cnlnssadj - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > cnlnssadj		Structured version Visualization version GIF version

Theorem cnlnssadj 32136

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 32135	. . . . 5 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧)))
2		df-rex 3060	. . . . 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 218	. . . 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 3928	. . . . . . . . 9 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → 𝑦 ∈ LinOp)
6		lnopf 31915	. . . . . . . . 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 3928	. . . . . . . . . 10 ⊢ (𝑡 ∈ (LinOp ∩ ContOp) → 𝑡 ∈ LinOp)
10		lnopf 31915	. . . . . . . . . 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 491	. . . . . . 7 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ((𝑡 ∈ (LinOp ∩ ContOp) ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧))) → 𝑡: ℋ⟶ ℋ))
14		eqcom 2742	. . . . . . . . . . 11 ⊢ (((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧)) ↔ (𝑥 ·_ih (𝑡‘𝑧)) = ((𝑦‘𝑥) ·_ih 𝑧))
15	14	biimpi 216	. . . . . . . . . 10 ⊢ (((𝑦‘𝑥) ·_ih 𝑧) = (𝑥 ·_ih (𝑡‘𝑧)) → (𝑥 ·_ih (𝑡‘𝑧)) = ((𝑦‘𝑥) ·_ih 𝑧))
16	15	2ralimi 3105	. . . . . . . . 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 31889	. . . . . . . . . 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 598	. . . . . . . . 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 244	. . . . . . . 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 453	. . . . . . 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 1130	. . . . . 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 31941	. . . . . . . 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 2827	. . . . . . 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 3443	. . . . . . . 8 ⊢ 𝑦 ∈ V
25		vex 3443	. . . . . . . 8 ⊢ 𝑡 ∈ V
26		feq1 6639	. . . . . . . . 9 ⊢ (𝑢 = 𝑦 → (𝑢: ℋ⟶ ℋ ↔ 𝑦: ℋ⟶ ℋ))
27		fveq1 6832	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑦 → (𝑢‘𝑧) = (𝑦‘𝑧))
28	27	oveq2d 7374	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑦 → (𝑥 ·_ih (𝑢‘𝑧)) = (𝑥 ·_ih (𝑦‘𝑧)))
29	28	eqeq1d 2737	. . . . . . . . . 10 ⊢ (𝑢 = 𝑦 → ((𝑥 ·_ih (𝑢‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧) ↔ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧)))
30	29	2ralbidv 3199	. . . . . . . . 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 1441	. . . . . . . 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 6639	. . . . . . . . 9 ⊢ (𝑣 = 𝑡 → (𝑣: ℋ⟶ ℋ ↔ 𝑡: ℋ⟶ ℋ))
33		fveq1 6832	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑡 → (𝑣‘𝑥) = (𝑡‘𝑥))
34	33	oveq1d 7373	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑡 → ((𝑣‘𝑥) ·_ih 𝑧) = ((𝑡‘𝑥) ·_ih 𝑧))
35	34	eqeq2d 2746	. . . . . . . . . 10 ⊢ (𝑣 = 𝑡 → ((𝑥 ·_ih (𝑦‘𝑧)) = ((𝑣‘𝑥) ·_ih 𝑧) ↔ (𝑥 ·_ih (𝑦‘𝑧)) = ((𝑡‘𝑥) ·_ih 𝑧)))
36	35	2ralbidv 3199	. . . . . . . . 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 1442	. . . . . . . 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 5489	. . . . . . 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 276	. . . . . 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 251	. . . . 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 1919	. . . 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 5849	. . 3 ⊢ (𝑦 ∈ dom adj_ℎ ↔ ∃𝑡⟨𝑦, 𝑡⟩ ∈ adj_ℎ)
44	42, 43	sylibr 234	. 2 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → 𝑦 ∈ dom adj_ℎ)
45	44	ssriv 3936	1 ⊢ (LinOp ∩ ContOp) ⊆ dom adj_ℎ

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ∀wral 3050 ∃wrex 3059 ∩ cin 3899 ⊆ wss 3900 ⟨cop 4585 {copab 5159 dom cdm 5623 ⟶wf 6487 ‘cfv 6491 (class class class)co 7358 ℋchba 30975 ·_ih csp 30978 ContOpccop 31002 LinOpclo 31003 adj_ℎcado 31011

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-inf2 9552 ax-cc 10347 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107 ax-mulf 11108 ax-hilex 31055 ax-hfvadd 31056 ax-hvcom 31057 ax-hvass 31058 ax-hv0cl 31059 ax-hvaddid 31060 ax-hfvmul 31061 ax-hvmulid 31062 ax-hvmulass 31063 ax-hvdistr1 31064 ax-hvdistr2 31065 ax-hvmul0 31066 ax-hfi 31135 ax-his1 31138 ax-his2 31139 ax-his3 31140 ax-his4 31141 ax-hcompl 31258

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3349 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4902 df-iun 4947 df-iin 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-omul 8402 df-er 8635 df-map 8767 df-pm 8768 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fsupp 9267 df-fi 9316 df-sup 9347 df-inf 9348 df-oi 9417 df-card 9853 df-acn 9856 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-ioo 13267 df-ico 13269 df-icc 13270 df-fz 13426 df-fzo 13573 df-fl 13714 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-rlim 15414 df-sum 15612 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17425 df-qtop 17430 df-imas 17431 df-xps 17433 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19248 df-cmn 19713 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-fbas 21308 df-fg 21309 df-cnfld 21312 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22892 df-cld 22965 df-ntr 22966 df-cls 22967 df-nei 23044 df-cn 23173 df-cnp 23174 df-lm 23175 df-t1 23260 df-haus 23261 df-tx 23508 df-hmeo 23701 df-fil 23792 df-fm 23884 df-flim 23885 df-flf 23886 df-xms 24266 df-ms 24267 df-tms 24268 df-cfil 25213 df-cau 25214 df-cmet 25215 df-grpo 30549 df-gid 30550 df-ginv 30551 df-gdiv 30552 df-ablo 30601 df-vc 30615 df-nv 30648 df-va 30651 df-ba 30652 df-sm 30653 df-0v 30654 df-vs 30655 df-nmcv 30656 df-ims 30657 df-dip 30757 df-ssp 30778 df-ph 30869 df-cbn 30919 df-hnorm 31024 df-hba 31025 df-hvsub 31027 df-hlim 31028 df-hcau 31029 df-sh 31263 df-ch 31277 df-oc 31308 df-ch0 31309 df-shs 31364 df-pjh 31451 df-h0op 31804 df-nmop 31895 df-cnop 31896 df-lnop 31897 df-unop 31899 df-hmop 31900 df-nmfn 31901 df-nlfn 31902 df-cnfn 31903 df-lnfn 31904 df-adjh 31905

This theorem is referenced by: bdopssadj 32137

W3C validator