cnlnssadj - Hilbert Space Explorer

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Theorem cnlnssadj 32285

Description: Every continuous linear Hilbert space operator has an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
cnlnssadj	⊢ (LinOp ∩ ContOp) ⊆ dom adj_ℎ

Proof of Theorem cnlnssadj Dummy variables 𝑢 𝑡 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnlnadj 32284	. . . . 5 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧)))
2		df-rex 3089	. . . . 5 ⊢ (∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧)) ↔ ∃𝑡(𝑡 ∈ (LinOp ∩ ContOp) ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧))))
3	1, 2	sylib 220	. . . 4 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ∃𝑡(𝑡 ∈ (LinOp ∩ ContOp) ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧))))
4		inss1 4190	. . . . . . . . . 10 ⊢ (LinOp ∩ ContOp) ⊆ LinOp
5	4	sseli 3934	. . . . . . . . 9 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → 𝑦 ∈ LinOp)
6		lnopf 32064	. . . . . . . . 9 ⊢ (𝑦 ∈ LinOp → 𝑦: ℋ⟶ ℋ)
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → 𝑦: ℋ⟶ ℋ)
8	7	a1d 25	. . . . . . 7 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ((𝑡 ∈ (LinOp ∩ ContOp) ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧))) → 𝑦: ℋ⟶ ℋ))
9	4	sseli 3934	. . . . . . . . . 10 ⊢ (𝑡 ∈ (LinOp ∩ ContOp) → 𝑡 ∈ LinOp)
10		lnopf 32064	. . . . . . . . . 10 ⊢ (𝑡 ∈ LinOp → 𝑡: ℋ⟶ ℋ)
11	9, 10	syl 17	. . . . . . . . 9 ⊢ (𝑡 ∈ (LinOp ∩ ContOp) → 𝑡: ℋ⟶ ℋ)
12	11	a1i 11	. . . . . . . 8 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → (𝑡 ∈ (LinOp ∩ ContOp) → 𝑡: ℋ⟶ ℋ))
13	12	adantrd 495	. . . . . . 7 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ((𝑡 ∈ (LinOp ∩ ContOp) ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧))) → 𝑡: ℋ⟶ ℋ))
14		eqcom 2771	. . . . . . . . . . 11 ⊢ (((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧)) ↔ (𝑥 ·_ih (𝑡‘𝑧)) = ((𝑦‘𝑥) ·_ih 𝑧))
15	14	biimpi 218	. . . . . . . . . 10 ⊢ (((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧)) → (𝑥 ·_ih (𝑡‘𝑧)) = ((𝑦‘𝑥) ·_ih 𝑧))
16	15	2ralimi 3134	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧)) → ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑡‘𝑧)) = ((𝑦‘𝑥) ·_ih 𝑧))
17		adjsym 32038	. . . . . . . . . 10 ⊢ ((𝑡: ℋ⟶ ℋ ∧ 𝑦: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑡‘𝑧)) = ((𝑦‘𝑥) ·_ih 𝑧) ↔ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑡‘𝑥) ·_ih 𝑧)))
18	11, 7, 17	syl2anr 606	. . . . . . . . 9 ⊢ ((𝑦 ∈ (LinOp ∩ ContOp) ∧ 𝑡 ∈ (LinOp ∩ ContOp)) → (∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑡‘𝑧)) = ((𝑦‘𝑥) ·_ih 𝑧) ↔ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑡‘𝑥) ·_ih 𝑧)))
19	16, 18	imbitrid 246	. . . . . . . 8 ⊢ ((𝑦 ∈ (LinOp ∩ ContOp) ∧ 𝑡 ∈ (LinOp ∩ ContOp)) → (∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧)) → ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑡‘𝑥) ·_ih 𝑧)))
20	19	expimpd 457	. . . . . . 7 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ((𝑡 ∈ (LinOp ∩ ContOp) ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧))) → ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑡‘𝑥) ·_ih 𝑧)))
21	8, 13, 20	3jcad 1143	. . . . . 6 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ((𝑡 ∈ (LinOp ∩ ContOp) ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧))) → (𝑦: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑡‘𝑥) ·_ih 𝑧))))
22		dfadj2 32090	. . . . . . . 8 ⊢ adj_ℎ = {⟨𝑢, 𝑣⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧))}
23	22	eleq2i 2856	. . . . . . 7 ⊢ (⟨𝑦, 𝑡⟩ ∈ adj_ℎ ↔ ⟨𝑦, 𝑡⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧))})
24		vex 3460	. . . . . . . 8 ⊢ 𝑦 ∈ V
25		vex 3460	. . . . . . . 8 ⊢ 𝑡 ∈ V
26		feq1 6671	. . . . . . . . 9 ⊢ (𝑢 = 𝑦 → (𝑢: ℋ⟶ ℋ ↔ 𝑦: ℋ⟶ ℋ))
27		fveq1 6868	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑦 → (𝑢‘𝑧) = (𝑦‘𝑧))
28	27	oveq2d 7414	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑦 → (𝑥 ·_ih (𝑢‘𝑧)) = (𝑥 ·_ih (𝑦‘𝑧)))
29	28	eqeq1d 2766	. . . . . . . . . 10 ⊢ (𝑢 = 𝑦 → ((𝑥 ·_ih (𝑢‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧) ↔ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧)))
30	29	2ralbidv 3228	. . . . . . . . 9 ⊢ (𝑢 = 𝑦 → (∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧) ↔ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧)))
31	26, 30	3anbi13d 1461	. . . . . . . 8 ⊢ (𝑢 = 𝑦 → ((𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧)) ↔ (𝑦: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧))))
32		feq1 6671	. . . . . . . . 9 ⊢ (𝑣 = 𝑡 → (𝑣: ℋ⟶ ℋ ↔ 𝑡: ℋ⟶ ℋ))
33		fveq1 6868	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑡 → (𝑣‘𝑥) = (𝑡‘𝑥))
34	33	oveq1d 7413	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑡 → ((𝑣‘𝑥) ·_ih 𝑧) = ((𝑡‘𝑥) ·_ih 𝑧))
35	34	eqeq2d 2775	. . . . . . . . . 10 ⊢ (𝑣 = 𝑡 → ((𝑥 ·_ih (𝑦‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧) ↔ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑡‘𝑥) ·_ih 𝑧)))
36	35	2ralbidv 3228	. . . . . . . . 9 ⊢ (𝑣 = 𝑡 → (∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧) ↔ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑡‘𝑥) ·_ih 𝑧)))
37	32, 36	3anbi23d 1462	. . . . . . . 8 ⊢ (𝑣 = 𝑡 → ((𝑦: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧)) ↔ (𝑦: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑡‘𝑥) ·_ih 𝑧))))
38	24, 25, 31, 37	opelopab 5515	. . . . . . 7 ⊢ (⟨𝑦, 𝑡⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧))} ↔ (𝑦: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑡‘𝑥) ·_ih 𝑧)))
39	23, 38	bitr2i 278	. . . . . 6 ⊢ ((𝑦: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑡‘𝑥) ·_ih 𝑧)) ↔ ⟨𝑦, 𝑡⟩ ∈ adj_ℎ)
40	21, 39	imbitrdi 253	. . . . 5 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ((𝑡 ∈ (LinOp ∩ ContOp) ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧))) → ⟨𝑦, 𝑡⟩ ∈ adj_ℎ))
41	40	eximdv 1939	. . . 4 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → (∃𝑡(𝑡 ∈ (LinOp ∩ ContOp) ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧))) → ∃𝑡⟨𝑦, 𝑡⟩ ∈ adj_ℎ))
42	3, 41	mpd 15	. . 3 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ∃𝑡⟨𝑦, 𝑡⟩ ∈ adj_ℎ)
43	24	eldm2 5879	. . 3 ⊢ (𝑦 ∈ dom adj_ℎ ↔ ∃𝑡⟨𝑦, 𝑡⟩ ∈ adj_ℎ)
44	42, 43	sylibr 236	. 2 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → 𝑦 ∈ dom adj_ℎ)
45	44	ssriv 3942	1 ⊢ (LinOp ∩ ContOp) ⊆ dom adj_ℎ

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1099 = wceq 1562 ∃wex 1801 ∈ wcel 2144 ∀wral 3078 ∃wrex 3088 ∩ cin 3905 ⊆ wss 3906 ⟨cop 4590 {copab 5164 dom cdm 5649 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 ℋchba 31124 ·_ih csp 31127 ContOpccop 31151 LinOpclo 31152 adj_ℎcado 31160

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-inf2 9598 ax-cc 10394 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155 ax-hilex 31204 ax-hfvadd 31205 ax-hvcom 31206 ax-hvass 31207 ax-hv0cl 31208 ax-hvaddid 31209 ax-hfvmul 31210 ax-hvmulid 31211 ax-hvmulass 31212 ax-hvdistr1 31213 ax-hvdistr2 31214 ax-hvmul0 31215 ax-hfi 31284 ax-his1 31287 ax-his2 31288 ax-his3 31289 ax-his4 31290 ax-hcompl 31407

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-of 7662 df-om 7849 df-1st 7972 df-2nd 7973 df-supp 8143 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-oadd 8443 df-omul 8444 df-er 8680 df-map 8812 df-pm 8813 df-ixp 8882 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9899 df-acn 9902 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-z 12571 df-dec 12691 df-uz 12842 df-q 12952 df-rp 12996 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13355 df-ico 13357 df-icc 13358 df-fz 13515 df-fzo 13662 df-fl 13804 df-seq 14017 df-exp 14077 df-hash 14346 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-clim 15517 df-rlim 15518 df-sum 15716 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-mulr 17302 df-starv 17303 df-sca 17304 df-vsca 17305 df-ip 17306 df-tset 17307 df-ple 17308 df-ds 17310 df-unif 17311 df-hom 17312 df-cco 17313 df-rest 17453 df-topn 17454 df-0g 17472 df-gsum 17473 df-topgen 17474 df-pt 17475 df-prds 17478 df-xrs 17534 df-qtop 17539 df-imas 17540 df-xps 17542 df-mre 17616 df-mrc 17617 df-acs 17619 df-mgm 18676 df-sgrp 18755 df-mnd 18771 df-submnd 18820 df-mulg 19112 df-cntz 19359 df-cmn 19824 df-psmet 21418 df-xmet 21419 df-met 21420 df-bl 21421 df-mopn 21422 df-fbas 21423 df-fg 21424 df-cnfld 21427 df-top 22956 df-topon 22973 df-topsp 22995 df-bases 23008 df-cld 23081 df-ntr 23082 df-cls 23083 df-nei 23160 df-cn 23289 df-cnp 23290 df-lm 23291 df-t1 23376 df-haus 23377 df-tx 23624 df-hmeo 23817 df-fil 23908 df-fm 24000 df-flim 24001 df-flf 24002 df-xms 24382 df-ms 24383 df-tms 24384 df-cfil 25319 df-cau 25320 df-cmet 25321 df-grpo 30698 df-gid 30699 df-ginv 30700 df-gdiv 30701 df-ablo 30750 df-vc 30764 df-nv 30797 df-va 30800 df-ba 30801 df-sm 30802 df-0v 30803 df-vs 30804 df-nmcv 30805 df-ims 30806 df-dip 30906 df-ssp 30927 df-ph 31018 df-cbn 31068 df-hnorm 31173 df-hba 31174 df-hvsub 31176 df-hlim 31177 df-hcau 31178 df-sh 31412 df-ch 31426 df-oc 31457 df-ch0 31458 df-shs 31513 df-pjh 31600 df-h0op 31953 df-nmop 32044 df-cnop 32045 df-lnop 32046 df-unop 32048 df-hmop 32049 df-nmfn 32050 df-nlfn 32051 df-cnfn 32052 df-lnfn 32053 df-adjh 32054

This theorem is referenced by: bdopssadj 32286

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