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Theorem ishmo 27050
 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)
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 27049 . . 3 (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡})
43eleq2d 2673 . 2 (𝑈 ∈ NrmCVec → (𝑇𝐻𝑇 ∈ {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡}))
5 fveq2 6103 . . . 4 (𝑡 = 𝑇 → (𝐴𝑡) = (𝐴𝑇))
6 id 22 . . . 4 (𝑡 = 𝑇𝑡 = 𝑇)
75, 6eqeq12d 2625 . . 3 (𝑡 = 𝑇 → ((𝐴𝑡) = 𝑡 ↔ (𝐴𝑇) = 𝑇))
87elrab 3331 . 2 (𝑇 ∈ {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡} ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴𝑇) = 𝑇))
94, 8syl6bb 275 1 (𝑈 ∈ NrmCVec → (𝑇𝐻 ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴𝑇) = 𝑇)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  dom cdm 5038  ‘cfv 5804  (class class class)co 6549  NrmCVeccnv 26823  adjcaj 26987  HmOpchmo 26988 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-hmo 26990 This theorem is referenced by: (None)
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