Theorem hmoval 30772

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 7362	. . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑢 = 𝑈) → (𝑢adj𝑢) = (𝑈adj𝑈))
3	2	anidms 566	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢adj𝑢) = (𝑈adj𝑈))
4		hmoval.9	. . . . . 6 ⊢ 𝐴 = (𝑈adj𝑈)
5	3, 4	eqtr4di 2782	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑢adj𝑢) = 𝐴)
6	5	dmeqd 5852	. . . 4 ⊢ (𝑢 = 𝑈 → dom (𝑢adj𝑢) = dom 𝐴)
7	5	fveq1d 6828	. . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑢adj𝑢)‘𝑡) = (𝐴‘𝑡))
8	7	eqeq1d 2731	. . . 4 ⊢ (𝑢 = 𝑈 → (((𝑢adj𝑢)‘𝑡) = 𝑡 ↔ (𝐴‘𝑡) = 𝑡))
9	6, 8	rabeqbidv 3415	. . 3 ⊢ (𝑢 = 𝑈 → {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡} = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})
10		df-hmo 30713	. . 3 ⊢ HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡})
11		ovex 7386	. . . . . 6 ⊢ (𝑈adj𝑈) ∈ V
12	4, 11	eqeltri 2824	. . . . 5 ⊢ 𝐴 ∈ V
13	12	dmex 7849	. . . 4 ⊢ dom 𝐴 ∈ V
14	13	rabex 5281	. . 3 ⊢ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡} ∈ V
15	9, 10, 14	fvmpt 6934	. 2 ⊢ (𝑈 ∈ NrmCVec → (HmOp‘𝑈) = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})
16	1, 15	eqtrid 2776	1 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3396  Vcvv 3438  dom cdm 5623  ‘cfv 6486  (class class class)co 7353  NrmCVeccnv 30546  adjcaj 30710  HmOpchmo 30711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-hmo 30713

This theorem is referenced by:  ishmo  30773