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Theorem ucnextcn 21397
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 21357 . . . . . 6  |-  ( ( V  e. UnifSp  /\  A  C_  X )  ->  T  e.  (UnifOn `  A )
)
1512, 10, 14syl2anc 673 . . . . 5  |-  ( ph  ->  T  e.  (UnifOn `  A ) )
16 cuspusp 21393 . . . . . . . 8  |-  ( W  e. CUnifSp  ->  W  e. UnifSp )
178, 16syl 17 . . . . . . 7  |-  ( ph  ->  W  e. UnifSp )
182, 5, 4isusp 21354 . . . . . . 7  |-  ( W  e. UnifSp 
<->  ( U  e.  (UnifOn `  Y )  /\  K  =  (unifTop `  U )
) )
1917, 18sylib 201 . . . . . 6  |-  ( ph  ->  ( U  e.  (UnifOn `  Y )  /\  K  =  (unifTop `  U )
) )
2019simpld 466 . . . . 5  |-  ( ph  ->  U  e.  (UnifOn `  Y ) )
21 isucn 21371 . . . . 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 673 . . . 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 215 . . 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 466 . 2  |-  ( ph  ->  F : A --> Y )
25 ucnextcn.c . 2  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  X )
2620adantr 472 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  U  e.  (UnifOn `  Y )
)
2726elfvexd 5907 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  Y  e.  _V )
28 simpr 468 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
2925adantr 472 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( cls `  J
) `  A )  =  X )
3028, 29eleqtrrd 2552 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  ( ( cls `  J
) `  A )
)
311, 3istps 20028 . . . . . . . . 9  |-  ( V  e.  TopSp 
<->  J  e.  (TopOn `  X ) )
326, 31sylib 201 . . . . . . . 8  |-  ( ph  ->  J  e.  (TopOn `  X ) )
3332adantr 472 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  J  e.  (TopOn `  X )
)
3410adantr 472 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  A  C_  X )
35 trnei 20985 . . . . . . 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 1292 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) ) )
3730, 36mpbid 215 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) )
38 filfbas 20941 . . . . 5  |-  ( ( ( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  ->  ( ( ( nei `  J ) `
 { x }
)t 
A )  e.  (
fBas `  A )
)
3937, 38syl 17 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  (
fBas `  A )
)
4024adantr 472 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  F : A --> Y )
41 fmval 21036 . . . 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 1292 . . 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 472 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  T  e.  (UnifOn `  A )
)
4411adantr 472 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F  e.  ( T Cnu U ) )
45 ucnextcn.s . . . . . . . . . . 11  |-  S  =  (UnifSt `  V )
461, 45, 3isusp 21354 . . . . . . . . . 10  |-  ( V  e. UnifSp 
<->  ( S  e.  (UnifOn `  X )  /\  J  =  (unifTop `  S )
) )
4712, 46sylib 201 . . . . . . . . 9  |-  ( ph  ->  ( S  e.  (UnifOn `  X )  /\  J  =  (unifTop `  S )
) )
4847simpld 466 . . . . . . . 8  |-  ( ph  ->  S  e.  (UnifOn `  X ) )
4948adantr 472 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  S  e.  (UnifOn `  X )
)
5012adantr 472 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  V  e. UnifSp )
516adantr 472 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  V  e.  TopSp )
521, 3, 45neipcfilu 21389 . . . . . . . 8  |-  ( ( V  e. UnifSp  /\  V  e. 
TopSp  /\  x  e.  X
)  ->  ( ( nei `  J ) `  { x } )  e.  (CauFilu `  S ) )
5350, 51, 28, 52syl3anc 1292 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( nei `  J
) `  { x } )  e.  (CauFilu `  S ) )
54 0nelfb 20924 . . . . . . . 8  |-  ( ( ( ( nei `  J
) `  { x } )t  A )  e.  (
fBas `  A )  ->  -.  (/)  e.  ( ( ( nei `  J
) `  { x } )t  A ) )
5539, 54syl 17 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  -.  (/) 
e.  ( ( ( nei `  J ) `
 { x }
)t 
A ) )
56 trcfilu 21387 . . . . . . 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 1297 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  (CauFilu `  ( St  ( A  X.  A ) ) ) )
5843elfvexd 5907 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  _V )
59 ressuss 21356 . . . . . . . . 9  |-  ( A  e.  _V  ->  (UnifSt `  ( Vs  A ) )  =  ( (UnifSt `  V
)t  ( A  X.  A
) ) )
6045oveq1i 6318 . . . . . . . . 9  |-  ( St  ( A  X.  A ) )  =  ( (UnifSt `  V )t  ( A  X.  A ) )
6159, 13, 603eqtr4g 2530 . . . . . . . 8  |-  ( A  e.  _V  ->  T  =  ( St  ( A  X.  A ) ) )
6261fveq2d 5883 . . . . . . 7  |-  ( A  e.  _V  ->  (CauFilu `  T )  =  (CauFilu `  ( St  ( A  X.  A ) ) ) )
6358, 62syl 17 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (CauFilu `  T )  =  (CauFilu `  ( St  ( A  X.  A ) ) ) )
6457, 63eleqtrrd 2552 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  (CauFilu `  T ) )
65 imaeq2 5170 . . . . . . 7  |-  ( a  =  b  ->  ( F " a )  =  ( F " b
) )
6665cbvmptv 4488 . . . . . 6  |-  ( a  e.  ( ( ( nei `  J ) `
 { x }
)t 
A )  |->  ( F
" a ) )  =  ( b  e.  ( ( ( nei `  J ) `  {
x } )t  A ) 
|->  ( F " b
) )
6766rneqi 5067 . . . . 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 21385 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ran  ( a  e.  ( ( ( nei `  J
) `  { x } )t  A )  |->  ( F
" a ) )  e.  (CauFilu `  U ) )
69 cfilufg 21386 . . . 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 673 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( Y filGen ran  ( a  e.  ( ( ( nei `  J ) `  {
x } )t  A ) 
|->  ( F " a
) ) )  e.  (CauFilu `  U ) )
7142, 70eqeltrd 2549 . 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 21396 1  |-  ( ph  ->  ( ( JCnExt K
) `  F )  e.  ( J  Cn  K
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757   _Vcvv 3031    C_ wss 3390   (/)c0 3722   {csn 3959   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   ran crn 4840   "cima 4842   -->wf 5585   ` cfv 5589  (class class class)co 6308   Basecbs 15199   ↾s cress 15200   ↾t crest 15397   TopOpenctopn 15398   fBascfbas 19035   filGencfg 19036  TopOnctopon 19995   TopSpctps 19996   clsccl 20110   neicnei 20190    Cn ccn 20317   Hauscha 20401   Filcfil 20938    FilMap cfm 21026  CnExtccnext 21152  UnifOncust 21292  unifTopcutop 21323  UnifStcuss 21346  UnifSpcusp 21347   Cnucucn 21368  CauFiluccfilu 21379  CUnifSpccusp 21390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-unif 15291  df-rest 15399  df-topgen 15420  df-fbas 19044  df-fg 19045  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-cn 20320  df-cnp 20321  df-haus 20408  df-reg 20409  df-tx 20654  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-cnext 21153  df-ust 21293  df-utop 21324  df-uss 21349  df-usp 21350  df-ucn 21369  df-cfilu 21380  df-cusp 21391
This theorem is referenced by:  rrhcn  28875
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