Theorems · Theorem · ring theory
Algebra.mul_smul_comm
∀ {R : Type u} {A : Type w} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] (s : R) (x y : A),
x * s • y = s • (x * y)This is just a special case of the global mul_smul_comm lemma that requires less typeclass
search (and was here first).
- Defined in
- Mathlib.Algebra.Algebra.Defs
- Cited by
- 24 results in Mathlib
- Foundations
- Depth 19 from the axioms · uses no axioms
- Assumes
- CommSemiringSemiringAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Algebra.algebraMapproof · cited by 4,706
- Algebra.smul_defproof · cited by 287
- Algebra.left_commproof · cited by 2
Cited by24
Results whose statement or proof uses this declaration.
- AddMonoidAlgebra.scalarTensorEquiv_tmulproof · cited by 8
- PowerSeries.IsWeierstrassFactorizationAt.smulproof · cited by 2
- Algebra.traceMatrix_of_matrix_vecMulproof · cited by 2
- IsStrictlyPositive.smulproof · cited by 1
- HahnSeries.SummableFamily.hsum_powerSeriesFamily_mulproof · cited by 1
- HahnSeries.SummableFamily.support_powerSeriesFamily_subsetproof · cited by 1
- PowerBasis.repr_mul_isIntegralproof · cited by 1
- Matrix.PosDef.smulproof · cited by 1
- PowerSeries.binomialSeries_addproof · cited by 1
- Ideal.FG.of_FG_map_of_faithfullyFlatproof · cited by 1
- FractionalIdeal.mem_singleton_mulproof · cited by 1
- CStarAlgebra.star_left_conjugate_le_norm_smulproof · cited by 1