Theorems · Theorem · functional analysis
spectrum.zero_mem_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
- 4 results in Mathlib
- Foundations
- Depth 19 from the axioms · uses propext
- Assumes
- CommSemiringRingAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- Algebra.algebraMapproof · cited by 4,706
- map_zeroproof · cited by 1,614
- IsUnitstatement and proof · cited by 1,602
- spectrumstatement · cited by 510
- zero_subproof · cited by 335
- spectrum.mem_iffproof · cited by 13
- IsUnit.neg_iffproof · cited by 7
Cited by4
Results whose statement or proof uses this declaration.
- spectrum.zero_notMem_iffproof · cited by 5
- Unitization.zero_mem_spectrum_inrproof · cited by 2
- spectrum.zero_memproof · cited by 1
- spectrum.not_isUnit_of_zero_memproof · cited by 0