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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 2798 | . 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 29010 | . . 3 ⊢ 𝐻 ∈ Sℋ |
6 | 2, 3, 5 | hhssmet 29059 | . 2 ⊢ 𝐷 ∈ (Met‘𝐻) |
7 | simpl 486 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ (Cau‘𝐷)) | |
8 | 2, 3, 5 | hhssims2 29058 | . . . . . . . . . . 11 ⊢ 𝐷 = ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)) |
9 | 8 | fveq2i 6648 | . . . . . . . . . 10 ⊢ (Cau‘𝐷) = (Cau‘((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻))) |
10 | 7, 9 | eleqtrdi 2900 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ (Cau‘((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)))) |
11 | eqid 2798 | . . . . . . . . . . 11 ⊢ (normℎ ∘ −ℎ ) = (normℎ ∘ −ℎ ) | |
12 | 11 | hilxmet 28978 | . . . . . . . . . 10 ⊢ (normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) |
13 | simpr 488 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓:ℕ⟶𝐻) | |
14 | causs 23902 | . . . . . . . . . 10 ⊢ (((normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) ∧ 𝑓:ℕ⟶𝐻) → (𝑓 ∈ (Cau‘(normℎ ∘ −ℎ )) ↔ 𝑓 ∈ (Cau‘((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻))))) | |
15 | 12, 13, 14 | sylancr 590 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → (𝑓 ∈ (Cau‘(normℎ ∘ −ℎ )) ↔ 𝑓 ∈ (Cau‘((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻))))) |
16 | 10, 15 | mpbird 260 | . . . . . . . 8 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ (Cau‘(normℎ ∘ −ℎ ))) |
17 | 4 | chssii 29014 | . . . . . . . . . 10 ⊢ 𝐻 ⊆ ℋ |
18 | fss 6501 | . . . . . . . . . 10 ⊢ ((𝑓:ℕ⟶𝐻 ∧ 𝐻 ⊆ ℋ) → 𝑓:ℕ⟶ ℋ) | |
19 | 13, 17, 18 | sylancl 589 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓:ℕ⟶ ℋ) |
20 | ax-hilex 28782 | . . . . . . . . . 10 ⊢ ℋ ∈ V | |
21 | nnex 11631 | . . . . . . . . . 10 ⊢ ℕ ∈ V | |
22 | 20, 21 | elmap 8418 | . . . . . . . . 9 ⊢ (𝑓 ∈ ( ℋ ↑m ℕ) ↔ 𝑓:ℕ⟶ ℋ) |
23 | 19, 22 | sylibr 237 | . . . . . . . 8 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ ( ℋ ↑m ℕ)) |
24 | eqid 2798 | . . . . . . . . . 10 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
25 | 24, 11 | hhims 28955 | . . . . . . . . . 10 ⊢ (normℎ ∘ −ℎ ) = (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
26 | 24, 25 | hhcau 28981 | . . . . . . . . 9 ⊢ Cauchy = ((Cau‘(normℎ ∘ −ℎ )) ∩ ( ℋ ↑m ℕ)) |
27 | 26 | elin2 4124 | . . . . . . . 8 ⊢ (𝑓 ∈ Cauchy ↔ (𝑓 ∈ (Cau‘(normℎ ∘ −ℎ )) ∧ 𝑓 ∈ ( ℋ ↑m ℕ))) |
28 | 16, 23, 27 | sylanbrc 586 | . . . . . . 7 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ Cauchy) |
29 | ax-hcompl 28985 | . . . . . . 7 ⊢ (𝑓 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝑓 ⇝𝑣 𝑥) | |
30 | vex 3444 | . . . . . . . . 9 ⊢ 𝑓 ∈ V | |
31 | vex 3444 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
32 | 30, 31 | breldm 5741 | . . . . . . . 8 ⊢ (𝑓 ⇝𝑣 𝑥 → 𝑓 ∈ dom ⇝𝑣 ) |
33 | 32 | rexlimivw 3241 | . . . . . . 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 29020 | . . . . . . 7 ⊢ ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ | |
36 | ffun 6490 | . . . . . . 7 ⊢ ( ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ → Fun ⇝𝑣 ) | |
37 | funfvbrb 6798 | . . . . . . 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 221 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ⇝𝑣 ( ⇝𝑣 ‘𝑓)) |
40 | eqid 2798 | . . . . . . . 8 ⊢ (MetOpen‘(normℎ ∘ −ℎ )) = (MetOpen‘(normℎ ∘ −ℎ )) | |
41 | 24, 25, 40 | hhlm 28982 | . . . . . . 7 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ)) |
42 | resss 5843 | . . . . . . 7 ⊢ ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ)) ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) | |
43 | 41, 42 | eqsstri 3949 | . . . . . 6 ⊢ ⇝𝑣 ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) |
44 | 43 | ssbri 5075 | . . . . 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 23131 | . . . . . . 7 ⊢ (((normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) ∧ 𝐻 ⊆ ℋ) → ((MetOpen‘(normℎ ∘ −ℎ )) ↾t 𝐻) = (MetOpen‘𝐷)) |
47 | 12, 17, 46 | mp2an 691 | . . . . . 6 ⊢ ((MetOpen‘(normℎ ∘ −ℎ )) ↾t 𝐻) = (MetOpen‘𝐷) |
48 | 47 | eqcomi 2807 | . . . . 5 ⊢ (MetOpen‘𝐷) = ((MetOpen‘(normℎ ∘ −ℎ )) ↾t 𝐻) |
49 | nnuz 12269 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
50 | 4 | a1i 11 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝐻 ∈ Cℋ ) |
51 | 40 | mopntop 23047 | . . . . . 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 6658 | . . . . . . 7 ⊢ ( ⇝𝑣 ‘𝑓) ∈ V | |
54 | 53 | chlimi 29017 | . . . . . 6 ⊢ ((𝐻 ∈ Cℋ ∧ 𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 ( ⇝𝑣 ‘𝑓)) → ( ⇝𝑣 ‘𝑓) ∈ 𝐻) |
55 | 50, 13, 39, 54 | syl3anc 1368 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → ( ⇝𝑣 ‘𝑓) ∈ 𝐻) |
56 | 1zzd 12001 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 1 ∈ ℤ) | |
57 | 48, 49, 50, 52, 55, 56, 13 | lmss 21903 | . . . 4 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → (𝑓(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))( ⇝𝑣 ‘𝑓) ↔ 𝑓(⇝𝑡‘(MetOpen‘𝐷))( ⇝𝑣 ‘𝑓))) |
58 | 45, 57 | mpbid 235 | . . 3 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓(⇝𝑡‘(MetOpen‘𝐷))( ⇝𝑣 ‘𝑓)) |
59 | 30, 53 | breldm 5741 | . . 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 23916 | 1 ⊢ 𝐷 ∈ (CMet‘𝐻) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 ⊆ wss 3881 〈cop 4531 class class class wbr 5030 × cxp 5517 dom cdm 5519 ↾ cres 5521 ∘ ccom 5523 Fun wfun 6318 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 ℂcc 10524 1c1 10527 ℕcn 11625 ↾t crest 16686 ∞Metcxmet 20076 MetOpencmopn 20081 Topctop 21498 ⇝𝑡clm 21831 Cauccau 23857 CMetccmet 23858 IndMetcims 28374 ℋchba 28702 +ℎ cva 28703 ·ℎ csm 28704 normℎcno 28706 −ℎ cmv 28708 Cauchyccauold 28709 ⇝𝑣 chli 28710 Cℋ cch 28712 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 ax-hilex 28782 ax-hfvadd 28783 ax-hvcom 28784 ax-hvass 28785 ax-hv0cl 28786 ax-hvaddid 28787 ax-hfvmul 28788 ax-hvmulid 28789 ax-hvmulass 28790 ax-hvdistr1 28791 ax-hvdistr2 28792 ax-hvmul0 28793 ax-hfi 28862 ax-his1 28865 ax-his2 28866 ax-his3 28867 ax-his4 28868 ax-hcompl 28985 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-omul 8090 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ico 12732 df-icc 12733 df-fz 12886 df-fl 13157 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-rest 16688 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-top 21499 df-topon 21516 df-bases 21551 df-ntr 21625 df-nei 21703 df-lm 21834 df-haus 21920 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-cfil 23859 df-cau 23860 df-cmet 23861 df-grpo 28276 df-gid 28277 df-ginv 28278 df-gdiv 28279 df-ablo 28328 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-vs 28382 df-nmcv 28383 df-ims 28384 df-ssp 28505 df-hnorm 28751 df-hba 28752 df-hvsub 28754 df-hlim 28755 df-hcau 28756 df-sh 28990 df-ch 29004 df-ch0 29036 |
This theorem is referenced by: hhssbnOLD 29062 |
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