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Mirrors > Home > HSE Home > Th. List > elunop | Structured version Visualization version GIF version |
Description: Property defining a unitary Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elunop | ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3458 | . 2 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ V) | |
2 | fof 6465 | . . . 4 ⊢ (𝑇: ℋ–onto→ ℋ → 𝑇: ℋ⟶ ℋ) | |
3 | ax-hilex 28463 | . . . 4 ⊢ ℋ ∈ V | |
4 | fex 6862 | . . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ℋ ∈ V) → 𝑇 ∈ V) | |
5 | 2, 3, 4 | sylancl 586 | . . 3 ⊢ (𝑇: ℋ–onto→ ℋ → 𝑇 ∈ V) |
6 | 5 | adantr 481 | . 2 ⊢ ((𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)) → 𝑇 ∈ V) |
7 | foeq1 6461 | . . . 4 ⊢ (𝑡 = 𝑇 → (𝑡: ℋ–onto→ ℋ ↔ 𝑇: ℋ–onto→ ℋ)) | |
8 | fveq1 6544 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥)) | |
9 | fveq1 6544 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦)) | |
10 | 8, 9 | oveq12d 7041 | . . . . . 6 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = ((𝑇‘𝑥) ·ih (𝑇‘𝑦))) |
11 | 10 | eqeq1d 2799 | . . . . 5 ⊢ (𝑡 = 𝑇 → (((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = (𝑥 ·ih 𝑦) ↔ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦))) |
12 | 11 | 2ralbidv 3168 | . . . 4 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = (𝑥 ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦))) |
13 | 7, 12 | anbi12d 630 | . . 3 ⊢ (𝑡 = 𝑇 → ((𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = (𝑥 ·ih 𝑦)) ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)))) |
14 | df-unop 29307 | . . 3 ⊢ UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = (𝑥 ·ih 𝑦))} | |
15 | 13, 14 | elab2g 3610 | . 2 ⊢ (𝑇 ∈ V → (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)))) |
16 | 1, 6, 15 | pm5.21nii 380 | 1 ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ∀wral 3107 Vcvv 3440 ⟶wf 6228 –onto→wfo 6230 ‘cfv 6232 (class class class)co 7023 ℋchba 28383 ·ih csp 28386 UniOpcuo 28413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pr 5228 ax-hilex 28463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-ov 7026 df-unop 29307 |
This theorem is referenced by: unop 29379 unopf1o 29380 cnvunop 29382 counop 29385 idunop 29442 lnopunii 29476 elunop2 29477 |
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