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Mirrors > Home > HSE Home > Th. List > speccl | Structured version Visualization version GIF version |
Description: The spectrum of an operator is a set of complex numbers. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
speccl | ⊢ (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) ⊆ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | specval 31139 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ}) | |
2 | ssrab2 4077 | . 2 ⊢ {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ⊆ ℂ | |
3 | 1, 2 | eqsstrdi 4036 | 1 ⊢ (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 {crab 3433 ⊆ wss 3948 I cid 5573 ↾ cres 5678 ⟶wf 6537 –1-1→wf1 6538 ‘cfv 6541 (class class class)co 7406 ℂcc 11105 ℋchba 30160 ·op chot 30180 −op chod 30181 Lambdacspc 30202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-hilex 30240 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-map 8819 df-spec 31096 |
This theorem is referenced by: (None) |
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