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Mirrors > Home > HSE Home > Th. List > hhcms | Structured version Visualization version GIF version |
Description: The Hilbert space induced metric determines a complete metric space. (Contributed by NM, 10-Apr-2008.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhcms.1 | ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
hhcms.2 | ⊢ 𝐷 = (IndMet‘𝑈) |
Ref | Expression |
---|---|
hhcms | ⊢ 𝐷 ∈ (CMet‘ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . 2 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
2 | hhcms.1 | . . 3 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
3 | hhcms.2 | . . 3 ⊢ 𝐷 = (IndMet‘𝑈) | |
4 | 2, 3 | hhmet 28031 | . 2 ⊢ 𝐷 ∈ (Met‘ ℋ) |
5 | 2, 3 | hhcau 28055 | . . . . . 6 ⊢ Cauchy = ((Cau‘𝐷) ∩ ( ℋ ↑𝑚 ℕ)) |
6 | 5 | eleq2i 2693 | . . . . 5 ⊢ (𝑓 ∈ Cauchy ↔ 𝑓 ∈ ((Cau‘𝐷) ∩ ( ℋ ↑𝑚 ℕ))) |
7 | elin 3796 | . . . . . 6 ⊢ (𝑓 ∈ ((Cau‘𝐷) ∩ ( ℋ ↑𝑚 ℕ)) ↔ (𝑓 ∈ (Cau‘𝐷) ∧ 𝑓 ∈ ( ℋ ↑𝑚 ℕ))) | |
8 | ax-hilex 27856 | . . . . . . . 8 ⊢ ℋ ∈ V | |
9 | nnex 11026 | . . . . . . . 8 ⊢ ℕ ∈ V | |
10 | 8, 9 | elmap 7886 | . . . . . . 7 ⊢ (𝑓 ∈ ( ℋ ↑𝑚 ℕ) ↔ 𝑓:ℕ⟶ ℋ) |
11 | 10 | anbi2i 730 | . . . . . 6 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓 ∈ ( ℋ ↑𝑚 ℕ)) ↔ (𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶ ℋ)) |
12 | 7, 11 | bitri 264 | . . . . 5 ⊢ (𝑓 ∈ ((Cau‘𝐷) ∩ ( ℋ ↑𝑚 ℕ)) ↔ (𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶ ℋ)) |
13 | 6, 12 | bitri 264 | . . . 4 ⊢ (𝑓 ∈ Cauchy ↔ (𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶ ℋ)) |
14 | ax-hcompl 28059 | . . . 4 ⊢ (𝑓 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝑓 ⇝𝑣 𝑥) | |
15 | 13, 14 | sylbir 225 | . . 3 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶ ℋ) → ∃𝑥 ∈ ℋ 𝑓 ⇝𝑣 𝑥) |
16 | 2, 3, 1 | hhlm 28056 | . . . . . . 7 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘𝐷)) ↾ ( ℋ ↑𝑚 ℕ)) |
17 | 16 | breqi 4659 | . . . . . 6 ⊢ (𝑓 ⇝𝑣 𝑥 ↔ 𝑓((⇝𝑡‘(MetOpen‘𝐷)) ↾ ( ℋ ↑𝑚 ℕ))𝑥) |
18 | vex 3203 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
19 | 18 | brres 5402 | . . . . . 6 ⊢ (𝑓((⇝𝑡‘(MetOpen‘𝐷)) ↾ ( ℋ ↑𝑚 ℕ))𝑥 ↔ (𝑓(⇝𝑡‘(MetOpen‘𝐷))𝑥 ∧ 𝑓 ∈ ( ℋ ↑𝑚 ℕ))) |
20 | 17, 19 | bitri 264 | . . . . 5 ⊢ (𝑓 ⇝𝑣 𝑥 ↔ (𝑓(⇝𝑡‘(MetOpen‘𝐷))𝑥 ∧ 𝑓 ∈ ( ℋ ↑𝑚 ℕ))) |
21 | vex 3203 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
22 | 21, 18 | breldm 5329 | . . . . . 6 ⊢ (𝑓(⇝𝑡‘(MetOpen‘𝐷))𝑥 → 𝑓 ∈ dom (⇝𝑡‘(MetOpen‘𝐷))) |
23 | 22 | adantr 481 | . . . . 5 ⊢ ((𝑓(⇝𝑡‘(MetOpen‘𝐷))𝑥 ∧ 𝑓 ∈ ( ℋ ↑𝑚 ℕ)) → 𝑓 ∈ dom (⇝𝑡‘(MetOpen‘𝐷))) |
24 | 20, 23 | sylbi 207 | . . . 4 ⊢ (𝑓 ⇝𝑣 𝑥 → 𝑓 ∈ dom (⇝𝑡‘(MetOpen‘𝐷))) |
25 | 24 | rexlimivw 3029 | . . 3 ⊢ (∃𝑥 ∈ ℋ 𝑓 ⇝𝑣 𝑥 → 𝑓 ∈ dom (⇝𝑡‘(MetOpen‘𝐷))) |
26 | 15, 25 | syl 17 | . 2 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶ ℋ) → 𝑓 ∈ dom (⇝𝑡‘(MetOpen‘𝐷))) |
27 | 1, 4, 26 | iscmet3i 23110 | 1 ⊢ 𝐷 ∈ (CMet‘ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ∩ cin 3573 〈cop 4183 class class class wbr 4653 dom cdm 5114 ↾ cres 5116 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 ℕcn 11020 MetOpencmopn 19736 ⇝𝑡clm 21030 Caucca 23051 CMetcms 23052 IndMetcims 27446 ℋchil 27776 +ℎ cva 27777 ·ℎ csm 27778 normℎcno 27780 Cauchyccau 27783 ⇝𝑣 chli 27784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 ax-hilex 27856 ax-hfvadd 27857 ax-hvcom 27858 ax-hvass 27859 ax-hv0cl 27860 ax-hvaddid 27861 ax-hfvmul 27862 ax-hvmulid 27863 ax-hvmulass 27864 ax-hvdistr1 27865 ax-hvdistr2 27866 ax-hvmul0 27867 ax-hfi 27936 ax-his1 27939 ax-his2 27940 ax-his3 27941 ax-his4 27942 ax-hcompl 28059 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ico 12181 df-fz 12327 df-fl 12593 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-rest 16083 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-top 20699 df-topon 20716 df-bases 20750 df-ntr 20824 df-nei 20902 df-lm 21033 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-cfil 23053 df-cau 23054 df-cmet 23055 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ims 27456 df-hnorm 27825 df-hvsub 27828 df-hlim 27829 df-hcau 27830 |
This theorem is referenced by: hhhl 28061 hilcms 28062 |
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