Theorems · Theorem · linear algebra

lift_rank_range_le

∀ {R : Type u} {M : Type v} {M' : Type v'} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
  [inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] (f : M →ₗ[R] M'),
  Cardinal.lift.{v, v'} (Module.rank R ↥f.range) ≤ Cardinal.lift.{v', v} (Module.rank R M)

The rank of the range of a linear map is at most the rank of the source.

Defined in

Mathlib.LinearAlgebra.Dimension.Basic

Cited by

7 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidModuleAddCommMonoidModule