Theorems · Theorem · functional analysis
spectrum.unit_mem_mul_comm
∀ {R : Type u} {A : Type v} [inst : CommSemiring R] [inst_1 : Ring A] [inst_2 : Algebra R A] {a b : A} {r : Rˣ},
↑r ∈ spectrum R (a * b) ↔ ↑r ∈ spectrum R (b * a)- Defined in
- Mathlib.Algebra.Algebra.Spectrum.Basic
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 22 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringRingAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites25
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Ringstatement and proof · cited by 7,463
- mul_oneproof · cited by 3,885
- one_mulproof · cited by 2,841
- Unitsstatement and proof · cited by 2,804
- Units.valstatement and proof · cited by 1,966
- mul_assocproof · cited by 1,667
- IsUnitproof · cited by 1,602
- sub_eq_add_negproof · cited by 1,023
- neg_mulproof · cited by 654
Cited by3
Results whose statement or proof uses this declaration.
- spectrum.setOfPred_isUnit_inter_mul_commproof · cited by 2
- spectrum.nonzero_mul_commproof · cited by 0
- spectrum.preimage_units_mul_commproof · cited by 0