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Theorem elunop 27921
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 3184 . 2 (𝑇 ∈ UniOp → 𝑇 ∈ V)
2 fof 6013 . . . 4 (𝑇: ℋ–onto→ ℋ → 𝑇: ℋ⟶ ℋ)
3 ax-hilex 27046 . . . 4 ℋ ∈ V
4 fex 6372 . . . 4 ((𝑇: ℋ⟶ ℋ ∧ ℋ ∈ V) → 𝑇 ∈ V)
52, 3, 4sylancl 692 . . 3 (𝑇: ℋ–onto→ ℋ → 𝑇 ∈ V)
65adantr 479 . 2 ((𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)) → 𝑇 ∈ V)
7 foeq1 6009 . . . 4 (𝑡 = 𝑇 → (𝑡: ℋ–onto→ ℋ ↔ 𝑇: ℋ–onto→ ℋ))
8 fveq1 6087 . . . . . . 7 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
9 fveq1 6087 . . . . . . 7 (𝑡 = 𝑇 → (𝑡𝑦) = (𝑇𝑦))
108, 9oveq12d 6545 . . . . . 6 (𝑡 = 𝑇 → ((𝑡𝑥) ·ih (𝑡𝑦)) = ((𝑇𝑥) ·ih (𝑇𝑦)))
1110eqeq1d 2611 . . . . 5 (𝑡 = 𝑇 → (((𝑡𝑥) ·ih (𝑡𝑦)) = (𝑥 ·ih 𝑦) ↔ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)))
12112ralbidv 2971 . . . 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 742 . . 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 27892 . . 3 UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡𝑥) ·ih (𝑡𝑦)) = (𝑥 ·ih 𝑦))}
1513, 14elab2g 3321 . 2 (𝑇 ∈ V → (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦))))
161, 6, 15pm5.21nii 366 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 194  wa 382   = wceq 1474  wcel 1976  wral 2895  Vcvv 3172  wf 5786  ontowfo 5788  cfv 5790  (class class class)co 6527  chil 26966   ·ih csp 26969  UniOpcuo 26996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-hilex 27046
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-unop 27892
This theorem is referenced by:  unop  27964  unopf1o  27965  cnvunop  27967  counop  27970  idunop  28027  lnopunii  28061  elunop2  28062
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