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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 2737 | . . . . . . . 8 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
2 | eqid 2737 | . . . . . . . 8 ⊢ (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
3 | eqid 2737 | . . . . . . . 8 ⊢ (MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) = (MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) | |
4 | 1, 2, 3 | hhlm 29280 | . . . . . . 7 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ)) |
5 | resss 5876 | . . . . . . 7 ⊢ ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ)) ⊆ (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) | |
6 | 4, 5 | eqsstri 3935 | . . . . . 6 ⊢ ⇝𝑣 ⊆ (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) |
7 | dmss 5771 | . . . . . 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 29256 | . . . . . 6 ⊢ (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∈ (∞Met‘ ℋ) |
10 | 3 | lmcau 24210 | . . . . . 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 3910 | . . . 4 ⊢ dom ⇝𝑣 ⊆ (Cau‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) |
13 | 4 | dmeqi 5773 | . . . . . 6 ⊢ dom ⇝𝑣 = dom ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ)) |
14 | dmres 5873 | . . . . . 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 2765 | . . . . 5 ⊢ dom ⇝𝑣 = (( ℋ ↑m ℕ) ∩ dom (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)))) |
16 | inss1 4143 | . . . . 5 ⊢ (( ℋ ↑m ℕ) ∩ dom (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)))) ⊆ ( ℋ ↑m ℕ) | |
17 | 15, 16 | eqsstri 3935 | . . . 4 ⊢ dom ⇝𝑣 ⊆ ( ℋ ↑m ℕ) |
18 | 12, 17 | ssini 4146 | . . 3 ⊢ dom ⇝𝑣 ⊆ ((Cau‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) ∩ ( ℋ ↑m ℕ)) |
19 | 1, 2 | hhcau 29279 | . . 3 ⊢ Cauchy = ((Cau‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) ∩ ( ℋ ↑m ℕ)) |
20 | 18, 19 | sseqtrri 3938 | . 2 ⊢ dom ⇝𝑣 ⊆ Cauchy |
21 | relres 5880 | . . . 4 ⊢ Rel ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ)) | |
22 | 4 | releqi 5649 | . . . 4 ⊢ (Rel ⇝𝑣 ↔ Rel ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑m ℕ))) |
23 | 21, 22 | mpbir 234 | . . 3 ⊢ Rel ⇝𝑣 |
24 | 23 | releldmi 5817 | . 2 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ dom ⇝𝑣 ) |
25 | 20, 24 | sseldi 3899 | 1 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ Cauchy) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∩ cin 3865 ⊆ wss 3866 〈cop 4547 class class class wbr 5053 dom cdm 5551 ↾ cres 5553 Rel wrel 5556 ‘cfv 6380 (class class class)co 7213 ↑m cmap 8508 ℕcn 11830 ∞Metcxmet 20348 MetOpencmopn 20353 ⇝𝑡clm 22123 Cauccau 24150 IndMetcims 28672 ℋchba 29000 +ℎ cva 29001 ·ℎ csm 29002 normℎcno 29004 Cauchyccauold 29007 ⇝𝑣 chli 29008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809 ax-hilex 29080 ax-hfvadd 29081 ax-hvcom 29082 ax-hvass 29083 ax-hv0cl 29084 ax-hvaddid 29085 ax-hfvmul 29086 ax-hvmulid 29087 ax-hvmulass 29088 ax-hvdistr1 29089 ax-hvdistr2 29090 ax-hvmul0 29091 ax-hfi 29160 ax-his1 29163 ax-his2 29164 ax-his3 29165 ax-his4 29166 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-map 8510 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-sup 9058 df-inf 9059 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-n0 12091 df-z 12177 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-icc 12942 df-seq 13575 df-exp 13636 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-topgen 16948 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-top 21791 df-topon 21808 df-bases 21843 df-lm 22126 df-haus 22212 df-cau 24153 df-grpo 28574 df-gid 28575 df-ginv 28576 df-gdiv 28577 df-ablo 28626 df-vc 28640 df-nv 28673 df-va 28676 df-ba 28677 df-sm 28678 df-0v 28679 df-vs 28680 df-nmcv 28681 df-ims 28682 df-hnorm 29049 df-hvsub 29052 df-hlim 29053 df-hcau 29054 |
This theorem is referenced by: isch3 29322 chscllem2 29719 |
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