Theorems · Definition · geometry

AffineEquiv.arrowCongrEquiv

{k : Type u_1} →
  {P₁ : Type u_2} →
    {P₂ : Type u_3} →
      {P₃ : Type u_4} →
        {P₄ : Type u_5} →
          {V₁ : Type u_6} →
            {V₂ : Type u_7} →
              {V₃ : Type u_8} →
                {V₄ : Type u_9} →
                  [inst : Ring k] →
                    [inst_1 : AddCommGroup V₁] →
                      [inst_2 : AddCommGroup V₂] →
                        [inst_3 : AddCommGroup V₃] →
                          [inst_4 : AddCommGroup V₄] →
                            [inst_5 : Module k V₁] →
                              [inst_6 : Module k V₂] →
                                [inst_7 : Module k V₃] →
                                  [inst_8 : Module k V₄] →
                                    [inst_9 : AddTorsor V₁ P₁] →
                                      [inst_10 : AddTorsor V₂ P₂] →
                                        [inst_11 : AddTorsor V₃ P₃] →
                                          [inst_12 : AddTorsor V₄ P₄] →
                                            (P₁ ≃ᵃ[k] P₂) → (P₃ ≃ᵃ[k] P₄) → (P₁ →ᵃ[k] P₃) ≃ (P₂ →ᵃ[k] P₄)

Affine isomorphisms between the domains and codomains of two spaces of affine maps give a bijection between the two function spaces. See AffineEquiv.arrowCongr and AffineEquiv.arrowCongrₗ for the affine and linear versions of this bijection.

Defined in

Mathlib.LinearAlgebra.AffineSpace.AffineEquiv

Cited by

2 results in Mathlib

Foundations

Depth 34 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupAddCommGroupAddCommGroupAddCommGroupModuleModuleModuleModuleAddTorsorAddTorsorAddTorsorAddTorsor