![]() |
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 2734 | . . . . . . . 8 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
2 | eqid 2734 | . . . . . . . 8 ⊢ (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
3 | eqid 2734 | . . . . . . . 8 ⊢ (MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) = (MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) | |
4 | 1, 2, 3 | hhlm 31227 | . . . . . . 7 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ)) |
5 | resss 6021 | . . . . . . 7 ⊢ ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ)) ⊆ (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) | |
6 | 4, 5 | eqsstri 4029 | . . . . . 6 ⊢ ⇝𝑣 ⊆ (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) |
7 | dmss 5915 | . . . . . 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 31203 | . . . . . 6 ⊢ (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∈ (∞Met‘ ℋ) |
10 | 3 | lmcau 25360 | . . . . . 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 4004 | . . . 4 ⊢ dom ⇝𝑣 ⊆ (Cau‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) |
13 | 4 | dmeqi 5917 | . . . . . 6 ⊢ dom ⇝𝑣 = dom ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ)) |
14 | dmres 6031 | . . . . . 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 2762 | . . . . 5 ⊢ dom ⇝𝑣 = (( ℋ ↑m ℕ) ∩ dom (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)))) |
16 | inss1 4244 | . . . . 5 ⊢ (( ℋ ↑m ℕ) ∩ dom (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)))) ⊆ ( ℋ ↑m ℕ) | |
17 | 15, 16 | eqsstri 4029 | . . . 4 ⊢ dom ⇝𝑣 ⊆ ( ℋ ↑m ℕ) |
18 | 12, 17 | ssini 4247 | . . 3 ⊢ dom ⇝𝑣 ⊆ ((Cau‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) ∩ ( ℋ ↑m ℕ)) |
19 | 1, 2 | hhcau 31226 | . . 3 ⊢ Cauchy = ((Cau‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) ∩ ( ℋ ↑m ℕ)) |
20 | 18, 19 | sseqtrri 4032 | . 2 ⊢ dom ⇝𝑣 ⊆ Cauchy |
21 | relres 6025 | . . . 4 ⊢ Rel ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ)) | |
22 | 4 | releqi 5789 | . . . 4 ⊢ (Rel ⇝𝑣 ↔ Rel ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ))) |
23 | 21, 22 | mpbir 231 | . . 3 ⊢ Rel ⇝𝑣 |
24 | 23 | releldmi 5961 | . 2 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ dom ⇝𝑣 ) |
25 | 20, 24 | sselid 3992 | 1 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ Cauchy) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∩ cin 3961 ⊆ wss 3962 〈cop 4636 class class class wbr 5147 dom cdm 5688 ↾ cres 5690 Rel wrel 5693 ‘cfv 6562 (class class class)co 7430 ↑m cmap 8864 ℕcn 12263 ∞Metcxmet 21366 MetOpencmopn 21371 ⇝𝑡clm 23249 Cauccau 25300 IndMetcims 30619 ℋchba 30947 +ℎ cva 30948 ·ℎ csm 30949 normℎcno 30951 Cauchyccauold 30954 ⇝𝑣 chli 30955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-mulf 11232 ax-hilex 31027 ax-hfvadd 31028 ax-hvcom 31029 ax-hvass 31030 ax-hv0cl 31031 ax-hvaddid 31032 ax-hfvmul 31033 ax-hvmulid 31034 ax-hvmulass 31035 ax-hvdistr1 31036 ax-hvdistr2 31037 ax-hvmul0 31038 ax-hfi 31107 ax-his1 31110 ax-his2 31111 ax-his3 31112 ax-his4 31113 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-map 8866 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-n0 12524 df-z 12611 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-icc 13390 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-topgen 17489 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-top 22915 df-topon 22932 df-bases 22968 df-lm 23252 df-haus 23338 df-cau 25303 df-grpo 30521 df-gid 30522 df-ginv 30523 df-gdiv 30524 df-ablo 30573 df-vc 30587 df-nv 30620 df-va 30623 df-ba 30624 df-sm 30625 df-0v 30626 df-vs 30627 df-nmcv 30628 df-ims 30629 df-hnorm 30996 df-hvsub 30999 df-hlim 31000 df-hcau 31001 |
This theorem is referenced by: isch3 31269 chscllem2 31666 |
Copyright terms: Public domain | W3C validator |