Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > hlimcaui | Structured version Visualization version GIF version |
Description: If a sequence in Hilbert space subset converges to a limit, it is a Cauchy sequence. (Contributed by NM, 17-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlimcaui | ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ Cauchy) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . . . . . 8 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
2 | eqid 2818 | . . . . . . . 8 ⊢ (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
3 | eqid 2818 | . . . . . . . 8 ⊢ (MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) = (MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) | |
4 | 1, 2, 3 | hhlm 28903 | . . . . . . 7 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ)) |
5 | resss 5871 | . . . . . . 7 ⊢ ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ)) ⊆ (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) | |
6 | 4, 5 | eqsstri 3998 | . . . . . 6 ⊢ ⇝𝑣 ⊆ (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) |
7 | dmss 5764 | . . . . . 6 ⊢ ( ⇝𝑣 ⊆ (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) → dom ⇝𝑣 ⊆ dom (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)))) | |
8 | 6, 7 | ax-mp 5 | . . . . 5 ⊢ dom ⇝𝑣 ⊆ dom (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) |
9 | 1, 2 | hhxmet 28879 | . . . . . 6 ⊢ (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∈ (∞Met‘ ℋ) |
10 | 3 | lmcau 23843 | . . . . . 6 ⊢ ((IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∈ (∞Met‘ ℋ) → dom (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ⊆ (Cau‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) |
11 | 9, 10 | ax-mp 5 | . . . . 5 ⊢ dom (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ⊆ (Cau‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) |
12 | 8, 11 | sstri 3973 | . . . 4 ⊢ dom ⇝𝑣 ⊆ (Cau‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) |
13 | 4 | dmeqi 5766 | . . . . . 6 ⊢ dom ⇝𝑣 = dom ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ)) |
14 | dmres 5868 | . . . . . 6 ⊢ dom ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ)) = (( ℋ ↑m ℕ) ∩ dom (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)))) | |
15 | 13, 14 | eqtri 2841 | . . . . 5 ⊢ dom ⇝𝑣 = (( ℋ ↑m ℕ) ∩ dom (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)))) |
16 | inss1 4202 | . . . . 5 ⊢ (( ℋ ↑m ℕ) ∩ dom (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)))) ⊆ ( ℋ ↑m ℕ) | |
17 | 15, 16 | eqsstri 3998 | . . . 4 ⊢ dom ⇝𝑣 ⊆ ( ℋ ↑m ℕ) |
18 | 12, 17 | ssini 4205 | . . 3 ⊢ dom ⇝𝑣 ⊆ ((Cau‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) ∩ ( ℋ ↑m ℕ)) |
19 | 1, 2 | hhcau 28902 | . . 3 ⊢ Cauchy = ((Cau‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) ∩ ( ℋ ↑m ℕ)) |
20 | 18, 19 | sseqtrri 4001 | . 2 ⊢ dom ⇝𝑣 ⊆ Cauchy |
21 | relres 5875 | . . . 4 ⊢ Rel ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ)) | |
22 | 4 | releqi 5645 | . . . 4 ⊢ (Rel ⇝𝑣 ↔ Rel ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ))) |
23 | 21, 22 | mpbir 232 | . . 3 ⊢ Rel ⇝𝑣 |
24 | 23 | releldmi 5811 | . 2 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ dom ⇝𝑣 ) |
25 | 20, 24 | sseldi 3962 | 1 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ Cauchy) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∩ cin 3932 ⊆ wss 3933 〈cop 4563 class class class wbr 5057 dom cdm 5548 ↾ cres 5550 Rel wrel 5553 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 ℕcn 11626 ∞Metcxmet 20458 MetOpencmopn 20463 ⇝𝑡clm 21762 Cauccau 23783 IndMetcims 28295 ℋchba 28623 +ℎ cva 28624 ·ℎ csm 28625 normℎcno 28627 Cauchyccauold 28630 ⇝𝑣 chli 28631 |
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-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 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 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-iun 4912 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-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-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 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-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-icc 12733 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-topgen 16705 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-top 21430 df-topon 21447 df-bases 21482 df-lm 21765 df-haus 21851 df-cau 23786 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-hnorm 28672 df-hvsub 28675 df-hlim 28676 df-hcau 28677 |
This theorem is referenced by: isch3 28945 chscllem2 29342 |
Copyright terms: Public domain | W3C validator |