Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > hhsscms | Structured version Visualization version GIF version |
Description: The induced metric of a closed subspace is complete. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhssims2.1 | ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 |
hhssims2.3 | ⊢ 𝐷 = (IndMet‘𝑊) |
hhsscms.3 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
hhsscms | ⊢ 𝐷 ∈ (CMet‘𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2824 | . 2 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
2 | hhssims2.1 | . . 3 ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
3 | hhssims2.3 | . . 3 ⊢ 𝐷 = (IndMet‘𝑊) | |
4 | hhsscms.3 | . . . 4 ⊢ 𝐻 ∈ Cℋ | |
5 | 4 | chshii 29007 | . . 3 ⊢ 𝐻 ∈ Sℋ |
6 | 2, 3, 5 | hhssmet 29056 | . 2 ⊢ 𝐷 ∈ (Met‘𝐻) |
7 | simpl 485 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ (Cau‘𝐷)) | |
8 | 2, 3, 5 | hhssims2 29055 | . . . . . . . . . . 11 ⊢ 𝐷 = ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)) |
9 | 8 | fveq2i 6676 | . . . . . . . . . 10 ⊢ (Cau‘𝐷) = (Cau‘((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻))) |
10 | 7, 9 | eleqtrdi 2926 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ (Cau‘((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)))) |
11 | eqid 2824 | . . . . . . . . . . 11 ⊢ (normℎ ∘ −ℎ ) = (normℎ ∘ −ℎ ) | |
12 | 11 | hilxmet 28975 | . . . . . . . . . 10 ⊢ (normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) |
13 | simpr 487 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓:ℕ⟶𝐻) | |
14 | causs 23904 | . . . . . . . . . 10 ⊢ (((normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) ∧ 𝑓:ℕ⟶𝐻) → (𝑓 ∈ (Cau‘(normℎ ∘ −ℎ )) ↔ 𝑓 ∈ (Cau‘((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻))))) | |
15 | 12, 13, 14 | sylancr 589 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → (𝑓 ∈ (Cau‘(normℎ ∘ −ℎ )) ↔ 𝑓 ∈ (Cau‘((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻))))) |
16 | 10, 15 | mpbird 259 | . . . . . . . 8 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ (Cau‘(normℎ ∘ −ℎ ))) |
17 | 4 | chssii 29011 | . . . . . . . . . 10 ⊢ 𝐻 ⊆ ℋ |
18 | fss 6530 | . . . . . . . . . 10 ⊢ ((𝑓:ℕ⟶𝐻 ∧ 𝐻 ⊆ ℋ) → 𝑓:ℕ⟶ ℋ) | |
19 | 13, 17, 18 | sylancl 588 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓:ℕ⟶ ℋ) |
20 | ax-hilex 28779 | . . . . . . . . . 10 ⊢ ℋ ∈ V | |
21 | nnex 11647 | . . . . . . . . . 10 ⊢ ℕ ∈ V | |
22 | 20, 21 | elmap 8438 | . . . . . . . . 9 ⊢ (𝑓 ∈ ( ℋ ↑m ℕ) ↔ 𝑓:ℕ⟶ ℋ) |
23 | 19, 22 | sylibr 236 | . . . . . . . 8 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ ( ℋ ↑m ℕ)) |
24 | eqid 2824 | . . . . . . . . . 10 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
25 | 24, 11 | hhims 28952 | . . . . . . . . . 10 ⊢ (normℎ ∘ −ℎ ) = (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
26 | 24, 25 | hhcau 28978 | . . . . . . . . 9 ⊢ Cauchy = ((Cau‘(normℎ ∘ −ℎ )) ∩ ( ℋ ↑m ℕ)) |
27 | 26 | elin2 4177 | . . . . . . . 8 ⊢ (𝑓 ∈ Cauchy ↔ (𝑓 ∈ (Cau‘(normℎ ∘ −ℎ )) ∧ 𝑓 ∈ ( ℋ ↑m ℕ))) |
28 | 16, 23, 27 | sylanbrc 585 | . . . . . . 7 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ Cauchy) |
29 | ax-hcompl 28982 | . . . . . . 7 ⊢ (𝑓 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝑓 ⇝𝑣 𝑥) | |
30 | vex 3500 | . . . . . . . . 9 ⊢ 𝑓 ∈ V | |
31 | vex 3500 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
32 | 30, 31 | breldm 5780 | . . . . . . . 8 ⊢ (𝑓 ⇝𝑣 𝑥 → 𝑓 ∈ dom ⇝𝑣 ) |
33 | 32 | rexlimivw 3285 | . . . . . . 7 ⊢ (∃𝑥 ∈ ℋ 𝑓 ⇝𝑣 𝑥 → 𝑓 ∈ dom ⇝𝑣 ) |
34 | 28, 29, 33 | 3syl 18 | . . . . . 6 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ dom ⇝𝑣 ) |
35 | hlimf 29017 | . . . . . . 7 ⊢ ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ | |
36 | ffun 6520 | . . . . . . 7 ⊢ ( ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ → Fun ⇝𝑣 ) | |
37 | funfvbrb 6824 | . . . . . . 7 ⊢ (Fun ⇝𝑣 → (𝑓 ∈ dom ⇝𝑣 ↔ 𝑓 ⇝𝑣 ( ⇝𝑣 ‘𝑓))) | |
38 | 35, 36, 37 | mp2b 10 | . . . . . 6 ⊢ (𝑓 ∈ dom ⇝𝑣 ↔ 𝑓 ⇝𝑣 ( ⇝𝑣 ‘𝑓)) |
39 | 34, 38 | sylib 220 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ⇝𝑣 ( ⇝𝑣 ‘𝑓)) |
40 | eqid 2824 | . . . . . . . 8 ⊢ (MetOpen‘(normℎ ∘ −ℎ )) = (MetOpen‘(normℎ ∘ −ℎ )) | |
41 | 24, 25, 40 | hhlm 28979 | . . . . . . 7 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ)) |
42 | resss 5881 | . . . . . . 7 ⊢ ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ)) ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) | |
43 | 41, 42 | eqsstri 4004 | . . . . . 6 ⊢ ⇝𝑣 ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) |
44 | 43 | ssbri 5114 | . . . . 5 ⊢ (𝑓 ⇝𝑣 ( ⇝𝑣 ‘𝑓) → 𝑓(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))( ⇝𝑣 ‘𝑓)) |
45 | 39, 44 | syl 17 | . . . 4 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))( ⇝𝑣 ‘𝑓)) |
46 | 8, 40, 1 | metrest 23137 | . . . . . . 7 ⊢ (((normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) ∧ 𝐻 ⊆ ℋ) → ((MetOpen‘(normℎ ∘ −ℎ )) ↾t 𝐻) = (MetOpen‘𝐷)) |
47 | 12, 17, 46 | mp2an 690 | . . . . . 6 ⊢ ((MetOpen‘(normℎ ∘ −ℎ )) ↾t 𝐻) = (MetOpen‘𝐷) |
48 | 47 | eqcomi 2833 | . . . . 5 ⊢ (MetOpen‘𝐷) = ((MetOpen‘(normℎ ∘ −ℎ )) ↾t 𝐻) |
49 | nnuz 12284 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
50 | 4 | a1i 11 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝐻 ∈ Cℋ ) |
51 | 40 | mopntop 23053 | . . . . . 6 ⊢ ((normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) → (MetOpen‘(normℎ ∘ −ℎ )) ∈ Top) |
52 | 12, 51 | mp1i 13 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → (MetOpen‘(normℎ ∘ −ℎ )) ∈ Top) |
53 | fvex 6686 | . . . . . . 7 ⊢ ( ⇝𝑣 ‘𝑓) ∈ V | |
54 | 53 | chlimi 29014 | . . . . . 6 ⊢ ((𝐻 ∈ Cℋ ∧ 𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 ( ⇝𝑣 ‘𝑓)) → ( ⇝𝑣 ‘𝑓) ∈ 𝐻) |
55 | 50, 13, 39, 54 | syl3anc 1367 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → ( ⇝𝑣 ‘𝑓) ∈ 𝐻) |
56 | 1zzd 12016 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 1 ∈ ℤ) | |
57 | 48, 49, 50, 52, 55, 56, 13 | lmss 21909 | . . . 4 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → (𝑓(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))( ⇝𝑣 ‘𝑓) ↔ 𝑓(⇝𝑡‘(MetOpen‘𝐷))( ⇝𝑣 ‘𝑓))) |
58 | 45, 57 | mpbid 234 | . . 3 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓(⇝𝑡‘(MetOpen‘𝐷))( ⇝𝑣 ‘𝑓)) |
59 | 30, 53 | breldm 5780 | . . 3 ⊢ (𝑓(⇝𝑡‘(MetOpen‘𝐷))( ⇝𝑣 ‘𝑓) → 𝑓 ∈ dom (⇝𝑡‘(MetOpen‘𝐷))) |
60 | 58, 59 | syl 17 | . 2 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ dom (⇝𝑡‘(MetOpen‘𝐷))) |
61 | 1, 6, 60 | iscmet3i 23918 | 1 ⊢ 𝐷 ∈ (CMet‘𝐻) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3142 ⊆ wss 3939 〈cop 4576 class class class wbr 5069 × cxp 5556 dom cdm 5558 ↾ cres 5560 ∘ ccom 5562 Fun wfun 6352 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ↑m cmap 8409 ℂcc 10538 1c1 10541 ℕcn 11641 ↾t crest 16697 ∞Metcxmet 20533 MetOpencmopn 20538 Topctop 21504 ⇝𝑡clm 21837 Cauccau 23859 CMetccmet 23860 IndMetcims 28371 ℋchba 28699 +ℎ cva 28700 ·ℎ csm 28701 normℎcno 28703 −ℎ cmv 28705 Cauchyccauold 28706 ⇝𝑣 chli 28707 Cℋ cch 28709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cc 9860 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 ax-hilex 28779 ax-hfvadd 28780 ax-hvcom 28781 ax-hvass 28782 ax-hv0cl 28783 ax-hvaddid 28784 ax-hfvmul 28785 ax-hvmulid 28786 ax-hvmulass 28787 ax-hvdistr1 28788 ax-hvdistr2 28789 ax-hvmul0 28790 ax-hfi 28859 ax-his1 28862 ax-his2 28863 ax-his3 28864 ax-his4 28865 ax-hcompl 28982 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-omul 8110 df-er 8292 df-map 8411 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-acn 9374 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ico 12747 df-icc 12748 df-fz 12896 df-fl 13165 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-rlim 14849 df-rest 16699 df-topgen 16720 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-fbas 20545 df-fg 20546 df-top 21505 df-topon 21522 df-bases 21557 df-ntr 21631 df-nei 21709 df-lm 21840 df-haus 21926 df-fil 22457 df-fm 22549 df-flim 22550 df-flf 22551 df-cfil 23861 df-cau 23862 df-cmet 23863 df-grpo 28273 df-gid 28274 df-ginv 28275 df-gdiv 28276 df-ablo 28325 df-vc 28339 df-nv 28372 df-va 28375 df-ba 28376 df-sm 28377 df-0v 28378 df-vs 28379 df-nmcv 28380 df-ims 28381 df-ssp 28502 df-hnorm 28748 df-hba 28749 df-hvsub 28751 df-hlim 28752 df-hcau 28753 df-sh 28987 df-ch 29001 df-ch0 29033 |
This theorem is referenced by: hhssbnOLD 29059 |
Copyright terms: Public domain | W3C validator |