Theorem lnophm 32341

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 2858	. 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 2858	. . . . . 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 23	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
4		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑇‘𝑥) = (𝑇‘𝑦))
5	3, 4	oveq12d 7432	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ·ih (𝑇‘𝑥)) = (𝑦 ·ih (𝑇‘𝑦)))
6	5	eleq1d 2855	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ ↔ (𝑦 ·ih (𝑇‘𝑦)) ∈ ℝ))
7	6	cbvralvw 3250	. . . . . . 7 ⊢ (∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ ↔ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑇‘𝑦)) ∈ ℝ)
8		fveq1 6884	. . . . . . . . . 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 7430	. . . . . . . . 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 2855	. . . . . . . 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 3195	. . . . . . 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 286	. . . . . 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 643	. . . . 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 2858	. . . . . 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 6884	. . . . . . . . 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 7430	. . . . . . . 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 2855	. . . . . . 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 3195	. . . . . 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 643	. . . . 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 32314	. . . . . 6 ⊢ ( I ↾ ℋ) ∈ LinOp
21		fvresi 7175	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦)
22	21	oveq2d 7430	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑦 ·ih (( I ↾ ℋ)‘𝑦)) = (𝑦 ·ih 𝑦))
23		hiidrcl 31417	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑦 ·ih 𝑦) ∈ ℝ)
24	22, 23	eqeltrd 2870	. . . . . . 7 ⊢ (𝑦 ∈ ℋ → (𝑦 ·ih (( I ↾ ℋ)‘𝑦)) ∈ ℝ)
25	24	rgen 3088	. . . . . 6 ⊢ ∀𝑦 ∈ ℋ (𝑦 ·ih (( I ↾ ℋ)‘𝑦)) ∈ ℝ
26	20, 25	pm3.2i 475	. . . . 5 ⊢ (( I ↾ ℋ) ∈ LinOp ∧ ∀𝑦 ∈ ℋ (𝑦 ·ih (( I ↾ ℋ)‘𝑦)) ∈ ℝ)
27	13, 19, 26	elimhyp 4558	. . . 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 488	. . 3 ⊢ if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) ∈ LinOp
29	27	simpri 490	. . 3 ⊢ ∀𝑦 ∈ ℋ (𝑦 ·ih (if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ))‘𝑦)) ∈ ℝ
30	28, 29	lnophmi 32340	. 2 ⊢ if((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ), 𝑇, ( I ↾ ℋ)) ∈ HrmOp
31	1, 30	dedth 4551	1 ⊢ ((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ) → 𝑇 ∈ HrmOp)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2150  ∀wral 3086  ifcif 4492   I cid 5559   ↾ cres 5667  ‘cfv 6540  (class class class)co 7414  ℝcr 11102   ℋchba 31241   ·_ih csp 31244  LinOpclo 31269  HrmOpcho 31272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736  ax-resscn 11160  ax-1cn 11161  ax-icn 11162  ax-addcl 11163  ax-addrcl 11164  ax-mulcl 11165  ax-mulrcl 11166  ax-mulcom 11167  ax-addass 11168  ax-mulass 11169  ax-distr 11170  ax-i2m1 11171  ax-1ne0 11172  ax-1rid 11173  ax-rnegex 11174  ax-rrecex 11175  ax-cnre 11176  ax-pre-lttri 11177  ax-pre-lttrn 11178  ax-pre-ltadd 11179  ax-pre-mulgt0 11180  ax-hilex 31321  ax-hfvadd 31322  ax-hvcom 31323  ax-hvass 31324  ax-hv0cl 31325  ax-hvaddid 31326  ax-hfvmul 31327  ax-hvmulid 31328  ax-hvmulass 31329  ax-hvdistr1 31330  ax-hvdistr2 31331  ax-hvmul0 31332  ax-hfi 31401  ax-his1 31404  ax-his2 31405  ax-his3 31406  ax-his4 31407

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-nel 3072  df-ral 3087  df-rex 3097  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5560  df-eprel 5565  df-po 5573  df-so 5574  df-fr 5618  df-we 5620  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7866  df-2nd 7990  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-er 8697  df-map 8829  df-en 8947  df-dom 8948  df-sdom 8949  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11446  df-neg 11447  df-div 11875  df-nn 12237  df-2 12306  df-3 12307  df-4 12308  df-cj 15153  df-re 15154  df-im 15155  df-hvsub 31293  df-lnop 32163  df-unop 32165  df-hmop 32166

This theorem is referenced by: (None)