Theorem hmoval 31469

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 7460	. . . . . . 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 2816	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑢 adj 𝑢) = 𝐴)
6	5	dmeqd 5899	. . . 4 ⊢ (𝑢 = 𝑈 → dom (𝑢 adj 𝑢) = dom 𝐴)
7	5	fveq1d 6918	. . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑢 adj 𝑢)‘𝑡) = (𝐴‘𝑡))
8	7	eqeq1d 2765	. . . 4 ⊢ (𝑢 = 𝑈 → (((𝑢 adj 𝑢)‘𝑡) = 𝑡 ↔ (𝐴‘𝑡) = 𝑡))
9	6, 8	rabeqbidv 3433	. . 3 ⊢ (𝑢 = 𝑈 → {𝑡 ∈ dom (𝑢 adj 𝑢) ∣ ((𝑢 adj 𝑢)‘𝑡) = 𝑡} = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})
10		df-hmo 31410	. . 3 ⊢ HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢 adj 𝑢) ∣ ((𝑢 adj 𝑢)‘𝑡) = 𝑡})
11		ovex 7484	. . . . . 6 ⊢ (𝑈 adj 𝑈) ∈ V
12	4, 11	eqeltri 2860	. . . . 5 ⊢ 𝐴 ∈ V
13	12	dmex 7955	. . . 4 ⊢ dom 𝐴 ∈ V
14	13	rabex 5307	. . 3 ⊢ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡} ∈ V
15	9, 10, 14	fvmpt 7024	. 2 ⊢ (𝑈 ∈ NrmCVec → (HmOp‘𝑈) = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})
16	1, 15	eqtrid 2810	1 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454  dom cdm 5658  ‘cfv 6566  (class class class)co 7451  NrmCVeccnv 31243   adj caj 31407  HmOpchmo 31408

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-hmo 31410

This theorem is used by:  ishmo  31470