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Theorem elunop 28701
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 3207 . 2 (𝑇 ∈ UniOp → 𝑇 ∈ V)
2 fof 6102 . . . 4 (𝑇: ℋ–onto→ ℋ → 𝑇: ℋ⟶ ℋ)
3 ax-hilex 27826 . . . 4 ℋ ∈ V
4 fex 6475 . . . 4 ((𝑇: ℋ⟶ ℋ ∧ ℋ ∈ V) → 𝑇 ∈ V)
52, 3, 4sylancl 693 . . 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 6098 . . . 4 (𝑡 = 𝑇 → (𝑡: ℋ–onto→ ℋ ↔ 𝑇: ℋ–onto→ ℋ))
8 fveq1 6177 . . . . . . 7 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
9 fveq1 6177 . . . . . . 7 (𝑡 = 𝑇 → (𝑡𝑦) = (𝑇𝑦))
108, 9oveq12d 6653 . . . . . 6 (𝑡 = 𝑇 → ((𝑡𝑥) ·ih (𝑡𝑦)) = ((𝑇𝑥) ·ih (𝑇𝑦)))
1110eqeq1d 2622 . . . . 5 (𝑡 = 𝑇 → (((𝑡𝑥) ·ih (𝑡𝑦)) = (𝑥 ·ih 𝑦) ↔ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)))
12112ralbidv 2986 . . . 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 746 . . 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 28672 . . 3 UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡𝑥) ·ih (𝑡𝑦)) = (𝑥 ·ih 𝑦))}
1513, 14elab2g 3347 . 2 (𝑇 ∈ V → (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦))))
161, 6, 15pm5.21nii 368 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 196  wa 384   = wceq 1481  wcel 1988  wral 2909  Vcvv 3195  wf 5872  ontowfo 5874  cfv 5876  (class class class)co 6635  chil 27746   ·ih csp 27749  UniOpcuo 27776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-hilex 27826
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-unop 28672
This theorem is referenced by:  unop  28744  unopf1o  28745  cnvunop  28747  counop  28750  idunop  28807  lnopunii  28841  elunop2  28842
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