Theorems · Theorem · linear algebra

Submodule.mulLeftMap_eq_mulMap_comp

∀ {R : Type u} {S : Type v} [inst : CommSemiring R] [inst_1 : Semiring S] [inst_2 : Algebra R S] {M : Submodule R S}
  (N : Submodule R S) {ι : Type u_1} [inst_3 : DecidableEq ι] (m : ι → ↥M),
  Submodule.mulLeftMap N m =
    M.mulMap N ∘ₗ
      LinearMap.rTensor (↥N) (Finsupp.linearCombination R m) ∘ₗ ↑(TensorProduct.finsuppScalarLeft R (↥N) ι).symm

Defined in: Mathlib.LinearAlgebra.TensorProduct.Submodule
Cited by: 2 results in Mathlib
Foundations: Depth 97 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommSemiring Semiring Algebra DecidableEq

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Cites24

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coeproof · cited by 62,936
RingHom.idstatement · cited by 18,349
Semiringstatement and proof · cited by 13,802
Algebrastatement and proof · cited by 11,388
CommSemiringstatement and proof · cited by 10,911
LinearMapstatement · cited by 10,215
Submodulestatement and proof · cited by 7,192
Finsuppstatement · cited by 5,255
TensorProductstatement · cited by 2,545
LinearMap.compstatement · cited by 1,642
LinearEquiv.symmstatement · cited by 1,461
one_smulproof · cited by 1,374

Cited by2

Results whose statement or proof uses this declaration.

Submodule.LinearDisjoint.linearIndependent_left_of_flatproof · cited by 2
Submodule.LinearDisjoint.of_basis_left'proof · cited by 1