cnlnssadj - Hilbert Space Explorer

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Theorem cnlnssadj 30442

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 30441	. . . . 5 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧)))
2		df-rex 3070	. . . . 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 217	. . . 4 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ∃𝑡(𝑡 ∈ (LinOp ∩ ContOp) ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧))))
4		inss1 4162	. . . . . . . . . 10 ⊢ (LinOp ∩ ContOp) ⊆ LinOp
5	4	sseli 3917	. . . . . . . . 9 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → 𝑦 ∈ LinOp)
6		lnopf 30221	. . . . . . . . 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 3917	. . . . . . . . . 10 ⊢ (𝑡 ∈ (LinOp ∩ ContOp) → 𝑡 ∈ LinOp)
10		lnopf 30221	. . . . . . . . . 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 492	. . . . . . 7 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ((𝑡 ∈ (LinOp ∩ ContOp) ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧))) → 𝑡: ℋ⟶ ℋ))
14		eqcom 2745	. . . . . . . . . . 11 ⊢ (((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧)) ↔ (𝑥 ·_ih (𝑡‘𝑧)) = ((𝑦‘𝑥) ·_ih 𝑧))
15	14	biimpi 215	. . . . . . . . . 10 ⊢ (((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧)) → (𝑥 ·_ih (𝑡‘𝑧)) = ((𝑦‘𝑥) ·_ih 𝑧))
16	15	2ralimi 3088	. . . . . . . . 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 30195	. . . . . . . . . 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 597	. . . . . . . . 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	syl5ib 243	. . . . . . . 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 454	. . . . . . 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 1128	. . . . . 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 30247	. . . . . . . 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 2830	. . . . . . 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 3436	. . . . . . . 8 ⊢ 𝑦 ∈ V
25		vex 3436	. . . . . . . 8 ⊢ 𝑡 ∈ V
26		feq1 6581	. . . . . . . . 9 ⊢ (𝑢 = 𝑦 → (𝑢: ℋ⟶ ℋ ↔ 𝑦: ℋ⟶ ℋ))
27		fveq1 6773	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑦 → (𝑢‘𝑧) = (𝑦‘𝑧))
28	27	oveq2d 7291	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑦 → (𝑥 ·_ih (𝑢‘𝑧)) = (𝑥 ·_ih (𝑦‘𝑧)))
29	28	eqeq1d 2740	. . . . . . . . . 10 ⊢ (𝑢 = 𝑦 → ((𝑥 ·_ih (𝑢‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧) ↔ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧)))
30	29	2ralbidv 3129	. . . . . . . . 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 1437	. . . . . . . 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 6581	. . . . . . . . 9 ⊢ (𝑣 = 𝑡 → (𝑣: ℋ⟶ ℋ ↔ 𝑡: ℋ⟶ ℋ))
33		fveq1 6773	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑡 → (𝑣‘𝑥) = (𝑡‘𝑥))
34	33	oveq1d 7290	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑡 → ((𝑣‘𝑥) ·_ih 𝑧) = ((𝑡‘𝑥) ·_ih 𝑧))
35	34	eqeq2d 2749	. . . . . . . . . 10 ⊢ (𝑣 = 𝑡 → ((𝑥 ·_ih (𝑦‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧) ↔ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑡‘𝑥) ·_ih 𝑧)))
36	35	2ralbidv 3129	. . . . . . . . 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 1438	. . . . . . . 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 5455	. . . . . . 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 275	. . . . . 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	syl6ib 250	. . . . 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 1920	. . . 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 5810	. . 3 ⊢ (𝑦 ∈ dom adj_ℎ ↔ ∃𝑡⟨𝑦, 𝑡⟩ ∈ adj_ℎ)
44	42, 43	sylibr 233	. 2 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → 𝑦 ∈ dom adj_ℎ)
45	44	ssriv 3925	1 ⊢ (LinOp ∩ ContOp) ⊆ dom adj_ℎ

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ∩ cin 3886 ⊆ wss 3887 ⟨cop 4567 {copab 5136 dom cdm 5589 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℋchba 29281 ·_ih csp 29284 ContOpccop 29308 LinOpclo 29309 adj_ℎcado 29317

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cc 10191 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 ax-hilex 29361 ax-hfvadd 29362 ax-hvcom 29363 ax-hvass 29364 ax-hv0cl 29365 ax-hvaddid 29366 ax-hfvmul 29367 ax-hvmulid 29368 ax-hvmulass 29369 ax-hvdistr1 29370 ax-hvdistr2 29371 ax-hvmul0 29372 ax-hfi 29441 ax-his1 29444 ax-his2 29445 ax-his3 29446 ax-his4 29447 ax-hcompl 29564

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-omul 8302 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-cn 22378 df-cnp 22379 df-lm 22380 df-t1 22465 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cfil 24419 df-cau 24420 df-cmet 24421 df-grpo 28855 df-gid 28856 df-ginv 28857 df-gdiv 28858 df-ablo 28907 df-vc 28921 df-nv 28954 df-va 28957 df-ba 28958 df-sm 28959 df-0v 28960 df-vs 28961 df-nmcv 28962 df-ims 28963 df-dip 29063 df-ssp 29084 df-ph 29175 df-cbn 29225 df-hnorm 29330 df-hba 29331 df-hvsub 29333 df-hlim 29334 df-hcau 29335 df-sh 29569 df-ch 29583 df-oc 29614 df-ch0 29615 df-shs 29670 df-pjh 29757 df-h0op 30110 df-nmop 30201 df-cnop 30202 df-lnop 30203 df-unop 30205 df-hmop 30206 df-nmfn 30207 df-nlfn 30208 df-cnfn 30209 df-lnfn 30210 df-adjh 30211

This theorem is referenced by: bdopssadj 30443

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