proposition: Lemma abgr_DirectSumId (A B : abgr) : @abmonoidshombinop (abgrdirprod A B) (abgrdirprod A B) (abgr_DirectSumPr1 A B · abgr_DirectSumIn1 A B) (abgr_DirectSumPr2 A B · abgr_DirectSumIn2 A B) = ((idmonoidiso (abgrdirprod A B)) : monoidfun (abgrdirprod A B) (abgrdirprod A B)) .
proof: Proof. use monoidfun_paths. use funextfun. intros x. use dirprod_paths. - use (runax A). - use (lunax B). Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.Foundations.UnivalenceAxiom UniMath.Algebra.BinaryOperations UniMath.Algebra.Groups UniMath.Algebra.Monoids UniMath.NumberSystems.Integers UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Epis UniMath.CategoryTheory.CategoriesWithBinOps UniMath.CategoryTheory.PrecategoriesWithAbgrops UniMath.CategoryTheory.PreAdditive UniMath.CategoryTheory.Additive UniMath.CategoryTheory.Abelian UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Coequalizers UniMath.CategoryTheory.Limits.Kernels UniMath.CategoryTheory.Limits.Cokernels UniMath.CategoryTheory.Limits.BinDirectSums