Theorems · Theorem · ring theory
Algebra.mul_sub_algebraMap_commutes
∀ {R : Type u_1} {A : Type u_2} [inst : CommSemiring R] [inst_1 : Ring A] [inst_2 : Algebra R A] (x : A) (r : R),
x * (x - (algebraMap R A) r) = (x - (algebraMap R A) r) * x- Defined in
- Mathlib.Algebra.Algebra.Basic
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 18 from the axioms · uses no axioms
- Assumes
- CommSemiringRingAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- RingHomstatement · cited by 10,189
- Ringstatement and proof · cited by 7,463
- Algebra.algebraMapstatement and proof · cited by 4,706
- mul_subproof · cited by 201
- sub_mulproof · cited by 170
- Algebra.commutesproof · cited by 27
Cited by8
Results whose statement or proof uses this declaration.
- Module.End.disjoint_genEigenspaceproof · cited by 7
- Module.End.isNilpotent_restrict_genEigenspace_natstatement · cited by 4
- Algebra.mul_sub_algebraMap_pow_commutesproof · cited by 2
- Module.End.apply_eq_of_mem_of_comm_of_isFinitelySemisimple_of_isNilproof · cited by 2
- LinearMap.trace_comp_eq_zero_of_commute_of_trace_restrict_eq_zeroproof · cited by 1
- Module.End.isNilpotent_restrict_maxGenEigenspace_sub_algebraMapstatement and proof · cited by 1
- Module.End.isNilpotent_restrict_sub_algebraMapstatement and proof · cited by 1
- Module.End.isNilpotent_restrict_genEigenspace_topstatement and proof · cited by 0