Theorems · Theorem · linear algebra

LinearMap.rank_quot_submodule_map_eq

∀ {R : Type u_1} {M : Type u} {M' : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : AddCommGroup M']
  [inst_3 : Module R M] [inst_4 : Module R M'] [HasRankNullity.{v, u_1} R] {f : M →ₗ[R] M'} (p : Submodule R M),
  Module.rank R (M' ⧸ Submodule.map f p) =
    Module.rank R (M' ⧸ f.range) + Module.rank R (↥f.range ⧸ Submodule.map f.rangeRestrict p)

Defined in

Mathlib.LinearAlgebra.Dimension.RankNullity

Cited by

1 results in Mathlib

Foundations

Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupAddCommGroupModuleModuleHasRankNullity