Theorems · Theorem · functional analysis
spectrum.mem_iff
∀ {R : Type u} {A : Type v} [inst : CommSemiring R] [inst_1 : Ring A] [inst_2 : Algebra R A] {r : R} {a : A},
r ∈ spectrum R a ↔ ¬IsUnit ((algebraMap R A) r - a)- Defined in
- Mathlib.Algebra.Algebra.Spectrum.Basic
- Cited by
- 13 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 · cited by 62,936
- Setstatement · cited by 53,352
- 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 · cited by 4,706
- IsUnitstatement · cited by 1,602
- spectrumstatement · cited by 510
Cited by13
Results whose statement or proof uses this declaration.
- spectrum.zero_mem_iffproof · cited by 4
- spectrum.subset_polynomial_aevalproof · cited by 4
- spectrum.map_polynomial_aeval_of_degree_posproof · cited by 2
- spectrum.exists_mem_of_not_isUnit_aeval_prodproof · cited by 2
- spectrum.units_conjugateproof · cited by 2
- spectrum.resolvent_zero_of_mem_spectrumproof · cited by 2
- Module.End.hasUnifEigenvalue_iff_mem_spectrumproof · cited by 2
- spectrum.exp_mem_expproof · cited by 1
- spectrum.units_smul_resolventproof · cited by 1
- CategoryTheory.finrank_endomorphism_eq_oneproof · cited by 1
- Module.End.HasUnifEigenvalue.mem_spectrumproof · cited by 1
- ContinuousLinearMap.spectrum_eqproof · cited by 1