Theorems · Theorem · functional analysis
spectrum.zero_notMem_iff
∀ (R : Type u) {A : Type v} [inst : CommSemiring R] [inst_1 : Ring A] [inst_2 : Algebra R A] {a : A},
0 ∉ spectrum R a ↔ IsUnit a- Defined in
- Mathlib.Algebra.Algebra.Spectrum.Basic
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 20 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.
Cites7
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
- IsUnitstatement and proof · cited by 1,602
- spectrumstatement · cited by 510
- spectrum.zero_mem_iffproof · cited by 4
Cited by5
Results whose statement or proof uses this declaration.
- spectrum.zero_notMemproof · cited by 12
- CStarAlgebra.isUnit_of_leproof · cited by 3
- spectrum.isUnit_of_zero_notMemproof · cited by 2
- isUnit_cfc_iffproof · cited by 2
- StarSubalgebra.coe_isUnitproof · cited by 1