![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
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 30944 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ}) | |
2 | ssrab2 4064 | . 2 ⊢ {𝑥 ∈ ℂ ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ} ⊆ ℂ | |
3 | 1, 2 | eqsstrdi 4023 | 1 ⊢ (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 {crab 3425 ⊆ wss 3935 I cid 5557 ↾ cres 5662 ⟶wf 6519 –1-1→wf1 6520 ‘cfv 6523 (class class class)co 7384 ℂcc 11080 ℋchba 29965 ·op chot 29985 −op chod 29986 Lambdacspc 30007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5283 ax-nul 5290 ax-pow 5347 ax-pr 5411 ax-un 7699 ax-cnex 11138 ax-hilex 30045 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3426 df-v 3468 df-sbc 3765 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-nul 4310 df-if 4514 df-pw 4589 df-sn 4614 df-pr 4616 df-op 4620 df-uni 4893 df-br 5133 df-opab 5195 df-mpt 5216 df-id 5558 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-iota 6475 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fv 6531 df-ov 7387 df-oprab 7388 df-mpo 7389 df-map 8796 df-spec 30901 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |