Theorems · Theorem · commutative algebra

LinearEquiv.eq_comp_toLinearMap_symm

∀ {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_8} {M₂ : Type u_9} {M₃ : Type u_10} [inst : Semiring R₁]
  [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] [inst_3 : AddCommMonoid M₁] [inst_4 : AddCommMonoid M₂]
  [inst_5 : AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃}
  {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} {σ₁₃ : R₁ →+* R₃} {σ₂₃ : R₂ →+* R₃} {re₁₂ : RingHomInvPair σ₁₂ σ₂₁}
  {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} [inst_6 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂}
  [inst_7 : RingHomCompTriple σ₂₁ σ₁₃ σ₂₃] (f : M₂ →ₛₗ[σ₂₃] M₃) (g : M₁ →ₛₗ[σ₁₃] M₃),
  f = g ∘ₛₗ ↑e₁₂.symm ↔ f ∘ₛₗ ↑e₁₂ = g

Defined in

Mathlib.Algebra.Module.Equiv.Defs

Cited by

5 results in Mathlib

Foundations

Depth 28 from the axioms · uses propext, Quot.sound

Assumes

SemiringSemiringSemiringAddCommMonoidAddCommMonoidAddCommMonoidRingHomCompTripleRingHomCompTriple