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Theorem ishmo 28515
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 28514 . . 3 (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡})
43eleq2d 2895 . 2 (𝑈 ∈ NrmCVec → (𝑇𝐻𝑇 ∈ {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡}))
5 fveq2 6663 . . . 4 (𝑡 = 𝑇 → (𝐴𝑡) = (𝐴𝑇))
6 id 22 . . . 4 (𝑡 = 𝑇𝑡 = 𝑇)
75, 6eqeq12d 2834 . . 3 (𝑡 = 𝑇 → ((𝐴𝑡) = 𝑡 ↔ (𝐴𝑇) = 𝑇))
87elrab 3677 . 2 (𝑇 ∈ {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡} ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴𝑇) = 𝑇))
94, 8syl6bb 288 1 (𝑈 ∈ NrmCVec → (𝑇𝐻 ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴𝑇) = 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  {crab 3139  dom cdm 5548  cfv 6348  (class class class)co 7145  NrmCVeccnv 28288  adjcaj 28452  HmOpchmo 28453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-hmo 28455
This theorem is referenced by: (None)
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