Theorems · Theorem · linear algebra

LinearEquiv.ofInjective_symm_apply

∀ {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [inst : Semiring R] [inst_1 : Semiring R₂]
  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] {module_M : Module R M} {module_M₂ : Module R₂ M₂}
  {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} (f : M →ₛₗ[σ₁₂] M₂) [inst_4 : RingHomInvPair σ₁₂ σ₂₁]
  [inst_5 : RingHomInvPair σ₂₁ σ₁₂] {h : Function.Injective ⇑f} (x : ↥f.range),
  f ((LinearEquiv.ofInjective f h).symm x) = ↑x

Defined in

Mathlib.Algebra.Module.Submodule.Equiv

Cited by

3 results in Mathlib

Foundations

Depth 31 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringSemiringAddCommMonoidAddCommMonoidRingHomInvPairRingHomInvPair