t : RBNode α✝
path : Path α✝
t' : RBNode α✝
path' : Path α✝
c₀ : RBColor
n₀ : Nat
c : RBColor
n✝ n : Nat
x✝¹ y✝ : RBNode α✝
v✝ : α✝
ha : x✝¹.Balanced black n
hb : y✝.Balanced black n
hp : Path.Balanced c₀ n₀ path red n
x✝ : Ordering
heq✝ : cut v✝ = Ordering.eq
e : (node red x✝¹ v✝ y✝, path) = (t', path')
⊢ ∃ c n, t'.Balanced c n ∧ Path.Balanced c₀ n₀ path' c n | cases e | case h_3.refl
α✝ : Type u_1
cut : α✝ → Ordering
t : RBNode α✝
path : Path α✝
c₀ : RBColor
n₀ : Nat
c : RBColor
n✝ n : Nat
x✝¹ y✝ : RBNode α✝
v✝ : α✝
ha : x✝¹.Balanced black n
hb : y✝.Balanced black n
hp : Path.Balanced c₀ n₀ path red n
x✝ : Ordering
heq✝ : cut v✝ = Ordering.eq
⊢ ∃ c n, (node red x✝¹ v✝ y✝).Balanced c n ∧ Path.Balanced c₀ n₀ path c n | cff44faeb7153c11 |
Polynomial.content_X_mul | Mathlib/RingTheory/Polynomial/Content.lean | theorem content_X_mul {p : R[X]} : content (X * p) = content p | case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
a : ℕ
⊢ ¬(X * p).coeff a = 0 ↔ ∃ a_1, ¬p.coeff a_1 = 0 ∧ a_1.succ = a | rcases a with - | a | case h.zero
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
⊢ ¬(X * p).coeff 0 = 0 ↔ ∃ a, ¬p.coeff a = 0 ∧ a.succ = 0
case h.succ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
a : ℕ
⊢ ¬(X * p).coeff (a + 1) = 0 ↔ ∃ a_1, ¬p.coeff a_1 = 0 ∧ a_1.succ = a + 1 | ac071a5a6c3a3683 |
Submodule.LinearDisjoint.of_linearDisjoint_fg_right | Mathlib/LinearAlgebra/LinearDisjoint.lean | theorem of_linearDisjoint_fg_right
(H : ∀ N' : Submodule R S, N' ≤ N → N'.FG → M.LinearDisjoint N') :
M.LinearDisjoint N := (linearDisjoint_iff _ _).2 fun x y hxy ↦ by
obtain ⟨N', hN, hFG, h⟩ :=
TensorProduct.exists_finite_submodule_right_of_finite' {x, y} (Set.toFinite _)
rw [Module.Finite.iff_fg] at hFG
obtain ⟨x', hx'⟩ := h (show x ∈ {x, y} by simp)
obtain ⟨y', hy'⟩ := h (show y ∈ {x, y} by simp)
rw [← hx', ← hy']; congr
exact (H N' hN hFG).injective (by simp [← mulMap_comp_lTensor _ hN, hx', hy', hxy])
| case intro.intro.intro
R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring S
inst✝ : Algebra R S
M N : Submodule R S
H : ∀ N' ≤ N, N'.FG → M.LinearDisjoint N'
x y : ↥M ⊗[R] ↥N
hxy : (M.mulMap N) x = (M.mulMap N) y
N' : Submodule R S
hN : N' ≤ N
hFG : N'.FG
h : {x, y} ⊆ ↑(LinearMap.range (LinearMap.lTensor (↥M) (inclusion hN)))
⊢ x = y | obtain ⟨x', hx'⟩ := h (show x ∈ {x, y} by simp) | case intro.intro.intro.intro
R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring S
inst✝ : Algebra R S
M N : Submodule R S
H : ∀ N' ≤ N, N'.FG → M.LinearDisjoint N'
x y : ↥M ⊗[R] ↥N
hxy : (M.mulMap N) x = (M.mulMap N) y
N' : Submodule R S
hN : N' ≤ N
hFG : N'.FG
h : {x, y} ⊆ ↑(LinearMap.range (LinearMap.lTensor (↥M) (inclusion hN)))
x' : ↥M ⊗[R] ↥N'
hx' : (LinearMap.lTensor (↥M) (inclusion hN)) x' = x
⊢ x = y | 3ea9d0fb125f2a5e |
WithSeminorms.isVonNBounded_iff_seminorm_bounded | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | theorem WithSeminorms.isVonNBounded_iff_seminorm_bounded {s : Set E} (hp : WithSeminorms p) :
Bornology.IsVonNBounded 𝕜 s ↔ ∀ i : ι, ∃ r > 0, ∀ x ∈ s, p i x < r | case neg
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Nonempty ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
s : Set E
hp : WithSeminorms p
hi : ∀ (i : ι), ∃ r > 0, ∀ x ∈ s, (p i) x < r
I : Finset ι
hI : ¬I.Nonempty
⊢ ∃ r > 0, ∀ x ∈ s, (I.sup p) x < r | simp only [Finset.not_nonempty_iff_eq_empty.mp hI, Finset.sup_empty, coe_bot, Pi.zero_apply,
exists_prop] | case neg
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Nonempty ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
s : Set E
hp : WithSeminorms p
hi : ∀ (i : ι), ∃ r > 0, ∀ x ∈ s, (p i) x < r
I : Finset ι
hI : ¬I.Nonempty
⊢ ∃ r > 0, ∀ x ∈ s, 0 < r | 24b13a155e892387 |
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