Theorems · Theorem · ring theory
Algebra.smul_def
∀ {R : Type u} {A : Type w} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] (r : R) (x : A),
r • x = (algebraMap R A) r * x- Defined in
- Mathlib.Algebra.Algebra.Defs
- Cited by
- 287 results in Mathlib
- Foundations
- Depth 17 from the axioms, rests on 134 definitions · 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.coestatement · 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
- RingHomstatement · cited by 10,189
- Algebra.algebraMapstatement · cited by 4,706
- Algebra.smul_def'proof · cited by 3
Cited by287
Results whose statement or proof uses this declaration.
- Algebra.algebraMap_eq_smul_oneproof · cited by 119
- IsScalarTower.of_algebraMap_eqproof · cited by 40
- Algebra.mul_smul_commproof · cited by 24
- RingHom.IsStableUnderBaseChange.mkproof · cited by 16
- IsIntegralClosure.isLocalizationproof · cited by 13
- Algebra.adjoin_eq_spanproof · cited by 13
- Derivation.map_algebraMapproof · cited by 12
- FractionalIdeal.spanSingleton_oneproof · cited by 12
- FractionalIdeal.coeIdeal_span_singletonproof · cited by 10
- Polynomial.aeval_eq_sum_rangeproof · cited by 9
- IsAlgebraic.exists_integral_multipleproof · cited by 8
- RCLike.real_smul_eq_coe_mulproof · cited by 8
Showing the 200 most cited of 287.