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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 2732 | . . . . . . . 8 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
2 | eqid 2732 | . . . . . . . 8 ⊢ (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
3 | eqid 2732 | . . . . . . . 8 ⊢ (MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) = (MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) | |
4 | 1, 2, 3 | hhlm 30439 | . . . . . . 7 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ)) |
5 | resss 6004 | . . . . . . 7 ⊢ ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ)) ⊆ (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) | |
6 | 4, 5 | eqsstri 4015 | . . . . . 6 ⊢ ⇝𝑣 ⊆ (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) |
7 | dmss 5900 | . . . . . 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 30415 | . . . . . 6 ⊢ (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∈ (∞Met‘ ℋ) |
10 | 3 | lmcau 24821 | . . . . . 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 3990 | . . . 4 ⊢ dom ⇝𝑣 ⊆ (Cau‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) |
13 | 4 | dmeqi 5902 | . . . . . 6 ⊢ dom ⇝𝑣 = dom ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ)) |
14 | dmres 6001 | . . . . . 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 2760 | . . . . 5 ⊢ dom ⇝𝑣 = (( ℋ ↑m ℕ) ∩ dom (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)))) |
16 | inss1 4227 | . . . . 5 ⊢ (( ℋ ↑m ℕ) ∩ dom (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)))) ⊆ ( ℋ ↑m ℕ) | |
17 | 15, 16 | eqsstri 4015 | . . . 4 ⊢ dom ⇝𝑣 ⊆ ( ℋ ↑m ℕ) |
18 | 12, 17 | ssini 4230 | . . 3 ⊢ dom ⇝𝑣 ⊆ ((Cau‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) ∩ ( ℋ ↑m ℕ)) |
19 | 1, 2 | hhcau 30438 | . . 3 ⊢ Cauchy = ((Cau‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) ∩ ( ℋ ↑m ℕ)) |
20 | 18, 19 | sseqtrri 4018 | . 2 ⊢ dom ⇝𝑣 ⊆ Cauchy |
21 | relres 6008 | . . . 4 ⊢ Rel ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ)) | |
22 | 4 | releqi 5775 | . . . 4 ⊢ (Rel ⇝𝑣 ↔ Rel ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ))) |
23 | 21, 22 | mpbir 230 | . . 3 ⊢ Rel ⇝𝑣 |
24 | 23 | releldmi 5945 | . 2 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ dom ⇝𝑣 ) |
25 | 20, 24 | sselid 3979 | 1 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ Cauchy) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∩ cin 3946 ⊆ wss 3947 〈cop 4633 class class class wbr 5147 dom cdm 5675 ↾ cres 5677 Rel wrel 5680 ‘cfv 6540 (class class class)co 7405 ↑m cmap 8816 ℕcn 12208 ∞Metcxmet 20921 MetOpencmopn 20926 ⇝𝑡clm 22721 Cauccau 24761 IndMetcims 29831 ℋchba 30159 +ℎ cva 30160 ·ℎ csm 30161 normℎcno 30163 Cauchyccauold 30166 ⇝𝑣 chli 30167 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 ax-hilex 30239 ax-hfvadd 30240 ax-hvcom 30241 ax-hvass 30242 ax-hv0cl 30243 ax-hvaddid 30244 ax-hfvmul 30245 ax-hvmulid 30246 ax-hvmulass 30247 ax-hvdistr1 30248 ax-hvdistr2 30249 ax-hvmul0 30250 ax-hfi 30319 ax-his1 30322 ax-his2 30323 ax-his3 30324 ax-his4 30325 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-icc 13327 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-topgen 17385 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-top 22387 df-topon 22404 df-bases 22440 df-lm 22724 df-haus 22810 df-cau 24764 df-grpo 29733 df-gid 29734 df-ginv 29735 df-gdiv 29736 df-ablo 29785 df-vc 29799 df-nv 29832 df-va 29835 df-ba 29836 df-sm 29837 df-0v 29838 df-vs 29839 df-nmcv 29840 df-ims 29841 df-hnorm 30208 df-hvsub 30211 df-hlim 30212 df-hcau 30213 |
This theorem is referenced by: isch3 30481 chscllem2 30878 |
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