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Theorem ishmo 30620
Description: The predicate "is a hermitian operator." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
hmoval.8 𝐻 = (HmOp‘𝑈)
hmoval.9 𝐴 = (𝑈adj𝑈)
Assertion
Ref Expression
ishmo (𝑈 ∈ NrmCVec → (𝑇𝐻 ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴𝑇) = 𝑇)))

Proof of Theorem ishmo
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 hmoval.8 . . . 4 𝐻 = (HmOp‘𝑈)
2 hmoval.9 . . . 4 𝐴 = (𝑈adj𝑈)
31, 2hmoval 30619 . . 3 (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡})
43eleq2d 2815 . 2 (𝑈 ∈ NrmCVec → (𝑇𝐻𝑇 ∈ {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡}))
5 fveq2 6897 . . . 4 (𝑡 = 𝑇 → (𝐴𝑡) = (𝐴𝑇))
6 id 22 . . . 4 (𝑡 = 𝑇𝑡 = 𝑇)
75, 6eqeq12d 2744 . . 3 (𝑡 = 𝑇 → ((𝐴𝑡) = 𝑡 ↔ (𝐴𝑇) = 𝑇))
87elrab 3682 . 2 (𝑇 ∈ {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡} ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴𝑇) = 𝑇))
94, 8bitrdi 287 1 (𝑈 ∈ NrmCVec → (𝑇𝐻 ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴𝑇) = 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  {crab 3429  dom cdm 5678  cfv 6548  (class class class)co 7420  NrmCVeccnv 30393  adjcaj 30557  HmOpchmo 30558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-hmo 30560
This theorem is referenced by: (None)
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