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Theorem ucnextcn 20675
Description: Extension by continuity. Theorem 2 of [BourbakiTop1] p. II.20. Given an uniform space on a set  X, a subset  A dense in  X, and a function  F uniformly continuous from  A to  Y, that function can be extended by continuity to the whole  X, and its extension is uniformly continuous. (Contributed by Thierry Arnoux, 25-Jan-2018.)
Hypotheses
Ref Expression
ucnextcn.x  |-  X  =  ( Base `  V
)
ucnextcn.y  |-  Y  =  ( Base `  W
)
ucnextcn.j  |-  J  =  ( TopOpen `  V )
ucnextcn.k  |-  K  =  ( TopOpen `  W )
ucnextcn.s  |-  S  =  (UnifSt `  V )
ucnextcn.t  |-  T  =  (UnifSt `  ( Vs  A
) )
ucnextcn.u  |-  U  =  (UnifSt `  W )
ucnextcn.v  |-  ( ph  ->  V  e.  TopSp )
ucnextcn.r  |-  ( ph  ->  V  e. UnifSp )
ucnextcn.w  |-  ( ph  ->  W  e.  TopSp )
ucnextcn.z  |-  ( ph  ->  W  e. CUnifSp )
ucnextcn.h  |-  ( ph  ->  K  e.  Haus )
ucnextcn.a  |-  ( ph  ->  A  C_  X )
ucnextcn.f  |-  ( ph  ->  F  e.  ( T Cnu U ) )
ucnextcn.c  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  X )
Assertion
Ref Expression
ucnextcn  |-  ( ph  ->  ( ( JCnExt K
) `  F )  e.  ( J  Cn  K
) )

Proof of Theorem ucnextcn
Dummy variables  a 
b  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ucnextcn.x . 2  |-  X  =  ( Base `  V
)
2 ucnextcn.y . 2  |-  Y  =  ( Base `  W
)
3 ucnextcn.j . 2  |-  J  =  ( TopOpen `  V )
4 ucnextcn.k . 2  |-  K  =  ( TopOpen `  W )
5 ucnextcn.u . 2  |-  U  =  (UnifSt `  W )
6 ucnextcn.v . 2  |-  ( ph  ->  V  e.  TopSp )
7 ucnextcn.w . 2  |-  ( ph  ->  W  e.  TopSp )
8 ucnextcn.z . 2  |-  ( ph  ->  W  e. CUnifSp )
9 ucnextcn.h . 2  |-  ( ph  ->  K  e.  Haus )
10 ucnextcn.a . 2  |-  ( ph  ->  A  C_  X )
11 ucnextcn.f . . . 4  |-  ( ph  ->  F  e.  ( T Cnu U ) )
12 ucnextcn.r . . . . . 6  |-  ( ph  ->  V  e. UnifSp )
13 ucnextcn.t . . . . . . 7  |-  T  =  (UnifSt `  ( Vs  A
) )
141, 13ressust 20635 . . . . . 6  |-  ( ( V  e. UnifSp  /\  A  C_  X )  ->  T  e.  (UnifOn `  A )
)
1512, 10, 14syl2anc 661 . . . . 5  |-  ( ph  ->  T  e.  (UnifOn `  A ) )
16 cuspusp 20671 . . . . . . . 8  |-  ( W  e. CUnifSp  ->  W  e. UnifSp )
178, 16syl 16 . . . . . . 7  |-  ( ph  ->  W  e. UnifSp )
182, 5, 4isusp 20632 . . . . . . 7  |-  ( W  e. UnifSp 
<->  ( U  e.  (UnifOn `  Y )  /\  K  =  (unifTop `  U )
) )
1917, 18sylib 196 . . . . . 6  |-  ( ph  ->  ( U  e.  (UnifOn `  Y )  /\  K  =  (unifTop `  U )
) )
2019simpld 459 . . . . 5  |-  ( ph  ->  U  e.  (UnifOn `  Y ) )
21 isucn 20649 . . . . 5  |-  ( ( T  e.  (UnifOn `  A )  /\  U  e.  (UnifOn `  Y )
)  ->  ( F  e.  ( T Cnu U )  <->  ( F : A --> Y  /\  A. w  e.  U  E. v  e.  T  A. y  e.  A  A. z  e.  A  (
y v z  -> 
( F `  y
) w ( F `
 z ) ) ) ) )
2215, 20, 21syl2anc 661 . . . 4  |-  ( ph  ->  ( F  e.  ( T Cnu U )  <->  ( F : A --> Y  /\  A. w  e.  U  E. v  e.  T  A. y  e.  A  A. z  e.  A  (
y v z  -> 
( F `  y
) w ( F `
 z ) ) ) ) )
2311, 22mpbid 210 . . 3  |-  ( ph  ->  ( F : A --> Y  /\  A. w  e.  U  E. v  e.  T  A. y  e.  A  A. z  e.  A  ( y v z  ->  ( F `  y ) w ( F `  z ) ) ) )
2423simpld 459 . 2  |-  ( ph  ->  F : A --> Y )
25 ucnextcn.c . 2  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  X )
2620adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  U  e.  (UnifOn `  Y )
)
2726elfvexd 5900 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  Y  e.  _V )
28 simpr 461 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
2925adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( cls `  J
) `  A )  =  X )
3028, 29eleqtrrd 2558 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  ( ( cls `  J
) `  A )
)
311, 3istps 19306 . . . . . . . . 9  |-  ( V  e.  TopSp 
<->  J  e.  (TopOn `  X ) )
326, 31sylib 196 . . . . . . . 8  |-  ( ph  ->  J  e.  (TopOn `  X ) )
3332adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  J  e.  (TopOn `  X )
)
3410adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  A  C_  X )
35 trnei 20261 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X  /\  x  e.  X )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) ) )
3633, 34, 28, 35syl3anc 1228 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) ) )
3730, 36mpbid 210 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) )
38 filfbas 20217 . . . . 5  |-  ( ( ( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  ->  ( ( ( nei `  J ) `
 { x }
)t 
A )  e.  (
fBas `  A )
)
3937, 38syl 16 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  (
fBas `  A )
)
4024adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  F : A --> Y )
41 fmval 20312 . . . 4  |-  ( ( Y  e.  _V  /\  ( ( ( nei `  J ) `  {
x } )t  A )  e.  ( fBas `  A
)  /\  F : A
--> Y )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  =  ( Y filGen ran  (
a  e.  ( ( ( nei `  J
) `  { x } )t  A )  |->  ( F
" a ) ) ) )
4227, 39, 40, 41syl3anc 1228 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  =  ( Y filGen ran  (
a  e.  ( ( ( nei `  J
) `  { x } )t  A )  |->  ( F
" a ) ) ) )
4315adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  T  e.  (UnifOn `  A )
)
4411adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F  e.  ( T Cnu U ) )
45 ucnextcn.s . . . . . . . . . . 11  |-  S  =  (UnifSt `  V )
461, 45, 3isusp 20632 . . . . . . . . . 10  |-  ( V  e. UnifSp 
<->  ( S  e.  (UnifOn `  X )  /\  J  =  (unifTop `  S )
) )
4712, 46sylib 196 . . . . . . . . 9  |-  ( ph  ->  ( S  e.  (UnifOn `  X )  /\  J  =  (unifTop `  S )
) )
4847simpld 459 . . . . . . . 8  |-  ( ph  ->  S  e.  (UnifOn `  X ) )
4948adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  S  e.  (UnifOn `  X )
)
5012adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  V  e. UnifSp )
516adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  V  e.  TopSp )
521, 3, 45neipcfilu 20667 . . . . . . . 8  |-  ( ( V  e. UnifSp  /\  V  e. 
TopSp  /\  x  e.  X
)  ->  ( ( nei `  J ) `  { x } )  e.  (CauFilu `  S ) )
5350, 51, 28, 52syl3anc 1228 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( nei `  J
) `  { x } )  e.  (CauFilu `  S ) )
54 0nelfb 20200 . . . . . . . 8  |-  ( ( ( ( nei `  J
) `  { x } )t  A )  e.  (
fBas `  A )  ->  -.  (/)  e.  ( ( ( nei `  J
) `  { x } )t  A ) )
5539, 54syl 16 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  -.  (/) 
e.  ( ( ( nei `  J ) `
 { x }
)t 
A ) )
56 trcfilu 20665 . . . . . . 7  |-  ( ( S  e.  (UnifOn `  X )  /\  (
( ( nei `  J
) `  { x } )  e.  (CauFilu `  S )  /\  -.  (/) 
e.  ( ( ( nei `  J ) `
 { x }
)t 
A ) )  /\  A  C_  X )  -> 
( ( ( nei `  J ) `  {
x } )t  A )  e.  (CauFilu `  ( St  ( A  X.  A ) ) ) )
5749, 53, 55, 34, 56syl121anc 1233 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  (CauFilu `  ( St  ( A  X.  A ) ) ) )
5843elfvexd 5900 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  _V )
59 ressuss 20634 . . . . . . . . 9  |-  ( A  e.  _V  ->  (UnifSt `  ( Vs  A ) )  =  ( (UnifSt `  V
)t  ( A  X.  A
) ) )
6045oveq1i 6305 . . . . . . . . 9  |-  ( St  ( A  X.  A ) )  =  ( (UnifSt `  V )t  ( A  X.  A ) )
6159, 13, 603eqtr4g 2533 . . . . . . . 8  |-  ( A  e.  _V  ->  T  =  ( St  ( A  X.  A ) ) )
6261fveq2d 5876 . . . . . . 7  |-  ( A  e.  _V  ->  (CauFilu `  T )  =  (CauFilu `  ( St  ( A  X.  A ) ) ) )
6358, 62syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (CauFilu `  T )  =  (CauFilu `  ( St  ( A  X.  A ) ) ) )
6457, 63eleqtrrd 2558 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  (CauFilu `  T ) )
65 imaeq2 5339 . . . . . . 7  |-  ( a  =  b  ->  ( F " a )  =  ( F " b
) )
6665cbvmptv 4544 . . . . . 6  |-  ( a  e.  ( ( ( nei `  J ) `
 { x }
)t 
A )  |->  ( F
" a ) )  =  ( b  e.  ( ( ( nei `  J ) `  {
x } )t  A ) 
|->  ( F " b
) )
6766rneqi 5235 . . . . 5  |-  ran  (
a  e.  ( ( ( nei `  J
) `  { x } )t  A )  |->  ( F
" a ) )  =  ran  ( b  e.  ( ( ( nei `  J ) `
 { x }
)t 
A )  |->  ( F
" b ) )
6843, 26, 44, 64, 67fmucnd 20663 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ran  ( a  e.  ( ( ( nei `  J
) `  { x } )t  A )  |->  ( F
" a ) )  e.  (CauFilu `  U ) )
69 cfilufg 20664 . . . 4  |-  ( ( U  e.  (UnifOn `  Y )  /\  ran  ( a  e.  ( ( ( nei `  J
) `  { x } )t  A )  |->  ( F
" a ) )  e.  (CauFilu `  U ) )  ->  ( Y filGen ran  ( a  e.  ( ( ( nei `  J
) `  { x } )t  A )  |->  ( F
" a ) ) )  e.  (CauFilu `  U
) )
7026, 68, 69syl2anc 661 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( Y filGen ran  ( a  e.  ( ( ( nei `  J ) `  {
x } )t  A ) 
|->  ( F " a
) ) )  e.  (CauFilu `  U ) )
7142, 70eqeltrd 2555 . 2  |-  ( (
ph  /\  x  e.  X )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  (CauFilu `  U ) )
721, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 25, 71cnextucn 20674 1  |-  ( ph  ->  ( ( JCnExt K
) `  F )  e.  ( J  Cn  K
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   _Vcvv 3118    C_ wss 3481   (/)c0 3790   {csn 4033   class class class wbr 4453    |-> cmpt 4511    X. cxp 5003   ran crn 5006   "cima 5008   -->wf 5590   ` cfv 5594  (class class class)co 6295   Basecbs 14507   ↾s cress 14508   ↾t crest 14693   TopOpenctopn 14694   fBascfbas 18276   filGencfg 18277  TopOnctopon 19264   TopSpctps 19266   clsccl 19387   neicnei 19466    Cn ccn 19593   Hauscha 19677   Filcfil 20214    FilMap cfm 20302  CnExtccnext 20427  UnifOncust 20570  unifTopcutop 20601  UnifStcuss 20624  UnifSpcusp 20625   Cnucucn 20646  CauFiluccfilu 20657  CUnifSpccusp 20668
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-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fi 7883  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-unif 14595  df-rest 14695  df-topgen 14716  df-fbas 18286  df-fg 18287  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-cn 19596  df-cnp 19597  df-haus 19684  df-reg 19685  df-tx 19931  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-cnext 20428  df-ust 20571  df-utop 20602  df-uss 20627  df-usp 20628  df-ucn 20647  df-cfilu 20658  df-cusp 20669
This theorem is referenced by:  rrhcn  27794
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