ModuleCat.cokernel_π_imageSubobject_ext

theorem ModuleCat.cokernel_π_imageSubobject_ext : ∀ {R : Type u} [inst : Ring R] {L M N : ModuleCat R} (f : L ⟶ M) [inst_1 : CategoryTheory.Limits.HasImage f] (g : CategoryTheory.Subobject.underlying.obj (CategoryTheory.Limits.imageSubobject f) ⟶ N) [inst_2 : CategoryTheory.Limits.HasCokernel g] {x y : ↑N} (l : ↑L), x = y + g ((CategoryTheory.Limits.factorThruImageSubobject f) l) → (CategoryTheory.Limits.cokernel.π g) x = (CategoryTheory.Limits.cokernel.π g) y

The theorem `ModuleCat.cokernel_π_imageSubobject_ext` states that for a given ring `R` and its modules `L`, `M`, and `N`, given a morphism `f` from `L` to `M` that has an image, and another morphism `g` from the image of `f` to `N` that has a cokernel, for any two elements `x` and `y` in the module `N` and an element `l` in module `L`, if `x` is equal to `y` plus `g` applied to the result of factoring `f` through its image applied to `l`, then the morphism from the cokernel of `g` applied to `x` is equal to the morphism from the cokernel of `g` applied to `y`. This theorem is significant in homology, where it implies that two elements in homology are equal if they differ by a boundary.

More concisely: Given rings R, modules L, M, and N, morphisms f : L → M with image I, and g : I → N with cokernel, if for all l in L and x, y in N, x = y + g(f(l)), then the morphisms from the cokernel of g applied to x and y are equal.