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Theorem cmetcusp1 20883
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 20878 . . . 4  |-  ( F  e. CMetSp  ->  F  e.  MetSp )
2 msxms 20048 . . . 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 20180 . . 3  |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnif `  D )
)  ->  F  e. UnifSp )
83, 7syl3an2 1252 . 2  |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  ->  F  e. UnifSp )
9 simpl3 993 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  ->  U  =  (metUnif `  D
) )
109fveq2d 5714 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
(CauFilu
`  U )  =  (CauFilu `  (metUnif `  D
) ) )
1110eleq2d 2510 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  -> 
( c  e.  (CauFilu `  U )  <->  c  e.  (CauFilu `  (metUnif `  D )
) ) )
12 simpl1 991 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  /\  c  e.  ( Fil `  X ) )  ->  X  =/=  (/) )
134, 5cmscmet 20876 . . . . . . . . 9  |-  ( F  e. CMetSp  ->  D  e.  (
CMet `  X )
)
14 cmetmet 20816 . . . . . . . . 9  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
15 metxmet 19928 . . . . . . . . 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  =  (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 20851 . . . . . 6  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X )  /\  c  e.  ( Fil `  X
) )  ->  (
c  e.  (CauFilu `  (metUnif `  D ) )  <->  c  e.  (CauFil `  D ) ) )
2112, 18, 19, 20syl3anc 1218 . . . . 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 20814 . . . . . . . . . . 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 20045 . . . . . . . . . . . . 13  |-  ( F  e.  *MetSp  ->  ( TopOpen
`  F )  =  ( MetOpen `  D )
)
293, 28syl 16 . . . . . . . . . . . 12  |-  ( F  e. CMetSp  ->  ( TopOpen `  F
)  =  ( MetOpen `  D ) )
3029oveq1d 6125 . . . . . . . . . . 11  |-  ( F  e. CMetSp  ->  ( ( TopOpen `  F )  fLim  c
)  =  ( (
MetOpen `  D )  fLim  c ) )
3130neeq1d 2637 . . . . . . . . . 10  |-  ( F  e. CMetSp  ->  ( ( (
TopOpen `  F )  fLim  c )  =/=  (/)  <->  ( ( MetOpen
`  D )  fLim  c )  =/=  (/) ) )
3231ralbidv 2754 . . . . . . . . 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 1010 . . . . 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 2818 . 2  |-  ( ( X  =/=  (/)  /\  F  e. CMetSp  /\  U  =  (metUnif `  D ) )  ->  A. c  e.  ( Fil `  X ) ( c  e.  (CauFilu `  U
)  ->  ( ( TopOpen
`  F )  fLim  c )  =/=  (/) ) )
404, 6, 27iscusp2 19896 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2734   (/)c0 3656    X. cxp 4857    |` cres 4861   ` cfv 5437  (class class class)co 6110   Basecbs 14193   distcds 14266   TopOpenctopn 14379   *Metcxmt 17820   Metcme 17821   MetOpencmopn 17825  metUnifcmetu 17827   Filcfil 19437    fLim cflim 19526  UnifStcuss 19847  UnifSpcusp 19848  CauFiluccfilu 19880  CUnifSpccusp 19891   *MetSpcxme 19911   MetSpcmt 19912  CauFilccfil 20782   CMetcms 20784  CMetSpccms 20862
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 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379
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 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-er 7120  df-map 7235  df-en 7330  df-dom 7331  df-sdom 7332  df-sup 7710  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-n0 10599  df-z 10666  df-uz 10881  df-q 10973  df-rp 11011  df-xneg 11108  df-xadd 11109  df-xmul 11110  df-ico 11325  df-topgen 14401  df-psmet 17828  df-xmet 17829  df-met 17830  df-bl 17831  df-mopn 17832  df-fbas 17833  df-fg 17834  df-metu 17836  df-top 18522  df-bases 18524  df-topon 18525  df-topsp 18526  df-fil 19438  df-ust 19794  df-utop 19825  df-usp 19851  df-cfilu 19881  df-cusp 19892  df-xms 19914  df-ms 19915  df-cfil 20785  df-cmet 20787  df-cms 20865
This theorem is referenced by:  cnfldcusp  20888  recusp  20905
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