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Theorem hmoval 28514
Description: The set of Hermitian (self-adjoint) operators on a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
hmoval.8 𝐻 = (HmOp‘𝑈)
hmoval.9 𝐴 = (𝑈adj𝑈)
Assertion
Ref Expression
hmoval (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡})
Distinct variable groups:   𝑡,𝐴   𝑡,𝑈
Allowed substitution hint:   𝐻(𝑡)

Proof of Theorem hmoval
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 hmoval.8 . 2 𝐻 = (HmOp‘𝑈)
2 oveq12 7154 . . . . . . 7 ((𝑢 = 𝑈𝑢 = 𝑈) → (𝑢adj𝑢) = (𝑈adj𝑈))
32anidms 567 . . . . . 6 (𝑢 = 𝑈 → (𝑢adj𝑢) = (𝑈adj𝑈))
4 hmoval.9 . . . . . 6 𝐴 = (𝑈adj𝑈)
53, 4syl6eqr 2871 . . . . 5 (𝑢 = 𝑈 → (𝑢adj𝑢) = 𝐴)
65dmeqd 5767 . . . 4 (𝑢 = 𝑈 → dom (𝑢adj𝑢) = dom 𝐴)
75fveq1d 6665 . . . . 5 (𝑢 = 𝑈 → ((𝑢adj𝑢)‘𝑡) = (𝐴𝑡))
87eqeq1d 2820 . . . 4 (𝑢 = 𝑈 → (((𝑢adj𝑢)‘𝑡) = 𝑡 ↔ (𝐴𝑡) = 𝑡))
96, 8rabeqbidv 3483 . . 3 (𝑢 = 𝑈 → {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡} = {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡})
10 df-hmo 28455 . . 3 HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡})
11 ovex 7178 . . . . . 6 (𝑈adj𝑈) ∈ V
124, 11eqeltri 2906 . . . . 5 𝐴 ∈ V
1312dmex 7605 . . . 4 dom 𝐴 ∈ V
1413rabex 5226 . . 3 {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡} ∈ V
159, 10, 14fvmpt 6761 . 2 (𝑈 ∈ NrmCVec → (HmOp‘𝑈) = {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡})
161, 15syl5eq 2865 1 (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  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:  ishmo  28515
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