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Mirrors > Home > MPE Home > Th. List > cmetcau | Structured version Visualization version GIF version |
Description: The convergence of a Cauchy sequence in a complete metric space. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Oct-2015.) |
Ref | Expression |
---|---|
cmetcau.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
Ref | Expression |
---|---|
cmetcau | ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ dom (⇝𝑡‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmetmet 23890 | . . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋)) | |
2 | metxmet 22941 | . . . . 5 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) |
4 | caun0 23885 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝑋 ≠ ∅) | |
5 | 3, 4 | sylan 583 | . . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝑋 ≠ ∅) |
6 | n0 4260 | . . 3 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑋) | |
7 | 5, 6 | sylib 221 | . 2 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → ∃𝑥 𝑥 ∈ 𝑋) |
8 | cmetcau.1 | . . 3 ⊢ 𝐽 = (MetOpen‘𝐷) | |
9 | simpll 766 | . . 3 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ (CMet‘𝑋)) | |
10 | simpr 488 | . . 3 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) | |
11 | simplr 768 | . . 3 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) ∧ 𝑥 ∈ 𝑋) → 𝐹 ∈ (Cau‘𝐷)) | |
12 | eqid 2798 | . . 3 ⊢ (𝑦 ∈ ℕ ↦ if(𝑦 ∈ dom 𝐹, (𝐹‘𝑦), 𝑥)) = (𝑦 ∈ ℕ ↦ if(𝑦 ∈ dom 𝐹, (𝐹‘𝑦), 𝑥)) | |
13 | 8, 9, 10, 11, 12 | cmetcaulem 23892 | . 2 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) ∧ 𝑥 ∈ 𝑋) → 𝐹 ∈ dom (⇝𝑡‘𝐽)) |
14 | 7, 13 | exlimddv 1936 | 1 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ dom (⇝𝑡‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 ifcif 4425 ↦ cmpt 5110 dom cdm 5519 ‘cfv 6324 ℕcn 11625 ∞Metcxmet 20076 Metcmet 20077 MetOpencmopn 20081 ⇝𝑡clm 21831 Cauccau 23857 CMetccmet 23858 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 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-ico 12732 df-rest 16688 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-top 21499 df-topon 21516 df-bases 21551 df-ntr 21625 df-nei 21703 df-lm 21834 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-cfil 23859 df-cau 23860 df-cmet 23861 |
This theorem is referenced by: iscmet3 23897 iscmet2 23898 bcthlem4 23931 minvecolem4a 28660 hlcompl 28698 heiborlem9 35257 bfplem1 35260 |
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