Theorems · Theorem · category theory

CategoryTheory.ShortComplex.moduleCat_pOpcycles_eq_iff

∀ {R : Type u} [inst : Ring R] (S : CategoryTheory.ShortComplex (ModuleCat R)) (x y : ↑S.X₂),
  (CategoryTheory.ConcreteCategory.hom S.pOpcycles) x = (CategoryTheory.ConcreteCategory.hom S.pOpcycles) y ↔
    x - y ∈ (ModuleCat.Hom.hom S.f).range

Defined in: Mathlib.Algebra.Homology.ShortComplex.ModuleCat
Cited by: 3 results in Mathlib
Foundations: Depth 99 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Ring

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Cites22

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
RingHom.idstatement · cited by 18,349
LinearMapstatement · cited by 10,215
CategoryTheory.Iso.homproof · cited by 7,684
Ringstatement and proof · cited by 7,463
Submodulestatement · cited by 7,192
CategoryTheory.ConcreteCategory.homstatement and proof · cited by 4,022
CategoryTheory.ShortComplexstatement and proof · cited by 1,850
ModuleCatstatement and proof · cited by 1,429
CategoryTheory.ShortComplex.X₂statement and proof · cited by 1,115
ModuleCat.carrierstatement and proof · cited by 997
LinearMap.rangestatement and proof · cited by 893

Cited by3

Results whose statement or proof uses this declaration.

groupHomology.comap_coinvariantsKer_pOpcycles_range_subtype_pOpcycles_eq_topproof · cited by 1
CategoryTheory.ShortComplex.moduleCat_pOpcycles_eq_zero_iffproof · cited by 0
groupHomology.H1CoresCoinf_exactproof · cited by 0