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Theorem cmetcusp1OLD 21618
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 21614 . . . 4  |-  ( F  e. CMetSp  ->  F  e.  MetSp )
2 msxms 20784 . . . 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 20915 . . 3  |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnifOLD
`  D ) )  ->  F  e. UnifSp )
83, 7syl3an2 1262 . 2  |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  ->  F  e. UnifSp )
9 simpl3 1001 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  /\  c  e.  ( Fil `  X ) )  ->  U  =  (metUnifOLD
`  D ) )
109fveq2d 5870 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
(CauFilu
`  U )  =  (CauFilu `  (metUnifOLD
`  D ) ) )
1110eleq2d 2537 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
( c  e.  (CauFilu `  U )  <->  c  e.  (CauFilu `  (metUnifOLD
`  D ) ) ) )
12 simpl1 999 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  /\  c  e.  ( Fil `  X ) )  ->  X  =/=  (/) )
134, 5cmscmet 21612 . . . . . . . . 9  |-  ( F  e. CMetSp  ->  D  e.  (
CMet `  X )
)
14 cmetmet 21552 . . . . . . . . 9  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
15 metxmet 20664 . . . . . . . . 9  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
1613, 14, 153syl 20 . . . . . . . 8  |-  ( F  e. CMetSp  ->  D  e.  ( *Met `  X
) )
17163ad2ant2 1018 . . . . . . 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 21586 . . . . . 6  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X )  /\  c  e.  ( Fil `  X
) )  ->  (
c  e.  (CauFilu `  (metUnifOLD `  D
) )  <->  c  e.  (CauFil `  D ) ) )
2112, 18, 19, 20syl3anc 1228 . . . . 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 2467 . . . . . . . . . . . 12  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
2423iscmet 21550 . . . . . . . . . . 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 2467 . . . . . . . . . . . . . 14  |-  ( TopOpen `  F )  =  (
TopOpen `  F )
2827, 4, 5xmstopn 20781 . . . . . . . . . . . . 13  |-  ( F  e.  *MetSp  ->  ( TopOpen
`  F )  =  ( MetOpen `  D )
)
293, 28syl 16 . . . . . . . . . . . 12  |-  ( F  e. CMetSp  ->  ( TopOpen `  F
)  =  ( MetOpen `  D ) )
3029oveq1d 6300 . . . . . . . . . . 11  |-  ( F  e. CMetSp  ->  ( ( TopOpen `  F )  fLim  c
)  =  ( (
MetOpen `  D )  fLim  c ) )
3130neeq1d 2744 . . . . . . . . . 10  |-  ( F  e. CMetSp  ->  ( ( (
TopOpen `  F )  fLim  c )  =/=  (/)  <->  ( ( MetOpen
`  D )  fLim  c )  =/=  (/) ) )
3231ralbidv 2903 . . . . . . . . 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 2833 . . . . . . 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 1018 . . . . 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 2878 . 2  |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnifOLD `  D ) )  ->  A. c  e.  ( Fil `  X ) ( c  e.  (CauFilu `  U
)  ->  ( ( TopOpen
`  F )  fLim  c )  =/=  (/) ) )
404, 6, 27iscusp2 20632 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   (/)c0 3785    X. cxp 4997    |` cres 5001   ` cfv 5588  (class class class)co 6285   Basecbs 14493   distcds 14567   TopOpenctopn 14680   *Metcxmt 18214   Metcme 18215   MetOpencmopn 18219  metUnifOLDcmetuOLD 18220   Filcfil 20173    fLim cflim 20262  UnifStcuss 20583  UnifSpcusp 20584  CauFiluccfilu 20616  CUnifSpccusp 20627   *MetSpcxme 20647   MetSpcmt 20648  CauFilccfil 21518   CMetcms 21520  CMetSpccms 21598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11084  df-q 11184  df-rp 11222  df-xneg 11319  df-xadd 11320  df-xmul 11321  df-ico 11536  df-topgen 14702  df-psmet 18222  df-xmet 18223  df-met 18224  df-bl 18225  df-mopn 18226  df-fbas 18227  df-fg 18228  df-metuOLD 18229  df-top 19206  df-bases 19208  df-topon 19209  df-topsp 19210  df-fil 20174  df-ust 20530  df-utop 20561  df-usp 20587  df-cfilu 20617  df-cusp 20628  df-xms 20650  df-ms 20651  df-cfil 21521  df-cmet 21523  df-cms 21601
This theorem is referenced by: (None)
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