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Theorem cmetcuspOLD 20992
Description: The uniform space generated by a complete metric is a complete uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
Assertion
Ref Expression
cmetcuspOLD  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (toUnifSp `  (metUnifOLD
`  D ) )  e. CUnifSp )

Proof of Theorem cmetcuspOLD
Dummy variables  x  c  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cmetmet 20924 . . . . 5  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
2 metxmet 20036 . . . . 5  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
31, 2syl 16 . . . 4  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( *Met `  X
) )
43anim2i 569 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) ) )
5 metuustOLD 20273 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
(metUnifOLD `  D )  e.  (UnifOn `  X ) )
6 eqid 2452 . . . 4  |-  (toUnifSp `  (metUnifOLD `  D
) )  =  (toUnifSp `  (metUnifOLD
`  D ) )
76tususp 19974 . . 3  |-  ( (metUnifOLD `  D )  e.  (UnifOn `  X )  ->  (toUnifSp `  (metUnifOLD
`  D ) )  e. UnifSp )
84, 5, 73syl 20 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (toUnifSp `  (metUnifOLD
`  D ) )  e. UnifSp )
9 simpll 753 . . . . 5  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  -> 
( X  =/=  (/)  /\  D  e.  ( CMet `  X
) ) )
109simprd 463 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  ->  D  e.  ( CMet `  X ) )
113ad3antlr 730 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  ->  D  e.  ( *Met `  X ) )
124, 5syl 16 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (metUnifOLD `  D
)  e.  (UnifOn `  X ) )
136tusbas 19970 . . . . . . . . . . . 12  |-  ( (metUnifOLD `  D )  e.  (UnifOn `  X )  ->  X  =  ( Base `  (toUnifSp `  (metUnifOLD
`  D ) ) ) )
1413fveq2d 5798 . . . . . . . . . . 11  |-  ( (metUnifOLD `  D )  e.  (UnifOn `  X )  ->  ( Fil `  X )  =  ( Fil `  ( Base `  (toUnifSp `  (metUnifOLD `  D
) ) ) ) )
1514eleq2d 2522 . . . . . . . . . 10  |-  ( (metUnifOLD `  D )  e.  (UnifOn `  X )  ->  (
c  e.  ( Fil `  X )  <->  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) ) )
1612, 15syl 16 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (
c  e.  ( Fil `  X )  <->  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) ) )
1716biimpar 485 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnifOLD `  D
) ) ) ) )  ->  c  e.  ( Fil `  X ) )
1817adantr 465 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  -> 
c  e.  ( Fil `  X ) )
196tususs 19972 . . . . . . . . . . . . 13  |-  ( (metUnifOLD `  D )  e.  (UnifOn `  X )  ->  (metUnifOLD `  D
)  =  (UnifSt `  (toUnifSp `  (metUnifOLD
`  D ) ) ) )
2019fveq2d 5798 . . . . . . . . . . . 12  |-  ( (metUnifOLD `  D )  e.  (UnifOn `  X )  ->  (CauFilu `  (metUnifOLD `  D ) )  =  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )
2112, 20syl 16 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (CauFilu `  (metUnifOLD `  D ) )  =  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )
2221eleq2d 2522 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (
c  e.  (CauFilu `  (metUnifOLD `  D
) )  <->  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnifOLD `  D
) ) ) ) ) )
2322biimpar 485 . . . . . . . . 9  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  -> 
c  e.  (CauFilu `  (metUnifOLD `  D
) ) )
2423adantlr 714 . . . . . . . 8  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  -> 
c  e.  (CauFilu `  (metUnifOLD `  D
) ) )
25 cfilucfil2OLD 20275 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( c  e.  (CauFilu `  (metUnifOLD
`  D ) )  <-> 
( c  e.  (
fBas `  X )  /\  A. x  e.  RR+  E. y  e.  c  ( D " ( y  X.  y ) ) 
C_  ( 0 [,) x ) ) ) )
264, 25syl 16 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (
c  e.  (CauFilu `  (metUnifOLD `  D
) )  <->  ( c  e.  ( fBas `  X
)  /\  A. x  e.  RR+  E. y  e.  c  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) )
2726simplbda 624 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  /\  c  e.  (CauFilu `  (metUnifOLD
`  D ) ) )  ->  A. x  e.  RR+  E. y  e.  c  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) )
289, 24, 27syl2anc 661 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  ->  A. x  e.  RR+  E. y  e.  c  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) )
29 iscfil 20903 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  (
c  e.  (CauFil `  D )  <->  ( c  e.  ( Fil `  X
)  /\  A. x  e.  RR+  E. y  e.  c  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) )
3029biimpar 485 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  ( c  e.  ( Fil `  X
)  /\  A. x  e.  RR+  E. y  e.  c  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) )  ->  c  e.  (CauFil `  D )
)
3111, 18, 28, 30syl12anc 1217 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  -> 
c  e.  (CauFil `  D ) )
32 eqid 2452 . . . . . . 7  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
3332cmetcvg 20923 . . . . . 6  |-  ( ( D  e.  ( CMet `  X )  /\  c  e.  (CauFil `  D )
)  ->  ( ( MetOpen
`  D )  fLim  c )  =/=  (/) )
3410, 31, 33syl2anc 661 . . . . 5  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  -> 
( ( MetOpen `  D
)  fLim  c )  =/=  (/) )
35 eqid 2452 . . . . . . . . . . 11  |-  (unifTop `  (metUnifOLD `  D
) )  =  (unifTop `  (metUnifOLD
`  D ) )
366, 35tustopn 19973 . . . . . . . . . 10  |-  ( (metUnifOLD `  D )  e.  (UnifOn `  X )  ->  (unifTop `  (metUnifOLD
`  D ) )  =  ( TopOpen `  (toUnifSp `  (metUnifOLD
`  D ) ) ) )
3712, 36syl 16 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (unifTop `  (metUnifOLD
`  D ) )  =  ( TopOpen `  (toUnifSp `  (metUnifOLD
`  D ) ) ) )
38 metutopOLD 20284 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
(unifTop `  (metUnifOLD
`  D ) )  =  ( MetOpen `  D
) )
394, 38syl 16 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (unifTop `  (metUnifOLD
`  D ) )  =  ( MetOpen `  D
) )
4037, 39eqtr3d 2495 . . . . . . . 8  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  ( TopOpen
`  (toUnifSp `  (metUnifOLD
`  D ) ) )  =  ( MetOpen `  D ) )
4140oveq1d 6210 . . . . . . 7  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (
( TopOpen `  (toUnifSp `  (metUnifOLD `  D
) ) )  fLim  c )  =  ( (
MetOpen `  D )  fLim  c ) )
4241neeq1d 2726 . . . . . 6  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (
( ( TopOpen `  (toUnifSp `  (metUnifOLD
`  D ) ) )  fLim  c )  =/=  (/)  <->  ( ( MetOpen `  D )  fLim  c
)  =/=  (/) ) )
4342biimpar 485 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  /\  (
( MetOpen `  D )  fLim  c )  =/=  (/) )  -> 
( ( TopOpen `  (toUnifSp `  (metUnifOLD
`  D ) ) )  fLim  c )  =/=  (/) )
449, 34, 43syl2anc 661 . . . 4  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) )  -> 
( ( TopOpen `  (toUnifSp `  (metUnifOLD
`  D ) ) )  fLim  c )  =/=  (/) )
4544ex 434 . . 3  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnifOLD `  D
) ) ) ) )  ->  ( c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnifOLD
`  D ) ) ) )  ->  (
( TopOpen `  (toUnifSp `  (metUnifOLD `  D
) ) )  fLim  c )  =/=  (/) ) )
4645ralrimiva 2827 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  A. c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnifOLD `  D
) ) ) ) ( c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnifOLD `  D
) ) ) )  ->  ( ( TopOpen `  (toUnifSp `  (metUnifOLD
`  D ) ) )  fLim  c )  =/=  (/) ) )
47 iscusp 20001 . 2  |-  ( (toUnifSp `  (metUnifOLD
`  D ) )  e. CUnifSp 
<->  ( (toUnifSp `  (metUnifOLD `  D
) )  e. UnifSp  /\  A. c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnifOLD
`  D ) ) ) ) ( c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnifOLD
`  D ) ) ) )  ->  (
( TopOpen `  (toUnifSp `  (metUnifOLD `  D
) ) )  fLim  c )  =/=  (/) ) ) )
488, 46, 47sylanbrc 664 1  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (toUnifSp `  (metUnifOLD
`  D ) )  e. CUnifSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645   A.wral 2796   E.wrex 2797    C_ wss 3431   (/)c0 3740    X. cxp 4941   "cima 4946   ` cfv 5521  (class class class)co 6195   0cc0 9388   RR+crp 11097   [,)cico 11408   Basecbs 14287   TopOpenctopn 14474   *Metcxmt 17921   Metcme 17922   fBascfbas 17924   MetOpencmopn 17926  metUnifOLDcmetuOLD 17927   Filcfil 19545    fLim cflim 19634  UnifOncust 19901  unifTopcutop 19932  UnifStcuss 19955  UnifSpcusp 19956  toUnifSpctus 19957  CauFiluccfilu 19988  CUnifSpccusp 19999  CauFilccfil 20890   CMetcms 20892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-z 10753  df-dec 10862  df-uz 10968  df-q 11060  df-rp 11098  df-xneg 11195  df-xadd 11196  df-xmul 11197  df-ico 11412  df-fz 11550  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-tset 14371  df-unif 14375  df-rest 14475  df-topn 14476  df-topgen 14496  df-psmet 17929  df-xmet 17930  df-met 17931  df-bl 17932  df-mopn 17933  df-fbas 17934  df-fg 17935  df-metuOLD 17936  df-fil 19546  df-ust 19902  df-utop 19933  df-uss 19958  df-usp 19959  df-tus 19960  df-cfilu 19989  df-cusp 20000  df-cfil 20893  df-cmet 20895
This theorem is referenced by: (None)
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