Theorems · Theorem · category theory

Function.Exact.iff_linearMap_rangeRestrict

∀ {R : Type u_1} {M : Type u_2} {N : Type u_4} {P : Type u_6} [inst : Semiring R] [inst_1 : AddCommMonoid M]
  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]
  [inst_6 : Module R P] {f : M →ₗ[R] N} {g : N →ₗ[R] P},
  Function.Exact ⇑f ⇑g ↔ Function.Exact ⇑f.range.subtype ⇑g.rangeRestrict

Two maps f : M →ₗ[R] N and g : N →ₗ[R] P are exact if and only if the induced maps LinearMap.range f → N → LinearMap.range g are exact.

Defined in

Mathlib.Algebra.Exact.Basic

Cited by

1 results in Mathlib

Foundations

Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidAddCommMonoidAddCommMonoidModuleModuleModule