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Mirrors > Home > HSE Home > Th. List > specval | Structured version Visualization version GIF version |
Description: The value of the spectrum of an operator. (Contributed by NM, 11-Apr-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
specval | ⊢ (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 10599 | . . 3 ⊢ ℂ ∈ V | |
2 | 1 | rabex 5216 | . 2 ⊢ {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ∈ V |
3 | ax-hilex 28755 | . 2 ⊢ ℋ ∈ V | |
4 | oveq1 7144 | . . . . 5 ⊢ (𝑡 = 𝑇 → (𝑡 −op (𝑥 ·op ( I ↾ ℋ))) = (𝑇 −op (𝑥 ·op ( I ↾ ℋ)))) | |
5 | f1eq1 6551 | . . . . 5 ⊢ ((𝑡 −op (𝑥 ·op ( I ↾ ℋ))) = (𝑇 −op (𝑥 ·op ( I ↾ ℋ))) → ((𝑡 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ ↔ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ)) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑡 = 𝑇 → ((𝑡 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ ↔ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ)) |
7 | 6 | notbid 320 | . . 3 ⊢ (𝑡 = 𝑇 → (¬ (𝑡 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ ↔ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ)) |
8 | 7 | rabbidv 3467 | . 2 ⊢ (𝑡 = 𝑇 → {𝑥 ∈ ℂ ∣ ¬ (𝑡 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} = {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ}) |
9 | df-spec 29611 | . 2 ⊢ Lambda = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ℂ ∣ ¬ (𝑡 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ}) | |
10 | 2, 3, 3, 8, 9 | fvmptmap 8426 | 1 ⊢ (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1537 {crab 3137 I cid 5440 ↾ cres 5538 ⟶wf 6332 –1-1→wf1 6333 ‘cfv 6336 (class class class)co 7137 ℂcc 10516 ℋchba 28675 ·op chot 28695 −op chod 28696 Lambdacspc 28717 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-hilex 28755 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ral 3138 df-rex 3139 df-rab 3142 df-v 3483 df-sbc 3759 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-op 4555 df-uni 4820 df-br 5048 df-opab 5110 df-mpt 5128 df-id 5441 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fv 6344 df-ov 7140 df-oprab 7141 df-mpo 7142 df-map 8389 df-spec 29611 |
This theorem is referenced by: speccl 29655 |
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