Theorems · Theorem · linear algebra

LinearMap.lift_rank_eq_of_surjective

∀ {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.{u, u_1} R] {f : M →ₗ[R] M'},
  Function.Surjective ⇑f →
    Cardinal.lift.{v, u} (Module.rank R M) =
      Cardinal.lift.{u, v} (Module.rank R M') + Cardinal.lift.{v, u} (Module.rank R ↥f.ker)

Defined in

Mathlib.LinearAlgebra.Dimension.RankNullity

Cited by

0 results in Mathlib

Foundations

Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupAddCommGroupModuleModuleHasRankNullity