Theorem hmoval 31418

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 = (𝑈 adj 𝑈)

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 7430	. . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑢 = 𝑈) → (𝑢 adj 𝑢) = (𝑈 adj 𝑈))
3	2	anidms 577	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 adj 𝑢) = (𝑈 adj 𝑈))
4		hmoval.9	. . . . . 6 ⊢ 𝐴 = (𝑈 adj 𝑈)
5	3, 4	eqtr4di 2813	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑢 adj 𝑢) = 𝐴)
6	5	dmeqd 5886	. . . 4 ⊢ (𝑢 = 𝑈 → dom (𝑢 adj 𝑢) = dom 𝐴)
7	5	fveq1d 6888	. . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑢 adj 𝑢)‘𝑡) = (𝐴‘𝑡))
8	7	eqeq1d 2762	. . . 4 ⊢ (𝑢 = 𝑈 → (((𝑢 adj 𝑢)‘𝑡) = 𝑡 ↔ (𝐴‘𝑡) = 𝑡))
9	6, 8	rabeqbidv 3429	. . 3 ⊢ (𝑢 = 𝑈 → {𝑡 ∈ dom (𝑢 adj 𝑢) ∣ ((𝑢 adj 𝑢)‘𝑡) = 𝑡} = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})
10		df-hmo 31359	. . 3 ⊢ HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢 adj 𝑢) ∣ ((𝑢 adj 𝑢)‘𝑡) = 𝑡})
11		ovex 7454	. . . . . 6 ⊢ (𝑈 adj 𝑈) ∈ V
12	4, 11	eqeltri 2856	. . . . 5 ⊢ 𝐴 ∈ V
13	12	dmex 7922	. . . 4 ⊢ dom 𝐴 ∈ V
14	13	rabex 5299	. . 3 ⊢ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡} ∈ V
15	9, 10, 14	fvmpt 6994	. 2 ⊢ (𝑈 ∈ NrmCVec → (HmOp‘𝑈) = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})
16	1, 15	eqtrid 2807	1 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  {crab 3412  Vcvv 3450  dom cdm 5650  ‘cfv 6539  (class class class)co 7421  NrmCVeccnv 31192   adj caj 31356  HmOpchmo 31357

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6541  df-fv 6547  df-ov 7424  df-hmo 31359

This theorem is used by:  ishmo  31419