Theorems · Definition · linear algebra

LinearEquiv.ofBijective

{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] → Function.Bijective ⇑f → M ≃ₛₗ[σ₁₂] M₂

A bijective linear map is a linear equivalence.

Defined in

Mathlib.Algebra.Module.Submodule.Equiv

Cited by

60 results in Mathlib

Foundations

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

Assumes

SemiringSemiringAddCommMonoidAddCommMonoidRingHomInvPairRingHomInvPair