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Theorem cmetcusp1OLD 20875
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
cmetcusp1OLD  |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  ->  F  e. CUnifSp )

Proof of Theorem cmetcusp1OLD
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 cmsms 20871 . . . 4  |-  ( F  e. CMetSp  ->  F  e.  MetSp )
2 msxms 20041 . . . 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, 6xmsuspOLD 20172 . . 3  |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnifOLD
`  D ) )  ->  F  e. UnifSp )
83, 7syl3an2 1252 . 2  |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  ->  F  e. UnifSp )
9 simpl3 993 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  /\  c  e.  ( Fil `  X ) )  ->  U  =  (metUnifOLD
`  D ) )
109fveq2d 5707 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
(CauFilu
`  U )  =  (CauFilu `  (metUnifOLD
`  D ) ) )
1110eleq2d 2510 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
( c  e.  (CauFilu `  U )  <->  c  e.  (CauFilu `  (metUnifOLD
`  D ) ) ) )
12 simpl1 991 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  /\  c  e.  ( Fil `  X ) )  ->  X  =/=  (/) )
134, 5cmscmet 20869 . . . . . . . . 9  |-  ( F  e. CMetSp  ->  D  e.  (
CMet `  X )
)
14 cmetmet 20809 . . . . . . . . 9  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
15 metxmet 19921 . . . . . . . . 9  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
1613, 14, 153syl 20 . . . . . . . 8  |-  ( F  e. CMetSp  ->  D  e.  ( *Met `  X
) )
17163ad2ant2 1010 . . . . . . 7  |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  ->  D  e.  ( *Met `  X ) )
1817adantr 465 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  /\  c  e.  ( Fil `  X ) )  ->  D  e.  ( *Met `  X ) )
19 simpr 461 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
c  e.  ( Fil `  X ) )
20 cfilucfil4OLD 20843 . . . . . 6  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X )  /\  c  e.  ( Fil `  X
) )  ->  (
c  e.  (CauFilu `  (metUnifOLD `  D
) )  <->  c  e.  (CauFil `  D ) ) )
2112, 18, 19, 20syl3anc 1218 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
( c  e.  (CauFilu `  (metUnifOLD
`  D ) )  <-> 
c  e.  (CauFil `  D ) ) )
2211, 21bitrd 253 . . . 4  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
( c  e.  (CauFilu `  U )  <->  c  e.  (CauFil `  D ) ) )
23 eqid 2443 . . . . . . . . . . . 12  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
2423iscmet 20807 . . . . . . . . . . 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 20038 . . . . . . . . . . . . 13  |-  ( F  e.  *MetSp  ->  ( TopOpen
`  F )  =  ( MetOpen `  D )
)
293, 28syl 16 . . . . . . . . . . . 12  |-  ( F  e. CMetSp  ->  ( TopOpen `  F
)  =  ( MetOpen `  D ) )
3029oveq1d 6118 . . . . . . . . . . 11  |-  ( F  e. CMetSp  ->  ( ( TopOpen `  F )  fLim  c
)  =  ( (
MetOpen `  D )  fLim  c ) )
3130neeq1d 2633 . . . . . . . . . 10  |-  ( F  e. CMetSp  ->  ( ( (
TopOpen `  F )  fLim  c )  =/=  (/)  <->  ( ( MetOpen
`  D )  fLim  c )  =/=  (/) ) )
3231ralbidv 2747 . . . . . . . . 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 2826 . . . . . . 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 1010 . . . . 5  |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  -> 
( c  e.  (CauFil `  D )  ->  (
( TopOpen `  F )  fLim  c )  =/=  (/) ) )
3736adantr 465 . . . 4  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
( c  e.  (CauFil `  D )  ->  (
( TopOpen `  F )  fLim  c )  =/=  (/) ) )
3822, 37sylbid 215 . . 3  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
( c  e.  (CauFilu `  U )  ->  (
( TopOpen `  F )  fLim  c )  =/=  (/) ) )
3938ralrimiva 2811 . 2  |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  ->  A. c  e.  ( Fil `  X ) ( c  e.  (CauFilu `  U
)  ->  ( ( TopOpen
`  F )  fLim  c )  =/=  (/) ) )
404, 6, 27iscusp2 19889 . 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  =  (metUnifOLD `  D ) )  ->  F  e. CUnifSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727   (/)c0 3649    X. cxp 4850    |` cres 4854   ` cfv 5430  (class class class)co 6103   Basecbs 14186   distcds 14259   TopOpenctopn 14372   *Metcxmt 17813   Metcme 17814   MetOpencmopn 17818  metUnifOLDcmetuOLD 17819   Filcfil 19430    fLim cflim 19519  UnifStcuss 19840  UnifSpcusp 19841  CauFiluccfilu 19873  CUnifSpccusp 19884   *MetSpcxme 19904   MetSpcmt 19905  CauFilccfil 20775   CMetcms 20777  CMetSpccms 20855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-n0 10592  df-z 10659  df-uz 10874  df-q 10966  df-rp 11004  df-xneg 11101  df-xadd 11102  df-xmul 11103  df-ico 11318  df-topgen 14394  df-psmet 17821  df-xmet 17822  df-met 17823  df-bl 17824  df-mopn 17825  df-fbas 17826  df-fg 17827  df-metuOLD 17828  df-top 18515  df-bases 18517  df-topon 18518  df-topsp 18519  df-fil 19431  df-ust 19787  df-utop 19818  df-usp 19844  df-cfilu 19874  df-cusp 19885  df-xms 19907  df-ms 19908  df-cfil 20778  df-cmet 20780  df-cms 20858
This theorem is referenced by: (None)
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