Theorems · Theorem · commutative algebra

DirectSum.coe_decomposeTensor_apply

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
  (ℳ : ι → Submodule R M) (N : Type u_5) [inst_3 : AddCommMonoid N] [inst_4 : Module R N] [inst_5 : DecidableEq ι]
  [inst_6 : DirectSum.Decomposition ℳ] (x : DirectSum ι fun i => ↥(DirectSum.decomposeTensor ℳ N i)),
  (DirectSum.coeAddMonoidHom (DirectSum.decomposeTensor ℳ N)) x =
    (LinearEquiv.rTensor N (DirectSum.decomposeLinearEquiv ℳ).symm)
      ((TensorProduct.directSumLeft R R (fun i => ↥(ℳ i)) N).symm
        ((DirectSum.congrLinearEquiv (DirectSum.decomposeTensorEquiv ℳ N)).symm x))

Defined in

Mathlib.LinearAlgebra.TensorProduct.Decomposition

Cited by

0 results in Mathlib

Foundations

Depth 99 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidModuleAddCommMonoidModuleDecidableEqDirectSum.Decomposition