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Theorem cmetcusp1 21520
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 21515 . . . 4  |-  ( F  e. CMetSp  ->  F  e.  MetSp )
2 msxms 20685 . . . 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 20817 . . 3  |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnif `  D )
)  ->  F  e. UnifSp )
83, 7syl3an2 1257 . 2  |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  ->  F  e. UnifSp )
9 simpl3 996 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  ->  U  =  (metUnif `  D
) )
109fveq2d 5861 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
(CauFilu
`  U )  =  (CauFilu `  (metUnif `  D
) ) )
1110eleq2d 2530 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
( c  e.  (CauFilu `  U )  <->  c  e.  (CauFilu `  (metUnif `  D )
) ) )
12 simpl1 994 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  ->  X  =/=  (/) )
134, 5cmscmet 21513 . . . . . . . . 9  |-  ( F  e. CMetSp  ->  D  e.  (
CMet `  X )
)
14 cmetmet 21453 . . . . . . . . 9  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
15 metxmet 20565 . . . . . . . . 9  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
1613, 14, 153syl 20 . . . . . . . 8  |-  ( F  e. CMetSp  ->  D  e.  ( *Met `  X
) )
17163ad2ant2 1013 . . . . . . 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 21488 . . . . . 6  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X )  /\  c  e.  ( Fil `  X
) )  ->  (
c  e.  (CauFilu `  (metUnif `  D ) )  <->  c  e.  (CauFil `  D ) ) )
2112, 18, 19, 20syl3anc 1223 . . . . 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 2460 . . . . . . . . . . . 12  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
2423iscmet 21451 . . . . . . . . . . 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 2460 . . . . . . . . . . . . . 14  |-  ( TopOpen `  F )  =  (
TopOpen `  F )
2827, 4, 5xmstopn 20682 . . . . . . . . . . . . 13  |-  ( F  e.  *MetSp  ->  ( TopOpen
`  F )  =  ( MetOpen `  D )
)
293, 28syl 16 . . . . . . . . . . . 12  |-  ( F  e. CMetSp  ->  ( TopOpen `  F
)  =  ( MetOpen `  D ) )
3029oveq1d 6290 . . . . . . . . . . 11  |-  ( F  e. CMetSp  ->  ( ( TopOpen `  F )  fLim  c
)  =  ( (
MetOpen `  D )  fLim  c ) )
3130neeq1d 2737 . . . . . . . . . 10  |-  ( F  e. CMetSp  ->  ( ( (
TopOpen `  F )  fLim  c )  =/=  (/)  <->  ( ( MetOpen
`  D )  fLim  c )  =/=  (/) ) )
3231ralbidv 2896 . . . . . . . . 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 1013 . . . . 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 2871 . 2  |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  ->  A. c  e.  ( Fil `  X ) ( c  e.  (CauFilu `  U
)  ->  ( ( TopOpen
`  F )  fLim  c )  =/=  (/) ) )
404, 6, 27iscusp2 20533 . 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 968    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   (/)c0 3778    X. cxp 4990    |` cres 4994   ` cfv 5579  (class class class)co 6275   Basecbs 14479   distcds 14553   TopOpenctopn 14666   *Metcxmt 18167   Metcme 18168   MetOpencmopn 18172  metUnifcmetu 18174   Filcfil 20074    fLim cflim 20163  UnifStcuss 20484  UnifSpcusp 20485  CauFiluccfilu 20517  CUnifSpccusp 20528   *MetSpcxme 20548   MetSpcmt 20549  CauFilccfil 21419   CMetcms 21421  CMetSpccms 21499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ico 11524  df-topgen 14688  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-fbas 18180  df-fg 18181  df-metu 18183  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-fil 20075  df-ust 20431  df-utop 20462  df-usp 20488  df-cfilu 20518  df-cusp 20529  df-xms 20551  df-ms 20552  df-cfil 21422  df-cmet 21424  df-cms 21502
This theorem is referenced by:  cnfldcusp  21525  recusp  21542
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