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Theorem ucnextcn 20785
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 20745 . . . . . 6  |-  ( ( V  e. UnifSp  /\  A  C_  X )  ->  T  e.  (UnifOn `  A )
)
1512, 10, 14syl2anc 661 . . . . 5  |-  ( ph  ->  T  e.  (UnifOn `  A ) )
16 cuspusp 20781 . . . . . . . 8  |-  ( W  e. CUnifSp  ->  W  e. UnifSp )
178, 16syl 16 . . . . . . 7  |-  ( ph  ->  W  e. UnifSp )
182, 5, 4isusp 20742 . . . . . . 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 20759 . . . . 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 5884 . . . 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 2534 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  ( ( cls `  J
) `  A )
)
311, 3istps 19415 . . . . . . . . 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 20371 . . . . . . 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 1229 . . . . . 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 20327 . . . . 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 20422 . . . 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 1229 . . 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 20742 . . . . . . . . . 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 20777 . . . . . . . 8  |-  ( ( V  e. UnifSp  /\  V  e. 
TopSp  /\  x  e.  X
)  ->  ( ( nei `  J ) `  { x } )  e.  (CauFilu `  S ) )
5350, 51, 28, 52syl3anc 1229 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( nei `  J
) `  { x } )  e.  (CauFilu `  S ) )
54 0nelfb 20310 . . . . . . . 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 20775 . . . . . . 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 1234 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  (CauFilu `  ( St  ( A  X.  A ) ) ) )
5843elfvexd 5884 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  _V )
59 ressuss 20744 . . . . . . . . 9  |-  ( A  e.  _V  ->  (UnifSt `  ( Vs  A ) )  =  ( (UnifSt `  V
)t  ( A  X.  A
) ) )
6045oveq1i 6291 . . . . . . . . 9  |-  ( St  ( A  X.  A ) )  =  ( (UnifSt `  V )t  ( A  X.  A ) )
6159, 13, 603eqtr4g 2509 . . . . . . . 8  |-  ( A  e.  _V  ->  T  =  ( St  ( A  X.  A ) ) )
6261fveq2d 5860 . . . . . . 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 2534 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  (CauFilu `  T ) )
65 imaeq2 5323 . . . . . . 7  |-  ( a  =  b  ->  ( F " a )  =  ( F " b
) )
6665cbvmptv 4528 . . . . . 6  |-  ( a  e.  ( ( ( nei `  J ) `
 { x }
)t 
A )  |->  ( F
" a ) )  =  ( b  e.  ( ( ( nei `  J ) `  {
x } )t  A ) 
|->  ( F " b
) )
6766rneqi 5219 . . . . 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 20773 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ran  ( a  e.  ( ( ( nei `  J
) `  { x } )t  A )  |->  ( F
" a ) )  e.  (CauFilu `  U ) )
69 cfilufg 20774 . . . 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 2531 . 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 20784 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 1383    e. wcel 1804   A.wral 2793   E.wrex 2794   _Vcvv 3095    C_ wss 3461   (/)c0 3770   {csn 4014   class class class wbr 4437    |-> cmpt 4495    X. cxp 4987   ran crn 4990   "cima 4992   -->wf 5574   ` cfv 5578  (class class class)co 6281   Basecbs 14614   ↾s cress 14615   ↾t crest 14800   TopOpenctopn 14801   fBascfbas 18385   filGencfg 18386  TopOnctopon 19373   TopSpctps 19375   clsccl 19497   neicnei 19576    Cn ccn 19703   Hauscha 19787   Filcfil 20324    FilMap cfm 20412  CnExtccnext 20537  UnifOncust 20680  unifTopcutop 20711  UnifStcuss 20734  UnifSpcusp 20735   Cnucucn 20756  CauFiluccfilu 20767  CUnifSpccusp 20778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fi 7873  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-unif 14702  df-rest 14802  df-topgen 14823  df-fbas 18395  df-fg 18396  df-top 19377  df-bases 19379  df-topon 19380  df-topsp 19381  df-cld 19498  df-ntr 19499  df-cls 19500  df-nei 19577  df-cn 19706  df-cnp 19707  df-haus 19794  df-reg 19795  df-tx 20041  df-fil 20325  df-fm 20417  df-flim 20418  df-flf 20419  df-cnext 20538  df-ust 20681  df-utop 20712  df-uss 20737  df-usp 20738  df-ucn 20757  df-cfilu 20768  df-cusp 20779
This theorem is referenced by:  rrhcn  27956
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