Theorems · Theorem · linear algebra

LinearMap.prod_ext

∀ {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [inst : Semiring R] [inst_1 : AddCommMonoid M]
  [inst_2 : AddCommMonoid M₂] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module R M₂]
  [inst_6 : Module R M₃] {f g : M × M₂ →ₗ[R] M₃},
  f ∘ₗ LinearMap.inl R M M₂ = g ∘ₗ LinearMap.inl R M M₂ → f ∘ₗ LinearMap.inr R M M₂ = g ∘ₗ LinearMap.inr R M M₂ → f = g

Split equality of linear maps from a product into linear maps over each component, to allow ext to apply lemmas specific to M →ₗ M₃ and M₂ →ₗ M₃. See note [partially-applied ext lemmas].

Defined in

Mathlib.LinearAlgebra.Prod

Cited by

14 results in Mathlib

Foundations

Depth 35 from the axioms · uses propext, Quot.sound

Assumes

SemiringAddCommMonoidAddCommMonoidAddCommMonoidModuleModuleModule