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Theorem elhmop 29058
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 6405 . . . . . 6 (𝑡 = 𝑇 → (𝑡𝑦) = (𝑇𝑦))
21oveq2d 6888 . . . . 5 (𝑡 = 𝑇 → (𝑥 ·ih (𝑡𝑦)) = (𝑥 ·ih (𝑇𝑦)))
3 fveq1 6405 . . . . . 6 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
43oveq1d 6887 . . . . 5 (𝑡 = 𝑇 → ((𝑡𝑥) ·ih 𝑦) = ((𝑇𝑥) ·ih 𝑦))
52, 4eqeq12d 2819 . . . 4 (𝑡 = 𝑇 → ((𝑥 ·ih (𝑡𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
652ralbidv 3175 . . 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 29029 . . 3 HrmOp = {𝑡 ∈ ( ℋ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑡𝑥) ·ih 𝑦)}
86, 7elrab2 3560 . 2 (𝑇 ∈ HrmOp ↔ (𝑇 ∈ ( ℋ ↑𝑚 ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
9 ax-hilex 28182 . . . 4 ℋ ∈ V
109, 9elmap 8119 . . 3 (𝑇 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ ℋ)
1110anbi1i 612 . 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 266 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 197  wa 384   = wceq 1637  wcel 2156  wral 3094  wf 6095  cfv 6099  (class class class)co 6872  𝑚 cmap 8090  chil 28102   ·ih csp 28105  HrmOpcho 28133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5094  ax-un 7177  ax-hilex 28182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-ral 3099  df-rex 3100  df-rab 3103  df-v 3391  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4115  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4843  df-opab 4905  df-id 5217  df-xp 5315  df-rel 5316  df-cnv 5317  df-co 5318  df-dm 5319  df-rn 5320  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-fv 6107  df-ov 6875  df-oprab 6876  df-mpt2 6877  df-map 8092  df-hmop 29029
This theorem is referenced by:  hmopf  29059  hmop  29107  hmopadj2  29126  idhmop  29167  0hmop  29168  lnophmi  29203  hmops  29205  hmopm  29206  hmopco  29208  pjhmopi  29331
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