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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 23303 | . . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋)) | |
2 | metxmet 22359 | . . . . 5 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) |
4 | caun0 23298 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝑋 ≠ ∅) | |
5 | 3, 4 | sylan 569 | . . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝑋 ≠ ∅) |
6 | n0 4079 | . . 3 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑋) | |
7 | 5, 6 | sylib 208 | . 2 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → ∃𝑥 𝑥 ∈ 𝑋) |
8 | cmetcau.1 | . . 3 ⊢ 𝐽 = (MetOpen‘𝐷) | |
9 | simpll 750 | . . 3 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ (CMet‘𝑋)) | |
10 | simpr 471 | . . 3 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) | |
11 | simplr 752 | . . 3 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) ∧ 𝑥 ∈ 𝑋) → 𝐹 ∈ (Cau‘𝐷)) | |
12 | eqid 2771 | . . 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 23305 | . 2 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) ∧ 𝑥 ∈ 𝑋) → 𝐹 ∈ dom (⇝𝑡‘𝐽)) |
14 | 7, 13 | exlimddv 2015 | 1 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ dom (⇝𝑡‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∃wex 1852 ∈ wcel 2145 ≠ wne 2943 ∅c0 4063 ifcif 4226 ↦ cmpt 4864 dom cdm 5250 ‘cfv 6030 ℕcn 11226 ∞Metcxmt 19946 Metcme 19947 MetOpencmopn 19951 ⇝𝑡clm 21251 Caucca 23270 CMetcms 23271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-er 7900 df-map 8015 df-pm 8016 df-en 8114 df-dom 8115 df-sdom 8116 df-sup 8508 df-inf 8509 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-n0 11500 df-z 11585 df-uz 11894 df-q 11997 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-ico 12386 df-rest 16291 df-topgen 16312 df-psmet 19953 df-xmet 19954 df-met 19955 df-bl 19956 df-mopn 19957 df-fbas 19958 df-fg 19959 df-top 20919 df-topon 20936 df-bases 20971 df-ntr 21045 df-nei 21123 df-lm 21254 df-fil 21870 df-fm 21962 df-flim 21963 df-flf 21964 df-cfil 23272 df-cau 23273 df-cmet 23274 |
This theorem is referenced by: iscmet3 23310 iscmet2 23311 bcthlem4 23343 minvecolem4a 28073 hlcompl 28111 heiborlem9 33950 bfplem1 33953 |
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