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Theorem elhmop 31703
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 6901 . . . . . 6 (𝑡 = 𝑇 → (𝑡𝑦) = (𝑇𝑦))
21oveq2d 7442 . . . . 5 (𝑡 = 𝑇 → (𝑥 ·ih (𝑡𝑦)) = (𝑥 ·ih (𝑇𝑦)))
3 fveq1 6901 . . . . . 6 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
43oveq1d 7441 . . . . 5 (𝑡 = 𝑇 → ((𝑡𝑥) ·ih 𝑦) = ((𝑇𝑥) ·ih 𝑦))
52, 4eqeq12d 2744 . . . 4 (𝑡 = 𝑇 → ((𝑥 ·ih (𝑡𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
652ralbidv 3216 . . 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 31674 . . 3 HrmOp = {𝑡 ∈ ( ℋ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑡𝑥) ·ih 𝑦)}
86, 7elrab2 3687 . 2 (𝑇 ∈ HrmOp ↔ (𝑇 ∈ ( ℋ ↑m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
9 ax-hilex 30829 . . . 4 ℋ ∈ V
109, 9elmap 8896 . . 3 (𝑇 ∈ ( ℋ ↑m ℋ) ↔ 𝑇: ℋ⟶ ℋ)
1110anbi1i 622 . 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 274 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 205  wa 394   = wceq 1533  wcel 2098  wral 3058  wf 6549  cfv 6553  (class class class)co 7426  m cmap 8851  chba 30749   ·ih csp 30752  HrmOpcho 30780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-hilex 30829
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-map 8853  df-hmop 31674
This theorem is referenced by:  hmopf  31704  hmop  31752  hmopadj2  31771  idhmop  31812  0hmop  31813  lnophmi  31848  hmops  31850  hmopm  31851  hmopco  31853  pjhmopi  31976
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