Theorems · Definition · linear algebra

LinearMap.lflip

{R : Type u_1} →
  {R₂ : Type u_2} →
    {S : Type u_3} →
      {S₂ : Type u_4} →
        [inst : Semiring R] →
          [inst_1 : Semiring R₂] →
            [inst_2 : Semiring S] →
              [inst_3 : Semiring S₂] →
                {M : Type u_5} →
                  {N : Type u_7} →
                    {P : Type u_9} →
                      [inst_4 : AddCommMonoid M] →
                        [inst_5 : AddCommMonoid N] →
                          [inst_6 : AddCommMonoid P] →
                            [inst_7 : Module R M] →
                              [inst_8 : Module S N] →
                                [inst_9 : Module R₂ P] →
                                  [inst_10 : Module S₂ P] →
                                    [inst_11 : SMulCommClass S₂ R₂ P] →
                                      {ρ₁₂ : R →+* R₂} →
                                        {σ₁₂ : S →+* S₂} →
                                          {R₀ : Type u_14} →
                                            [inst_12 : Semiring R₀] →
                                              [inst_13 : Module R₀ P] →
                                                [inst_14 : SMulCommClass S₂ R₀ P] →
                                                  [inst_15 : SMulCommClass R₂ R₀ P] →
                                                    (M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P) ≃ₗ[R₀] N →ₛₗ[σ₁₂] M →ₛₗ[ρ₁₂] P

LinearMap.flip as an isomorphism of modules.

Defined in

Mathlib.LinearAlgebra.BilinearMap

Cited by

6 results in Mathlib

Foundations

Depth 36 from the axioms · uses propext, Quot.sound

Assumes

SemiringSemiringSemiringSemiringAddCommMonoidAddCommMonoidAddCommMonoidModuleModuleModuleModuleSMulCommClassSemiringModuleSMulCommClassSMulCommClass