cnlnssadj - Hilbert Space Explorer

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Theorem cnlnssadj 32443

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 32442	. . . . 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 221	. . . 4 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ∃𝑡(𝑡 ∈ (LinOp ∩ ContOp) ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧))))
4		inss1 4188	. . . . . . . . . 10 ⊢ (LinOp ∩ ContOp) ⊆ LinOp
5	4	sseli 3932	. . . . . . . . 9 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → 𝑦 ∈ LinOp)
6		lnopf 32222	. . . . . . . . 9 ⊢ (𝑦 ∈ LinOp → 𝑦: ℋ⟶ ℋ)
7	5, 6	syl 18	. . . . . . . 8 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → 𝑦: ℋ⟶ ℋ)
8	7	a1d 26	. . . . . . 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 3932	. . . . . . . . . 10 ⊢ (𝑡 ∈ (LinOp ∩ ContOp) → 𝑡 ∈ LinOp)
10		lnopf 32222	. . . . . . . . . 10 ⊢ (𝑡 ∈ LinOp → 𝑡: ℋ⟶ ℋ)
11	9, 10	syl 18	. . . . . . . . 9 ⊢ (𝑡 ∈ (LinOp ∩ ContOp) → 𝑡: ℋ⟶ ℋ)
12	11	a1i 11	. . . . . . . 8 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → (𝑡 ∈ (LinOp ∩ ContOp) → 𝑡: ℋ⟶ ℋ))
13	12	adantrd 496	. . . . . . 7 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ((𝑡 ∈ (LinOp ∩ ContOp) ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧))) → 𝑡: ℋ⟶ ℋ))
14		eqcom 2769	. . . . . . . . . . 11 ⊢ (((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧)) ↔ (𝑥 ·_ih (𝑡‘𝑧)) = ((𝑦‘𝑥) ·_ih 𝑧))
15	14	biimpi 219	. . . . . . . . . 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 32196	. . . . . . . . . 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 608	. . . . . . . . 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 247	. . . . . . . 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 458	. . . . . . 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 1146	. . . . . 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 32248	. . . . . . . 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 2854	. . . . . . 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 3458	. . . . . . . 8 ⊢ 𝑦 ∈ V
25		vex 3458	. . . . . . . 8 ⊢ 𝑡 ∈ V
26		feq1 6683	. . . . . . . . 9 ⊢ (𝑢 = 𝑦 → (𝑢: ℋ⟶ ℋ ↔ 𝑦: ℋ⟶ ℋ))
27		fveq1 6880	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑦 → (𝑢‘𝑧) = (𝑦‘𝑧))
28	27	oveq2d 7428	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑦 → (𝑥 ·_ih (𝑢‘𝑧)) = (𝑥 ·_ih (𝑦‘𝑧)))
29	28	eqeq1d 2764	. . . . . . . . . 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 1465	. . . . . . . 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 6683	. . . . . . . . 9 ⊢ (𝑣 = 𝑡 → (𝑣: ℋ⟶ ℋ ↔ 𝑡: ℋ⟶ ℋ))
33		fveq1 6880	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑡 → (𝑣‘𝑥) = (𝑡‘𝑥))
34	33	oveq1d 7427	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑡 → ((𝑣‘𝑥) ·_ih 𝑧) = ((𝑡‘𝑥) ·_ih 𝑧))
35	34	eqeq2d 2773	. . . . . . . . . 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 1466	. . . . . . . 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 5526	. . . . . . 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 279	. . . . . 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 254	. . . . 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 1946	. . . 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 16	. . 3 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ∃𝑡⟨𝑦, 𝑡⟩ ∈ adj_ℎ)
43	24	eldm2 5890	. . 3 ⊢ (𝑦 ∈ dom adj_ℎ ↔ ∃𝑡⟨𝑦, 𝑡⟩ ∈ adj_ℎ)
44	42, 43	sylibr 237	. 2 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → 𝑦 ∈ dom adj_ℎ)
45	44	ssriv 3940	1 ⊢ (LinOp ∩ ContOp) ⊆ dom adj_ℎ

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1102 = wceq 1569 ∃wex 1808 ∈ wcel 2142 ∀wral 3078 ∃wrex 3088 ∩ cin 3903 ⊆ wss 3904 ⟨cop 4594 {copab 5172 dom cdm 5660 ⟶wf 6532 ‘cfv 6536 (class class class)co 7412 ℋchba 31282 ·_ih csp 31285 ContOpccop 31309 LinOpclo 31310 adj_ℎcado 31318

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cc 10425 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 ax-hilex 31362 ax-hfvadd 31363 ax-hvcom 31364 ax-hvass 31365 ax-hv0cl 31366 ax-hvaddid 31367 ax-hfvmul 31368 ax-hvmulid 31369 ax-hvmulass 31370 ax-hvdistr1 31371 ax-hvdistr2 31372 ax-hvmul0 31373 ax-hfi 31442 ax-his1 31445 ax-his2 31446 ax-his3 31447 ax-his4 31448 ax-hcompl 31565

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-oadd 8455 df-omul 8456 df-er 8692 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9932 df-acn 9935 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-ioo 13382 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-fl 13832 df-seq 14045 df-exp 14105 df-hash 14374 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-clim 15546 df-rlim 15547 df-sum 15745 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-rest 17481 df-topn 17482 df-0g 17500 df-gsum 17501 df-topgen 17502 df-pt 17503 df-prds 17506 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-submnd 18848 df-mulg 19140 df-cntz 19393 df-cmn 19858 df-psmet 21525 df-xmet 21526 df-met 21527 df-bl 21528 df-mopn 21529 df-fbas 21530 df-fg 21531 df-cnfld 21534 df-top 23062 df-topon 23079 df-topsp 23101 df-bases 23114 df-cld 23187 df-ntr 23188 df-cls 23189 df-nei 23266 df-cn 23395 df-cnp 23396 df-lm 23397 df-t1 23482 df-haus 23483 df-tx 23730 df-hmeo 23923 df-fil 24014 df-fm 24106 df-flim 24107 df-flf 24108 df-xms 24488 df-ms 24489 df-tms 24490 df-cfil 25425 df-cau 25426 df-cmet 25427 df-grpo 30856 df-gid 30857 df-ginv 30858 df-gdiv 30859 df-ablo 30908 df-vc 30922 df-nv 30955 df-va 30958 df-ba 30959 df-sm 30960 df-0v 30961 df-vs 30962 df-nmcv 30963 df-ims 30964 df-dip 31064 df-ssp 31085 df-ph 31176 df-cbn 31226 df-hnorm 31331 df-hba 31332 df-hvsub 31334 df-hlim 31335 df-hcau 31336 df-sh 31570 df-ch 31584 df-oc 31615 df-ch0 31616 df-shs 31671 df-pjh 31758 df-h0op 32111 df-nmop 32202 df-cnop 32203 df-lnop 32204 df-unop 32206 df-hmop 32207 df-nmfn 32208 df-nlfn 32209 df-cnfn 32210 df-lnfn 32211 df-adjh 32212

This theorem is used by: bdopssadj 32444

W3C validator