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Mirrors > Home > HSE Home > Th. List > cnlnssadj | Structured version Visualization version GIF version |
Description: Every continuous linear Hilbert space operator has an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnlnssadj | ⊢ (LinOp ∩ ContOp) ⊆ dom adjℎ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnlnadj 29783 | . . . . 5 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·ih 𝑧) = (𝑥 ·ih (𝑡‘𝑧))) | |
2 | df-rex 3141 | . . . . 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 219 | . . . 4 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ∃𝑡(𝑡 ∈ (LinOp ∩ ContOp) ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·ih 𝑧) = (𝑥 ·ih (𝑡‘𝑧)))) |
4 | inss1 4202 | . . . . . . . . . 10 ⊢ (LinOp ∩ ContOp) ⊆ LinOp | |
5 | 4 | sseli 3960 | . . . . . . . . 9 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → 𝑦 ∈ LinOp) |
6 | lnopf 29563 | . . . . . . . . 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 3960 | . . . . . . . . . 10 ⊢ (𝑡 ∈ (LinOp ∩ ContOp) → 𝑡 ∈ LinOp) |
10 | lnopf 29563 | . . . . . . . . . 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 2825 | . . . . . . . . . . 11 ⊢ (((𝑦‘𝑥) ·ih 𝑧) = (𝑥 ·ih (𝑡‘𝑧)) ↔ (𝑥 ·ih (𝑡‘𝑧)) = ((𝑦‘𝑥) ·ih 𝑧)) | |
15 | 14 | biimpi 217 | . . . . . . . . . 10 ⊢ (((𝑦‘𝑥) ·ih 𝑧) = (𝑥 ·ih (𝑡‘𝑧)) → (𝑥 ·ih (𝑡‘𝑧)) = ((𝑦‘𝑥) ·ih 𝑧)) |
16 | 15 | 2ralimi 3158 | . . . . . . . . 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 29537 | . . . . . . . . . 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 596 | . . . . . . . . 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 245 | . . . . . . . 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 1121 | . . . . . 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 29589 | . . . . . . . 8 ⊢ adjℎ = {〈𝑢, 𝑣〉 ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·ih (𝑢‘𝑧)) = ((𝑣‘𝑥) ·ih 𝑧))} | |
23 | 22 | eleq2i 2901 | . . . . . . 7 ⊢ (〈𝑦, 𝑡〉 ∈ adjℎ ↔ 〈𝑦, 𝑡〉 ∈ {〈𝑢, 𝑣〉 ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·ih (𝑢‘𝑧)) = ((𝑣‘𝑥) ·ih 𝑧))}) |
24 | vex 3495 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
25 | vex 3495 | . . . . . . . 8 ⊢ 𝑡 ∈ V | |
26 | feq1 6488 | . . . . . . . . 9 ⊢ (𝑢 = 𝑦 → (𝑢: ℋ⟶ ℋ ↔ 𝑦: ℋ⟶ ℋ)) | |
27 | fveq1 6662 | . . . . . . . . . . . 12 ⊢ (𝑢 = 𝑦 → (𝑢‘𝑧) = (𝑦‘𝑧)) | |
28 | 27 | oveq2d 7161 | . . . . . . . . . . 11 ⊢ (𝑢 = 𝑦 → (𝑥 ·ih (𝑢‘𝑧)) = (𝑥 ·ih (𝑦‘𝑧))) |
29 | 28 | eqeq1d 2820 | . . . . . . . . . 10 ⊢ (𝑢 = 𝑦 → ((𝑥 ·ih (𝑢‘𝑧)) = ((𝑣‘𝑥) ·ih 𝑧) ↔ (𝑥 ·ih (𝑦‘𝑧)) = ((𝑣‘𝑥) ·ih 𝑧))) |
30 | 29 | 2ralbidv 3196 | . . . . . . . . 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 1429 | . . . . . . . 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 6488 | . . . . . . . . 9 ⊢ (𝑣 = 𝑡 → (𝑣: ℋ⟶ ℋ ↔ 𝑡: ℋ⟶ ℋ)) | |
33 | fveq1 6662 | . . . . . . . . . . . 12 ⊢ (𝑣 = 𝑡 → (𝑣‘𝑥) = (𝑡‘𝑥)) | |
34 | 33 | oveq1d 7160 | . . . . . . . . . . 11 ⊢ (𝑣 = 𝑡 → ((𝑣‘𝑥) ·ih 𝑧) = ((𝑡‘𝑥) ·ih 𝑧)) |
35 | 34 | eqeq2d 2829 | . . . . . . . . . 10 ⊢ (𝑣 = 𝑡 → ((𝑥 ·ih (𝑦‘𝑧)) = ((𝑣‘𝑥) ·ih 𝑧) ↔ (𝑥 ·ih (𝑦‘𝑧)) = ((𝑡‘𝑥) ·ih 𝑧))) |
36 | 35 | 2ralbidv 3196 | . . . . . . . . 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 1430 | . . . . . . . 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 5420 | . . . . . . 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 277 | . . . . . 6 ⊢ ((𝑦: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ (𝑥 ·ih (𝑦‘𝑧)) = ((𝑡‘𝑥) ·ih 𝑧)) ↔ 〈𝑦, 𝑡〉 ∈ adjℎ) |
40 | 21, 39 | syl6ib 252 | . . . . 5 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ((𝑡 ∈ (LinOp ∩ ContOp) ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·ih 𝑧) = (𝑥 ·ih (𝑡‘𝑧))) → 〈𝑦, 𝑡〉 ∈ adjℎ)) |
41 | 40 | eximdv 1909 | . . . 4 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → (∃𝑡(𝑡 ∈ (LinOp ∩ ContOp) ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑦‘𝑥) ·ih 𝑧) = (𝑥 ·ih (𝑡‘𝑧))) → ∃𝑡〈𝑦, 𝑡〉 ∈ adjℎ)) |
42 | 3, 41 | mpd 15 | . . 3 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → ∃𝑡〈𝑦, 𝑡〉 ∈ adjℎ) |
43 | 24 | eldm2 5763 | . . 3 ⊢ (𝑦 ∈ dom adjℎ ↔ ∃𝑡〈𝑦, 𝑡〉 ∈ adjℎ) |
44 | 42, 43 | sylibr 235 | . 2 ⊢ (𝑦 ∈ (LinOp ∩ ContOp) → 𝑦 ∈ dom adjℎ) |
45 | 44 | ssriv 3968 | 1 ⊢ (LinOp ∩ ContOp) ⊆ dom adjℎ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 ∩ cin 3932 ⊆ wss 3933 〈cop 4563 {copab 5119 dom cdm 5548 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℋchba 28623 ·ih csp 28626 ContOpccop 28650 LinOpclo 28651 adjℎcado 28659 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cc 9845 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 ax-hilex 28703 ax-hfvadd 28704 ax-hvcom 28705 ax-hvass 28706 ax-hv0cl 28707 ax-hvaddid 28708 ax-hfvmul 28709 ax-hvmulid 28710 ax-hvmulass 28711 ax-hvdistr1 28712 ax-hvdistr2 28713 ax-hvmul0 28714 ax-hfi 28783 ax-his1 28786 ax-his2 28787 ax-his3 28788 ax-his4 28789 ax-hcompl 28906 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-omul 8096 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-acn 9359 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-rlim 14834 df-sum 15031 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-cn 21763 df-cnp 21764 df-lm 21765 df-t1 21850 df-haus 21851 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-xms 22857 df-ms 22858 df-tms 22859 df-cfil 23785 df-cau 23786 df-cmet 23787 df-grpo 28197 df-gid 28198 df-ginv 28199 df-gdiv 28200 df-ablo 28249 df-vc 28263 df-nv 28296 df-va 28299 df-ba 28300 df-sm 28301 df-0v 28302 df-vs 28303 df-nmcv 28304 df-ims 28305 df-dip 28405 df-ssp 28426 df-ph 28517 df-cbn 28567 df-hnorm 28672 df-hba 28673 df-hvsub 28675 df-hlim 28676 df-hcau 28677 df-sh 28911 df-ch 28925 df-oc 28956 df-ch0 28957 df-shs 29012 df-pjh 29099 df-h0op 29452 df-nmop 29543 df-cnop 29544 df-lnop 29545 df-unop 29547 df-hmop 29548 df-nmfn 29549 df-nlfn 29550 df-cnfn 29551 df-lnfn 29552 df-adjh 29553 |
This theorem is referenced by: bdopssadj 29785 |
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