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Theorem elunop 29065
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 3361 . 2 (𝑇 ∈ UniOp → 𝑇 ∈ V)
2 fof 6256 . . . 4 (𝑇: ℋ–onto→ ℋ → 𝑇: ℋ⟶ ℋ)
3 ax-hilex 28190 . . . 4 ℋ ∈ V
4 fex 6632 . . . 4 ((𝑇: ℋ⟶ ℋ ∧ ℋ ∈ V) → 𝑇 ∈ V)
52, 3, 4sylancl 566 . . 3 (𝑇: ℋ–onto→ ℋ → 𝑇 ∈ V)
65adantr 466 . 2 ((𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)) → 𝑇 ∈ V)
7 foeq1 6252 . . . 4 (𝑡 = 𝑇 → (𝑡: ℋ–onto→ ℋ ↔ 𝑇: ℋ–onto→ ℋ))
8 fveq1 6331 . . . . . . 7 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
9 fveq1 6331 . . . . . . 7 (𝑡 = 𝑇 → (𝑡𝑦) = (𝑇𝑦))
108, 9oveq12d 6810 . . . . . 6 (𝑡 = 𝑇 → ((𝑡𝑥) ·ih (𝑡𝑦)) = ((𝑇𝑥) ·ih (𝑇𝑦)))
1110eqeq1d 2772 . . . . 5 (𝑡 = 𝑇 → (((𝑡𝑥) ·ih (𝑡𝑦)) = (𝑥 ·ih 𝑦) ↔ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)))
12112ralbidv 3137 . . . 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 608 . . 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 29036 . . 3 UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡𝑥) ·ih (𝑡𝑦)) = (𝑥 ·ih 𝑦))}
1513, 14elab2g 3502 . 2 (𝑇 ∈ V → (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦))))
161, 6, 15pm5.21nii 367 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 382   = wceq 1630  wcel 2144  wral 3060  Vcvv 3349  wf 6027  ontowfo 6029  cfv 6031  (class class class)co 6792  chil 28110   ·ih csp 28113  UniOpcuo 28140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-hilex 28190
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-unop 29036
This theorem is referenced by:  unop  29108  unopf1o  29109  cnvunop  29111  counop  29114  idunop  29171  lnopunii  29205  elunop2  29206
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