Theorems · Theorem · commutative algebra

LinearEquiv.mk.injEq

∀ {R : Type u_14} {S : Type u_15} [inst : Semiring R] [inst_1 : Semiring S] {σ : R →+* S} {σ' : S →+* R}
  [inst_2 : RingHomInvPair σ σ'] [inst_3 : RingHomInvPair σ' σ] {M : Type u_16} {M₂ : Type u_17}
  [inst_4 : AddCommMonoid M] [inst_5 : AddCommMonoid M₂] [inst_6 : Module R M] [inst_7 : Module S M₂]
  (toLinearMap : M →ₛₗ[σ] M₂) (invFun : M₂ → M)
  (left_inv : autoParam (Function.LeftInverse invFun toLinearMap.toFun) Equiv.left_inv._autoParam)
  (right_inv : autoParam (Function.RightInverse invFun toLinearMap.toFun) Equiv.right_inv._autoParam)
  (toLinearMap_1 : M →ₛₗ[σ] M₂) (invFun_1 : M₂ → M)
  (left_inv_1 : autoParam (Function.LeftInverse invFun_1 toLinearMap_1.toFun) Equiv.left_inv._autoParam)
  (right_inv_1 : autoParam (Function.RightInverse invFun_1 toLinearMap_1.toFun) Equiv.right_inv._autoParam),
  ({ toLinearMap := toLinearMap, invFun := invFun, left_inv := left_inv, right_inv := right_inv } =
      { toLinearMap := toLinearMap_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1 }) =
    (toLinearMap = toLinearMap_1 ∧ invFun = invFun_1)

Defined in

Mathlib.Algebra.Module.Equiv.Defs

Cited by

3 results in Mathlib

Foundations

Depth 20 from the axioms · uses propext

Assumes

SemiringSemiringRingHomInvPairRingHomInvPairAddCommMonoidAddCommMonoidModuleModule