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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 2610 | . . . . . . . 8 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
2 | eqid 2610 | . . . . . . . 8 ⊢ (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
3 | eqid 2610 | . . . . . . . 8 ⊢ (MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) = (MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) | |
4 | 1, 2, 3 | hhlm 27440 | . . . . . . 7 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑𝑚 ℕ)) |
5 | resss 5342 | . . . . . . 7 ⊢ ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑𝑚 ℕ)) ⊆ (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) | |
6 | 4, 5 | eqsstri 3598 | . . . . . 6 ⊢ ⇝𝑣 ⊆ (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) |
7 | dmss 5245 | . . . . . 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 27416 | . . . . . 6 ⊢ (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) ∈ (∞Met‘ ℋ) |
10 | 3 | lmcau 22919 | . . . . . 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 3577 | . . . 4 ⊢ dom ⇝𝑣 ⊆ (Cau‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) |
13 | 4 | dmeqi 5247 | . . . . . 6 ⊢ dom ⇝𝑣 = dom ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑𝑚 ℕ)) |
14 | dmres 5339 | . . . . . 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 2632 | . . . . 5 ⊢ dom ⇝𝑣 = (( ℋ ↑𝑚 ℕ) ∩ dom (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)))) |
16 | inss1 3795 | . . . . 5 ⊢ (( ℋ ↑𝑚 ℕ) ∩ dom (⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)))) ⊆ ( ℋ ↑𝑚 ℕ) | |
17 | 15, 16 | eqsstri 3598 | . . . 4 ⊢ dom ⇝𝑣 ⊆ ( ℋ ↑𝑚 ℕ) |
18 | 12, 17 | ssini 3798 | . . 3 ⊢ dom ⇝𝑣 ⊆ ((Cau‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) ∩ ( ℋ ↑𝑚 ℕ)) |
19 | 1, 2 | hhcau 27439 | . . 3 ⊢ Cauchy = ((Cau‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) ∩ ( ℋ ↑𝑚 ℕ)) |
20 | 18, 19 | sseqtr4i 3601 | . 2 ⊢ dom ⇝𝑣 ⊆ Cauchy |
21 | relres 5346 | . . . 4 ⊢ Rel ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑𝑚 ℕ)) | |
22 | 4 | releqi 5125 | . . . 4 ⊢ (Rel ⇝𝑣 ↔ Rel ((⇝𝑡‘(MetOpen‘(IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉))) ↾ ( ℋ ↑𝑚 ℕ))) |
23 | 21, 22 | mpbir 220 | . . 3 ⊢ Rel ⇝𝑣 |
24 | 23 | releldmi 5283 | . 2 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ dom ⇝𝑣 ) |
25 | 20, 24 | sseldi 3566 | 1 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ Cauchy) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 〈cop 4131 class class class wbr 4583 dom cdm 5038 ↾ cres 5040 Rel wrel 5043 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 ℕcn 10897 ∞Metcxmt 19552 MetOpencmopn 19557 ⇝𝑡clm 20840 Caucca 22859 IndMetcims 26830 ℋchil 27160 +ℎ cva 27161 ·ℎ csm 27162 normℎcno 27164 Cauchyccau 27167 ⇝𝑣 chli 27168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 ax-hilex 27240 ax-hfvadd 27241 ax-hvcom 27242 ax-hvass 27243 ax-hv0cl 27244 ax-hvaddid 27245 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvmulass 27248 ax-hvdistr1 27249 ax-hvdistr2 27250 ax-hvmul0 27251 ax-hfi 27320 ax-his1 27323 ax-his2 27324 ax-his3 27325 ax-his4 27326 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-icc 12053 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-lm 20843 df-haus 20929 df-cau 22862 df-grpo 26731 df-gid 26732 df-ginv 26733 df-gdiv 26734 df-ablo 26783 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-vs 26838 df-nmcv 26839 df-ims 26840 df-hnorm 27209 df-hvsub 27212 df-hlim 27213 df-hcau 27214 |
This theorem is referenced by: isch3 27482 chscllem2 27881 |
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