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Theorem hmoval 27974
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 6822 . . . . . . 7 ((𝑢 = 𝑈𝑢 = 𝑈) → (𝑢adj𝑢) = (𝑈adj𝑈))
32anidms 680 . . . . . 6 (𝑢 = 𝑈 → (𝑢adj𝑢) = (𝑈adj𝑈))
4 hmoval.9 . . . . . 6 𝐴 = (𝑈adj𝑈)
53, 4syl6eqr 2812 . . . . 5 (𝑢 = 𝑈 → (𝑢adj𝑢) = 𝐴)
65dmeqd 5481 . . . 4 (𝑢 = 𝑈 → dom (𝑢adj𝑢) = dom 𝐴)
75fveq1d 6354 . . . . 5 (𝑢 = 𝑈 → ((𝑢adj𝑢)‘𝑡) = (𝐴𝑡))
87eqeq1d 2762 . . . 4 (𝑢 = 𝑈 → (((𝑢adj𝑢)‘𝑡) = 𝑡 ↔ (𝐴𝑡) = 𝑡))
96, 8rabeqbidv 3335 . . 3 (𝑢 = 𝑈 → {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡} = {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡})
10 df-hmo 27915 . . 3 HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡})
11 ovex 6841 . . . . . 6 (𝑈adj𝑈) ∈ V
124, 11eqeltri 2835 . . . . 5 𝐴 ∈ V
1312dmex 7264 . . . 4 dom 𝐴 ∈ V
1413rabex 4964 . . 3 {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡} ∈ V
159, 10, 14fvmpt 6444 . 2 (𝑈 ∈ NrmCVec → (HmOp‘𝑈) = {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡})
161, 15syl5eq 2806 1 (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  dom cdm 5266  cfv 6049  (class class class)co 6813  NrmCVeccnv 27748  adjcaj 27912  HmOpchmo 27913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-hmo 27915
This theorem is referenced by:  ishmo  27975
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