Theorems · Definition · linear algebra

LinearMap.coprodEquiv

{R : Type u} →
  {M : Type v} →
    {M₂ : Type w} →
      {M₃ : Type y} →
        (S : Type u_3) →
          [inst : Semiring R] →
            [inst_1 : Semiring S] →
              [inst_2 : AddCommMonoid M] →
                [inst_3 : AddCommMonoid M₂] →
                  [inst_4 : AddCommMonoid M₃] →
                    [inst_5 : Module R M] →
                      [inst_6 : Module R M₂] →
                        [inst_7 : Module R M₃] →
                          [inst_8 : Module S M₃] →
                            [inst_9 : SMulCommClass R S M₃] → ((M →ₗ[R] M₃) × (M₂ →ₗ[R] M₃)) ≃ₗ[S] M × M₂ →ₗ[R] M₃

Taking the product of two maps with the same codomain is equivalent to taking the product of their domains. See note [bundled maps over different rings] for why separate R and S semirings are used.

Defined in

Mathlib.LinearAlgebra.Prod

Cited by

4 results in Mathlib

Foundations

Depth 33 from the axioms · uses propext, Quot.sound

Assumes

SemiringSemiringAddCommMonoidAddCommMonoidAddCommMonoidModuleModuleModuleModuleSMulCommClass