Theorem lnophm 32038

Description: A linear operator is Hermitian if 𝑥 ·_ih (𝑇‘𝑥) takes only real values. Remark in [ReedSimon] p. 195. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
lnophm	⊢ ((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ) → 𝑇 ∈ HrmOp)

Distinct variable group:   𝑥,𝑇

Proof of Theorem lnophm

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2829	. 2 ⊢ (𝑇 = if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) → (𝑇 ∈ HrmOp ↔ if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) ∈ HrmOp))
2		eleq1 2829	. . . . . 6 ⊢ (𝑇 = if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) → (𝑇 ∈ LinOp ↔ if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) ∈ LinOp))
3		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
4		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑇‘𝑥) = (𝑇‘𝑦))
5	3, 4	oveq12d 7449	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ·ih (𝑇‘𝑥)) = (𝑦 ·ih (𝑇‘𝑦)))
6	5	eleq1d 2826	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ ↔ (𝑦 ·ih (𝑇‘𝑦)) ∈ ℝ))
7	6	cbvralvw 3237	. . . . . . 7 ⊢ (∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ ↔ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑇‘𝑦)) ∈ ℝ)
8		fveq1 6905	. . . . . . . . . 10 ⊢ (𝑇 = if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) → (𝑇‘𝑦) = (if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ))‘𝑦))
9	8	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑇 = if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) → (𝑦 ·ih (𝑇‘𝑦)) = (𝑦 ·ih (if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ))‘𝑦)))
10	9	eleq1d 2826	. . . . . . . 8 ⊢ (𝑇 = if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) → ((𝑦 ·ih (𝑇‘𝑦)) ∈ ℝ ↔ (𝑦 ·ih (if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ))‘𝑦)) ∈ ℝ))
11	10	ralbidv 3178	. . . . . . 7 ⊢ (𝑇 = if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) → (∀𝑦 ∈ ℋ (𝑦 ·ih (𝑇‘𝑦)) ∈ ℝ ↔ ∀𝑦 ∈ ℋ (𝑦 ·ih (if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ))‘𝑦)) ∈ ℝ))
12	7, 11	bitrid 283	. . . . . 6 ⊢ (𝑇 = if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) → (∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ ↔ ∀𝑦 ∈ ℋ (𝑦 ·ih (if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ))‘𝑦)) ∈ ℝ))
13	2, 12	anbi12d 632	. . . . 5 ⊢ (𝑇 = if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) → ((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ) ↔ (if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) ∈ LinOp ∧ ∀𝑦 ∈ ℋ (𝑦 ·ih (if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ))‘𝑦)) ∈ ℝ)))
14		eleq1 2829	. . . . . 6 ⊢ (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) → (( I ↾ ℋ) ∈ LinOp ↔ if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) ∈ LinOp))
15		fveq1 6905	. . . . . . . . 9 ⊢ (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) → (( I ↾ ℋ)‘𝑦) = (if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ))‘𝑦))
16	15	oveq2d 7447	. . . . . . . 8 ⊢ (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) → (𝑦 ·ih (( I ↾ ℋ)‘𝑦)) = (𝑦 ·ih (if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ))‘𝑦)))
17	16	eleq1d 2826	. . . . . . 7 ⊢ (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) → ((𝑦 ·ih (( I ↾ ℋ)‘𝑦)) ∈ ℝ ↔ (𝑦 ·ih (if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ))‘𝑦)) ∈ ℝ))
18	17	ralbidv 3178	. . . . . 6 ⊢ (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) → (∀𝑦 ∈ ℋ (𝑦 ·ih (( I ↾ ℋ)‘𝑦)) ∈ ℝ ↔ ∀𝑦 ∈ ℋ (𝑦 ·ih (if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ))‘𝑦)) ∈ ℝ))
19	14, 18	anbi12d 632	. . . . 5 ⊢ (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) → ((( I ↾ ℋ) ∈ LinOp ∧ ∀𝑦 ∈ ℋ (𝑦 ·ih (( I ↾ ℋ)‘𝑦)) ∈ ℝ) ↔ (if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) ∈ LinOp ∧ ∀𝑦 ∈ ℋ (𝑦 ·ih (if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ))‘𝑦)) ∈ ℝ)))
20		idlnop 32011	. . . . . 6 ⊢ ( I ↾ ℋ) ∈ LinOp
21		fvresi 7193	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦)
22	21	oveq2d 7447	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑦 ·ih (( I ↾ ℋ)‘𝑦)) = (𝑦 ·ih 𝑦))
23		hiidrcl 31114	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑦 ·ih 𝑦) ∈ ℝ)
24	22, 23	eqeltrd 2841	. . . . . . 7 ⊢ (𝑦 ∈ ℋ → (𝑦 ·ih (( I ↾ ℋ)‘𝑦)) ∈ ℝ)
25	24	rgen 3063	. . . . . 6 ⊢ ∀𝑦 ∈ ℋ (𝑦 ·ih (( I ↾ ℋ)‘𝑦)) ∈ ℝ
26	20, 25	pm3.2i 470	. . . . 5 ⊢ (( I ↾ ℋ) ∈ LinOp ∧ ∀𝑦 ∈ ℋ (𝑦 ·ih (( I ↾ ℋ)‘𝑦)) ∈ ℝ)
27	13, 19, 26	elimhyp 4591	. . . 4 ⊢ (if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) ∈ LinOp ∧ ∀𝑦 ∈ ℋ (𝑦 ·ih (if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ))‘𝑦)) ∈ ℝ)
28	27	simpli 483	. . 3 ⊢ if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) ∈ LinOp
29	27	simpri 485	. . 3 ⊢ ∀𝑦 ∈ ℋ (𝑦 ·ih (if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ))‘𝑦)) ∈ ℝ
30	28, 29	lnophmi 32037	. 2 ⊢ if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) ∈ HrmOp
31	1, 30	dedth 4584	1 ⊢ ((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ) → 𝑇 ∈ HrmOp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ifcif 4525   I cid 5577   ↾ cres 5687  ‘cfv 6561  (class class class)co 7431  ℝcr 11154   ℋchba 30938   ·_ih csp 30941  LinOpclo 30966  HrmOpcho 30969

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-hilex 31018  ax-hfvadd 31019  ax-hvcom 31020  ax-hvass 31021  ax-hv0cl 31022  ax-hvaddid 31023  ax-hfvmul 31024  ax-hvmulid 31025  ax-hvmulass 31026  ax-hvdistr1 31027  ax-hvdistr2 31028  ax-hvmul0 31029  ax-hfi 31098  ax-his1 31101  ax-his2 31102  ax-his3 31103  ax-his4 31104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-2 12329  df-3 12330  df-4 12331  df-cj 15138  df-re 15139  df-im 15140  df-hvsub 30990  df-lnop 31860  df-unop 31862  df-hmop 31863

This theorem is referenced by: (None)