Theorem hmoval 30746

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 ∣ (𝐴‘𝑡) = 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.

Step	Hyp	Ref	Expression
1		hmoval.8	. 2 ⊢ 𝐻 = (HmOp‘𝑈)
2		oveq12 7435	. . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑢 = 𝑈) → (𝑢adj𝑢) = (𝑈adj𝑈))
3	2	anidms 565	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢adj𝑢) = (𝑈adj𝑈))
4		hmoval.9	. . . . . 6 ⊢ 𝐴 = (𝑈adj𝑈)
5	3, 4	eqtr4di 2784	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑢adj𝑢) = 𝐴)
6	5	dmeqd 5914	. . . 4 ⊢ (𝑢 = 𝑈 → dom (𝑢adj𝑢) = dom 𝐴)
7	5	fveq1d 6905	. . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑢adj𝑢)‘𝑡) = (𝐴‘𝑡))
8	7	eqeq1d 2728	. . . 4 ⊢ (𝑢 = 𝑈 → (((𝑢adj𝑢)‘𝑡) = 𝑡 ↔ (𝐴‘𝑡) = 𝑡))
9	6, 8	rabeqbidv 3437	. . 3 ⊢ (𝑢 = 𝑈 → {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡} = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})
10		df-hmo 30687	. . 3 ⊢ HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡})
11		ovex 7459	. . . . . 6 ⊢ (𝑈adj𝑈) ∈ V
12	4, 11	eqeltri 2822	. . . . 5 ⊢ 𝐴 ∈ V
13	12	dmex 7924	. . . 4 ⊢ dom 𝐴 ∈ V
14	13	rabex 5341	. . 3 ⊢ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡} ∈ V
15	9, 10, 14	fvmpt 7011	. 2 ⊢ (𝑈 ∈ NrmCVec → (HmOp‘𝑈) = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})
16	1, 15	eqtrid 2778	1 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  {crab 3419  Vcvv 3462  dom cdm 5684  ‘cfv 6556  (class class class)co 7426  NrmCVeccnv 30520  adjcaj 30684  HmOpchmo 30685

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 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435  ax-un 7748

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5156  df-opab 5218  df-mpt 5239  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6508  df-fun 6558  df-fv 6564  df-ov 7429  df-hmo 30687

This theorem is referenced by:  ishmo  30747