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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 2823 | . 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 29006 | . . 3 ⊢ 𝐻 ∈ Sℋ |
6 | 2, 3, 5 | hhssmet 29055 | . 2 ⊢ 𝐷 ∈ (Met‘𝐻) |
7 | simpl 485 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ (Cau‘𝐷)) | |
8 | 2, 3, 5 | hhssims2 29054 | . . . . . . . . . . 11 ⊢ 𝐷 = ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)) |
9 | 8 | fveq2i 6675 | . . . . . . . . . 10 ⊢ (Cau‘𝐷) = (Cau‘((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻))) |
10 | 7, 9 | eleqtrdi 2925 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ (Cau‘((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)))) |
11 | eqid 2823 | . . . . . . . . . . 11 ⊢ (normℎ ∘ −ℎ ) = (normℎ ∘ −ℎ ) | |
12 | 11 | hilxmet 28974 | . . . . . . . . . 10 ⊢ (normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) |
13 | simpr 487 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓:ℕ⟶𝐻) | |
14 | causs 23903 | . . . . . . . . . 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 29010 | . . . . . . . . . 10 ⊢ 𝐻 ⊆ ℋ |
18 | fss 6529 | . . . . . . . . . 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 28778 | . . . . . . . . . 10 ⊢ ℋ ∈ V | |
21 | nnex 11646 | . . . . . . . . . 10 ⊢ ℕ ∈ V | |
22 | 20, 21 | elmap 8437 | . . . . . . . . 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 2823 | . . . . . . . . . 10 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
25 | 24, 11 | hhims 28951 | . . . . . . . . . 10 ⊢ (normℎ ∘ −ℎ ) = (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
26 | 24, 25 | hhcau 28977 | . . . . . . . . 9 ⊢ Cauchy = ((Cau‘(normℎ ∘ −ℎ )) ∩ ( ℋ ↑m ℕ)) |
27 | 26 | elin2 4176 | . . . . . . . 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 28981 | . . . . . . 7 ⊢ (𝑓 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝑓 ⇝𝑣 𝑥) | |
30 | vex 3499 | . . . . . . . . 9 ⊢ 𝑓 ∈ V | |
31 | vex 3499 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
32 | 30, 31 | breldm 5779 | . . . . . . . 8 ⊢ (𝑓 ⇝𝑣 𝑥 → 𝑓 ∈ dom ⇝𝑣 ) |
33 | 32 | rexlimivw 3284 | . . . . . . 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 29016 | . . . . . . 7 ⊢ ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ | |
36 | ffun 6519 | . . . . . . 7 ⊢ ( ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ → Fun ⇝𝑣 ) | |
37 | funfvbrb 6823 | . . . . . . 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 2823 | . . . . . . . 8 ⊢ (MetOpen‘(normℎ ∘ −ℎ )) = (MetOpen‘(normℎ ∘ −ℎ )) | |
41 | 24, 25, 40 | hhlm 28978 | . . . . . . 7 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ)) |
42 | resss 5880 | . . . . . . 7 ⊢ ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ)) ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) | |
43 | 41, 42 | eqsstri 4003 | . . . . . 6 ⊢ ⇝𝑣 ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) |
44 | 43 | ssbri 5113 | . . . . 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 23136 | . . . . . . 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 2832 | . . . . 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 23052 | . . . . . 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 6685 | . . . . . . 7 ⊢ ( ⇝𝑣 ‘𝑓) ∈ V | |
54 | 53 | chlimi 29013 | . . . . . 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 21908 | . . . 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 5779 | . . 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 23917 | 1 ⊢ 𝐷 ∈ (CMet‘𝐻) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 ⊆ wss 3938 〈cop 4575 class class class wbr 5068 × cxp 5555 dom cdm 5557 ↾ cres 5559 ∘ ccom 5561 Fun wfun 6351 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 ℂcc 10537 1c1 10540 ℕcn 11640 ↾t crest 16696 ∞Metcxmet 20532 MetOpencmopn 20537 Topctop 21503 ⇝𝑡clm 21836 Cauccau 23858 CMetccmet 23859 IndMetcims 28370 ℋchba 28698 +ℎ cva 28699 ·ℎ csm 28700 normℎcno 28702 −ℎ cmv 28704 Cauchyccauold 28705 ⇝𝑣 chli 28706 Cℋ cch 28708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cc 9859 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 ax-hilex 28778 ax-hfvadd 28779 ax-hvcom 28780 ax-hvass 28781 ax-hv0cl 28782 ax-hvaddid 28783 ax-hfvmul 28784 ax-hvmulid 28785 ax-hvmulass 28786 ax-hvdistr1 28787 ax-hvdistr2 28788 ax-hvmul0 28789 ax-hfi 28858 ax-his1 28861 ax-his2 28862 ax-his3 28863 ax-his4 28864 ax-hcompl 28981 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 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 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-rlim 14848 df-rest 16698 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-top 21504 df-topon 21521 df-bases 21556 df-ntr 21630 df-nei 21708 df-lm 21839 df-haus 21925 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-cfil 23860 df-cau 23861 df-cmet 23862 df-grpo 28272 df-gid 28273 df-ginv 28274 df-gdiv 28275 df-ablo 28324 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-vs 28378 df-nmcv 28379 df-ims 28380 df-ssp 28501 df-hnorm 28747 df-hba 28748 df-hvsub 28750 df-hlim 28751 df-hcau 28752 df-sh 28986 df-ch 29000 df-ch0 29032 |
This theorem is referenced by: hhssbnOLD 29058 |
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