Theorems · Definition · geometry

AffineEquiv.ofBijective

{k : Type u_1} →
  {P₁ : Type u_2} →
    {P₂ : Type u_3} →
      {V₁ : Type u_6} →
        {V₂ : Type u_7} →
          [inst : Ring k] →
            [inst_1 : AddCommGroup V₁] →
              [inst_2 : AddCommGroup V₂] →
                [inst_3 : Module k V₁] →
                  [inst_4 : Module k V₂] →
                    [inst_5 : AddTorsor V₁ P₁] →
                      [inst_6 : AddTorsor V₂ P₂] → {φ : P₁ →ᵃ[k] P₂} → Function.Bijective ⇑φ → P₁ ≃ᵃ[k] P₂

Bijective affine maps are affine isomorphisms.

Defined in

Mathlib.LinearAlgebra.AffineSpace.AffineEquiv

Cited by

3 results in Mathlib

Foundations

Depth 32 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupAddCommGroupModuleModuleAddTorsorAddTorsor