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Theorem cmetcusp1 21769
Description: If the uniform set of a complete metric space is the uniform structure generated by its metric, then it is a complete uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.)
Hypotheses
Ref Expression
cmetcusp1.x  |-  X  =  ( Base `  F
)
cmetcusp1.d  |-  D  =  ( ( dist `  F
)  |`  ( X  X.  X ) )
cmetcusp1.u  |-  U  =  (UnifSt `  F )
Assertion
Ref Expression
cmetcusp1  |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  ->  F  e. CUnifSp )

Proof of Theorem cmetcusp1
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 cmsms 21764 . . . 4  |-  ( F  e. CMetSp  ->  F  e.  MetSp )
2 msxms 20934 . . . 4  |-  ( F  e.  MetSp  ->  F  e.  *MetSp )
31, 2syl 16 . . 3  |-  ( F  e. CMetSp  ->  F  e.  *MetSp )
4 cmetcusp1.x . . . 4  |-  X  =  ( Base `  F
)
5 cmetcusp1.d . . . 4  |-  D  =  ( ( dist `  F
)  |`  ( X  X.  X ) )
6 cmetcusp1.u . . . 4  |-  U  =  (UnifSt `  F )
74, 5, 6xmsusp 21066 . . 3  |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnif `  D )
)  ->  F  e. UnifSp )
83, 7syl3an2 1263 . 2  |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  ->  F  e. UnifSp )
9 simpl3 1002 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  ->  U  =  (metUnif `  D
) )
109fveq2d 5860 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
(CauFilu
`  U )  =  (CauFilu `  (metUnif `  D
) ) )
1110eleq2d 2513 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
( c  e.  (CauFilu `  U )  <->  c  e.  (CauFilu `  (metUnif `  D )
) ) )
12 simpl1 1000 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  ->  X  =/=  (/) )
134, 5cmscmet 21762 . . . . . . . . 9  |-  ( F  e. CMetSp  ->  D  e.  (
CMet `  X )
)
14 cmetmet 21702 . . . . . . . . 9  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
15 metxmet 20814 . . . . . . . . 9  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
1613, 14, 153syl 20 . . . . . . . 8  |-  ( F  e. CMetSp  ->  D  e.  ( *Met `  X
) )
17163ad2ant2 1019 . . . . . . 7  |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  ->  D  e.  ( *Met `  X ) )
1817adantr 465 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  ->  D  e.  ( *Met `  X ) )
19 simpr 461 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
c  e.  ( Fil `  X ) )
20 cfilucfil4 21737 . . . . . 6  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X )  /\  c  e.  ( Fil `  X
) )  ->  (
c  e.  (CauFilu `  (metUnif `  D ) )  <->  c  e.  (CauFil `  D ) ) )
2112, 18, 19, 20syl3anc 1229 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
( c  e.  (CauFilu `  (metUnif `  D )
)  <->  c  e.  (CauFil `  D ) ) )
2211, 21bitrd 253 . . . 4  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
( c  e.  (CauFilu `  U )  <->  c  e.  (CauFil `  D ) ) )
23 eqid 2443 . . . . . . . . . . . 12  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
2423iscmet 21700 . . . . . . . . . . 11  |-  ( D  e.  ( CMet `  X
)  <->  ( D  e.  ( Met `  X
)  /\  A. c  e.  (CauFil `  D )
( ( MetOpen `  D
)  fLim  c )  =/=  (/) ) )
2524simprbi 464 . . . . . . . . . 10  |-  ( D  e.  ( CMet `  X
)  ->  A. c  e.  (CauFil `  D )
( ( MetOpen `  D
)  fLim  c )  =/=  (/) )
2613, 25syl 16 . . . . . . . . 9  |-  ( F  e. CMetSp  ->  A. c  e.  (CauFil `  D ) ( (
MetOpen `  D )  fLim  c )  =/=  (/) )
27 eqid 2443 . . . . . . . . . . . . . 14  |-  ( TopOpen `  F )  =  (
TopOpen `  F )
2827, 4, 5xmstopn 20931 . . . . . . . . . . . . 13  |-  ( F  e.  *MetSp  ->  ( TopOpen
`  F )  =  ( MetOpen `  D )
)
293, 28syl 16 . . . . . . . . . . . 12  |-  ( F  e. CMetSp  ->  ( TopOpen `  F
)  =  ( MetOpen `  D ) )
3029oveq1d 6296 . . . . . . . . . . 11  |-  ( F  e. CMetSp  ->  ( ( TopOpen `  F )  fLim  c
)  =  ( (
MetOpen `  D )  fLim  c ) )
3130neeq1d 2720 . . . . . . . . . 10  |-  ( F  e. CMetSp  ->  ( ( (
TopOpen `  F )  fLim  c )  =/=  (/)  <->  ( ( MetOpen
`  D )  fLim  c )  =/=  (/) ) )
3231ralbidv 2882 . . . . . . . . 9  |-  ( F  e. CMetSp  ->  ( A. c  e.  (CauFil `  D )
( ( TopOpen `  F
)  fLim  c )  =/=  (/)  <->  A. c  e.  (CauFil `  D ) ( (
MetOpen `  D )  fLim  c )  =/=  (/) ) )
3326, 32mpbird 232 . . . . . . . 8  |-  ( F  e. CMetSp  ->  A. c  e.  (CauFil `  D ) ( (
TopOpen `  F )  fLim  c )  =/=  (/) )
3433r19.21bi 2812 . . . . . . 7  |-  ( ( F  e. CMetSp  /\  c  e.  (CauFil `  D )
)  ->  ( ( TopOpen
`  F )  fLim  c )  =/=  (/) )
3534ex 434 . . . . . 6  |-  ( F  e. CMetSp  ->  ( c  e.  (CauFil `  D )  ->  ( ( TopOpen `  F
)  fLim  c )  =/=  (/) ) )
36353ad2ant2 1019 . . . . 5  |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  -> 
( c  e.  (CauFil `  D )  ->  (
( TopOpen `  F )  fLim  c )  =/=  (/) ) )
3736adantr 465 . . . 4  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
( c  e.  (CauFil `  D )  ->  (
( TopOpen `  F )  fLim  c )  =/=  (/) ) )
3822, 37sylbid 215 . . 3  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
( c  e.  (CauFilu `  U )  ->  (
( TopOpen `  F )  fLim  c )  =/=  (/) ) )
3938ralrimiva 2857 . 2  |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  ->  A. c  e.  ( Fil `  X ) ( c  e.  (CauFilu `  U
)  ->  ( ( TopOpen
`  F )  fLim  c )  =/=  (/) ) )
404, 6, 27iscusp2 20782 . 2  |-  ( F  e. CUnifSp 
<->  ( F  e. UnifSp  /\  A. c  e.  ( Fil `  X ) ( c  e.  (CauFilu `  U )  -> 
( ( TopOpen `  F
)  fLim  c )  =/=  (/) ) ) )
418, 39, 40sylanbrc 664 1  |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  ->  F  e. CUnifSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   (/)c0 3770    X. cxp 4987    |` cres 4991   ` cfv 5578  (class class class)co 6281   Basecbs 14613   distcds 14687   TopOpenctopn 14800   *Metcxmt 18381   Metcme 18382   MetOpencmopn 18386  metUnifcmetu 18388   Filcfil 20323    fLim cflim 20412  UnifStcuss 20733  UnifSpcusp 20734  CauFiluccfilu 20766  CUnifSpccusp 20777   *MetSpcxme 20797   MetSpcmt 20798  CauFilccfil 21668   CMetcms 21670  CMetSpccms 21748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11092  df-q 11193  df-rp 11231  df-xneg 11328  df-xadd 11329  df-xmul 11330  df-ico 11545  df-topgen 14822  df-psmet 18389  df-xmet 18390  df-met 18391  df-bl 18392  df-mopn 18393  df-fbas 18394  df-fg 18395  df-metu 18397  df-top 19376  df-bases 19378  df-topon 19379  df-topsp 19380  df-fil 20324  df-ust 20680  df-utop 20711  df-usp 20737  df-cfilu 20767  df-cusp 20778  df-xms 20800  df-ms 20801  df-cfil 21671  df-cmet 21673  df-cms 21751
This theorem is referenced by:  cnfldcusp  21774  recusp  21791
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