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Theorem elunop 29336
Description: Property defining a unitary Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
Assertion
Ref Expression
elunop (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)))
Distinct variable group:   𝑥,𝑦,𝑇

Proof of Theorem elunop
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef 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)
52, 3, 4sylancl 586 . . 3 (𝑇: ℋ–onto→ ℋ → 𝑇 ∈ V)
65adantr 481 . 2 ((𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (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 → (𝑡𝑥) = (𝑇𝑥))
9 fveq1 6544 . . . . . . 7 (𝑡 = 𝑇 → (𝑡𝑦) = (𝑇𝑦))
108, 9oveq12d 7041 . . . . . 6 (𝑡 = 𝑇 → ((𝑡𝑥) ·ih (𝑡𝑦)) = ((𝑇𝑥) ·ih (𝑇𝑦)))
1110eqeq1d 2799 . . . . 5 (𝑡 = 𝑇 → (((𝑡𝑥) ·ih (𝑡𝑦)) = (𝑥 ·ih 𝑦) ↔ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)))
12112ralbidv 3168 . . . 4 (𝑡 = 𝑇 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡𝑥) ·ih (𝑡𝑦)) = (𝑥 ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)))
137, 12anbi12d 630 . . 3 (𝑡 = 𝑇 → ((𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡𝑥) ·ih (𝑡𝑦)) = (𝑥 ·ih 𝑦)) ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦))))
14 df-unop 29307 . . 3 UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡𝑥) ·ih (𝑡𝑦)) = (𝑥 ·ih 𝑦))}
1513, 14elab2g 3610 . 2 (𝑇 ∈ V → (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦))))
161, 6, 15pm5.21nii 380 1 (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (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  ontowfo 6230  cfv 6232  (class class class)co 7023  chba 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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