Theorem hmoval 31191

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 7425	. . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑢 = 𝑈) → (𝑢adj𝑢) = (𝑈adj𝑈))
3	2	anidms 577	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢adj𝑢) = (𝑈adj𝑈))
4		hmoval.9	. . . . . 6 ⊢ 𝐴 = (𝑈adj𝑈)
5	3, 4	eqtr4di 2818	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑢adj𝑢) = 𝐴)
6	5	dmeqd 5897	. . . 4 ⊢ (𝑢 = 𝑈 → dom (𝑢adj𝑢) = dom 𝐴)
7	5	fveq1d 6887	. . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑢adj𝑢)‘𝑡) = (𝐴‘𝑡))
8	7	eqeq1d 2767	. . . 4 ⊢ (𝑢 = 𝑈 → (((𝑢adj𝑢)‘𝑡) = 𝑡 ↔ (𝐴‘𝑡) = 𝑡))
9	6, 8	rabeqbidv 3436	. . 3 ⊢ (𝑢 = 𝑈 → {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡} = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})
10		df-hmo 31132	. . 3 ⊢ HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡})
11		ovex 7449	. . . . . 6 ⊢ (𝑈adj𝑈) ∈ V
12	4, 11	eqeltri 2861	. . . . 5 ⊢ 𝐴 ∈ V
13	12	dmex 7908	. . . 4 ⊢ dom 𝐴 ∈ V
14	13	rabex 5311	. . 3 ⊢ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡} ∈ V
15	9, 10, 14	fvmpt 6993	. 2 ⊢ (𝑈 ∈ NrmCVec → (HmOp‘𝑈) = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})
16	1, 15	eqtrid 2812	1 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  {crab 3418  Vcvv 3457  dom cdm 5663  ‘cfv 6540  (class class class)co 7416  NrmCVeccnv 30965  adjcaj 31129  HmOpchmo 31130

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7419  df-hmo 31132

This theorem is used by:  ishmo  31192