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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 31926 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ}) | |
2 | ssrab2 4089 | . 2 ⊢ {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ⊆ ℂ | |
3 | 1, 2 | eqsstrdi 4049 | 1 ⊢ (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 {crab 3432 ⊆ wss 3962 I cid 5581 ↾ cres 5690 ⟶wf 6558 –1-1→wf1 6559 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℋchba 30947 ·op chot 30967 −op chod 30968 Lambdacspc 30989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-hilex 31027 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-map 8866 df-spec 31883 |
This theorem is referenced by: (None) |
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