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Theorem elhmop 28578
Description: Property defining a Hermitian Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
elhmop (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
Distinct variable group:   𝑥,𝑦,𝑇

Proof of Theorem elhmop
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 fveq1 6147 . . . . . 6 (𝑡 = 𝑇 → (𝑡𝑦) = (𝑇𝑦))
21oveq2d 6620 . . . . 5 (𝑡 = 𝑇 → (𝑥 ·ih (𝑡𝑦)) = (𝑥 ·ih (𝑇𝑦)))
3 fveq1 6147 . . . . . 6 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
43oveq1d 6619 . . . . 5 (𝑡 = 𝑇 → ((𝑡𝑥) ·ih 𝑦) = ((𝑇𝑥) ·ih 𝑦))
52, 4eqeq12d 2636 . . . 4 (𝑡 = 𝑇 → ((𝑥 ·ih (𝑡𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
652ralbidv 2983 . . 3 (𝑡 = 𝑇 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
7 df-hmop 28549 . . 3 HrmOp = {𝑡 ∈ ( ℋ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑡𝑥) ·ih 𝑦)}
86, 7elrab2 3348 . 2 (𝑇 ∈ HrmOp ↔ (𝑇 ∈ ( ℋ ↑𝑚 ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
9 ax-hilex 27702 . . . 4 ℋ ∈ V
109, 9elmap 7830 . . 3 (𝑇 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ ℋ)
1110anbi1i 730 . 2 ((𝑇 ∈ ( ℋ ↑𝑚 ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
128, 11bitri 264 1 (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802  chil 27622   ·ih csp 27625  HrmOpcho 27653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-hilex 27702
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-hmop 28549
This theorem is referenced by:  hmopf  28579  hmop  28627  hmopadj2  28646  idhmop  28687  0hmop  28688  lnophmi  28723  hmops  28725  hmopm  28726  hmopco  28728  pjhmopi  28851
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