Theorems · Theorem · ring theory
Algebra.commutes
∀ {R : Type u} {A : Type w} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] (r : R) (x : A),
(algebraMap R A) r * x = x * (algebraMap R A) rmul_comm for Algebras when one element is from the base ring.
- Defined in
- Mathlib.Algebra.Algebra.Defs
- Cited by
- 27 results in Mathlib
- Foundations
- Depth 17 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.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.commutes'proof · cited by 1
Cited by28
Results whose statement or proof uses this declaration.
- Algebra.commute_algebraMap_leftproof · cited by 9
- Algebra.mul_sub_algebraMap_commutesproof · cited by 8
- AddMonoidAlgebra.scalarTensorEquiv_tmulproof · cited by 8
- Algebra.commute_algebraMap_rightproof · cited by 5
- QuaternionAlgebra.coe_commutesproof · cited by 3
- Polynomial.scaleRoots_aeval_eq_zeroproof · cited by 3
- Quaternion.exp_eqproof · cited by 3
- CliffordAlgebra.map_mul_map_of_isOrtho_of_mem_evenOddproof · cited by 3
- Algebra.left_commproof · cited by 2
- Submodule.rTensorOne'_tmulproof · cited by 2
- CliffordAlgebra.ι_mul_ι_mul_ιproof · cited by 2
- Polynomial.aevalEquivproof · cited by 2