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Theorem cnextucn 21396
Description: Extension by continuity. Proposition 11 of [BourbakiTop1] p. II.20. Given a topology  J on  X, a subset  A dense in  X, this states a condition for  F from  A to a space  Y Hausdorff and complete to be extensible by continuity. (Contributed by Thierry Arnoux, 4-Dec-2017.)
Hypotheses
Ref Expression
cnextucn.x  |-  X  =  ( Base `  V
)
cnextucn.y  |-  Y  =  ( Base `  W
)
cnextucn.j  |-  J  =  ( TopOpen `  V )
cnextucn.k  |-  K  =  ( TopOpen `  W )
cnextucn.u  |-  U  =  (UnifSt `  W )
cnextucn.v  |-  ( ph  ->  V  e.  TopSp )
cnextucn.t  |-  ( ph  ->  W  e.  TopSp )
cnextucn.w  |-  ( ph  ->  W  e. CUnifSp )
cnextucn.h  |-  ( ph  ->  K  e.  Haus )
cnextucn.a  |-  ( ph  ->  A  C_  X )
cnextucn.f  |-  ( ph  ->  F : A --> Y )
cnextucn.c  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  X )
cnextucn.l  |-  ( (
ph  /\  x  e.  X )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  (CauFilu `  U ) )
Assertion
Ref Expression
cnextucn  |-  ( ph  ->  ( ( JCnExt K
) `  F )  e.  ( J  Cn  K
) )
Distinct variable groups:    x, A    x, F    x, J    x, K    ph, x
Allowed substitution hints:    U( x)    V( x)    W( x)    X( x)    Y( x)

Proof of Theorem cnextucn
StepHypRef Expression
1 eqid 2471 . 2  |-  U. J  =  U. J
2 eqid 2471 . 2  |-  U. K  =  U. K
3 cnextucn.v . . 3  |-  ( ph  ->  V  e.  TopSp )
4 cnextucn.j . . . 4  |-  J  =  ( TopOpen `  V )
54tpstop 20031 . . 3  |-  ( V  e.  TopSp  ->  J  e.  Top )
63, 5syl 17 . 2  |-  ( ph  ->  J  e.  Top )
7 cnextucn.h . 2  |-  ( ph  ->  K  e.  Haus )
8 cnextucn.f . . 3  |-  ( ph  ->  F : A --> Y )
9 cnextucn.t . . . . 5  |-  ( ph  ->  W  e.  TopSp )
10 cnextucn.y . . . . . 6  |-  Y  =  ( Base `  W
)
11 cnextucn.k . . . . . 6  |-  K  =  ( TopOpen `  W )
1210, 11tpsuni 20030 . . . . 5  |-  ( W  e.  TopSp  ->  Y  =  U. K )
139, 12syl 17 . . . 4  |-  ( ph  ->  Y  =  U. K
)
1413feq3d 5726 . . 3  |-  ( ph  ->  ( F : A --> Y 
<->  F : A --> U. K
) )
158, 14mpbid 215 . 2  |-  ( ph  ->  F : A --> U. K
)
16 cnextucn.a . . 3  |-  ( ph  ->  A  C_  X )
17 cnextucn.x . . . . 5  |-  X  =  ( Base `  V
)
1817, 4tpsuni 20030 . . . 4  |-  ( V  e.  TopSp  ->  X  =  U. J )
193, 18syl 17 . . 3  |-  ( ph  ->  X  =  U. J
)
2016, 19sseqtrd 3454 . 2  |-  ( ph  ->  A  C_  U. J )
21 cnextucn.c . . 3  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  X )
2221, 19eqtrd 2505 . 2  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  U. J )
2310, 11istps 20028 . . . . . 6  |-  ( W  e.  TopSp 
<->  K  e.  (TopOn `  Y ) )
249, 23sylib 201 . . . . 5  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
2524adantr 472 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  K  e.  (TopOn `  Y )
)
2619eleq2d 2534 . . . . . . 7  |-  ( ph  ->  ( x  e.  X  <->  x  e.  U. J ) )
2726biimpar 493 . . . . . 6  |-  ( (
ph  /\  x  e.  U. J )  ->  x  e.  X )
2821adantr 472 . . . . . 6  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( cls `  J
) `  A )  =  X )
2927, 28eleqtrrd 2552 . . . . 5  |-  ( (
ph  /\  x  e.  U. J )  ->  x  e.  ( ( cls `  J
) `  A )
)
301toptopon 20025 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
316, 30sylib 201 . . . . . . . 8  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
32 fveq2 5879 . . . . . . . . . 10  |-  ( X  =  U. J  -> 
(TopOn `  X )  =  (TopOn `  U. J ) )
3332eleq2d 2534 . . . . . . . . 9  |-  ( X  =  U. J  -> 
( J  e.  (TopOn `  X )  <->  J  e.  (TopOn `  U. J ) ) )
3419, 33syl 17 . . . . . . . 8  |-  ( ph  ->  ( J  e.  (TopOn `  X )  <->  J  e.  (TopOn `  U. J ) ) )
3531, 34mpbird 240 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  X ) )
3635adantr 472 . . . . . 6  |-  ( (
ph  /\  x  e.  U. J )  ->  J  e.  (TopOn `  X )
)
3716adantr 472 . . . . . 6  |-  ( (
ph  /\  x  e.  U. J )  ->  A  C_  X )
38 trnei 20985 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X  /\  x  e.  X )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) ) )
3936, 37, 27, 38syl3anc 1292 . . . . 5  |-  ( (
ph  /\  x  e.  U. J )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) ) )
4029, 39mpbid 215 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) )
418adantr 472 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  F : A --> Y )
42 flfval 21083 . . . 4  |-  ( ( K  e.  (TopOn `  Y )  /\  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  /\  F : A --> Y )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =  ( K  fLim  ( ( Y  FilMap  F ) `  ( ( ( nei `  J ) `  {
x } )t  A ) ) ) )
4325, 40, 41, 42syl3anc 1292 . . 3  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =  ( K  fLim  ( ( Y  FilMap  F ) `  ( ( ( nei `  J ) `  {
x } )t  A ) ) ) )
44 cnextucn.w . . . . 5  |-  ( ph  ->  W  e. CUnifSp )
4544adantr 472 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  W  e. CUnifSp )
46 cnextucn.l . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  (CauFilu `  U ) )
4727, 46syldan 478 . . . . 5  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  (CauFilu `  U ) )
48 cnextucn.u . . . . . 6  |-  U  =  (UnifSt `  W )
4948fveq2i 5882 . . . . 5  |-  (CauFilu `  U
)  =  (CauFilu `  (UnifSt `  W ) )
5047, 49syl6eleq 2559 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  (CauFilu `  (UnifSt `  W
) ) )
51 fvex 5889 . . . . . . 7  |-  ( Base `  W )  e.  _V
5210, 51eqeltri 2545 . . . . . 6  |-  Y  e. 
_V
5352a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  U. J )  ->  Y  e.  _V )
54 filfbas 20941 . . . . . 6  |-  ( ( ( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  ->  ( ( ( nei `  J ) `
 { x }
)t 
A )  e.  (
fBas `  A )
)
5540, 54syl 17 . . . . 5  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  (
fBas `  A )
)
56 fmfil 21037 . . . . 5  |-  ( ( Y  e.  _V  /\  ( ( ( nei `  J ) `  {
x } )t  A )  e.  ( fBas `  A
)  /\  F : A
--> Y )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  ( Fil `  Y
) )
5753, 55, 41, 56syl3anc 1292 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  ( Fil `  Y
) )
5810, 11cuspcvg 21394 . . . 4  |-  ( ( W  e. CUnifSp  /\  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  (CauFilu `  (UnifSt `  W
) )  /\  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  ( Fil `  Y
) )  ->  ( K  fLim  ( ( Y 
FilMap  F ) `  (
( ( nei `  J
) `  { x } )t  A ) ) )  =/=  (/) )
5945, 50, 57, 58syl3anc 1292 . . 3  |-  ( (
ph  /\  x  e.  U. J )  ->  ( K  fLim  ( ( Y 
FilMap  F ) `  (
( ( nei `  J
) `  { x } )t  A ) ) )  =/=  (/) )
6043, 59eqnetrd 2710 . 2  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/) )
61 cuspusp 21393 . . . 4  |-  ( W  e. CUnifSp  ->  W  e. UnifSp )
6244, 61syl 17 . . 3  |-  ( ph  ->  W  e. UnifSp )
6311uspreg 21367 . . 3  |-  ( ( W  e. UnifSp  /\  K  e. 
Haus )  ->  K  e.  Reg )
6462, 7, 63syl2anc 673 . 2  |-  ( ph  ->  K  e.  Reg )
651, 2, 6, 7, 15, 20, 22, 60, 64cnextcn 21160 1  |-  ( ph  ->  ( ( JCnExt K
) `  F )  e.  ( J  Cn  K
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    C_ wss 3390   (/)c0 3722   {csn 3959   U.cuni 4190   -->wf 5585   ` cfv 5589  (class class class)co 6308   Basecbs 15199   ↾t crest 15397   TopOpenctopn 15398   fBascfbas 19035   Topctop 19994  TopOnctopon 19995   TopSpctps 19996   clsccl 20110   neicnei 20190    Cn ccn 20317   Hauscha 20401   Regcreg 20402   Filcfil 20938    FilMap cfm 21026    fLim cflim 21027    fLimf cflf 21028  CnExtccnext 21152  UnifStcuss 21346  UnifSpcusp 21347  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
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-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-fin 7591  df-fi 7943  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-usp 21350  df-cusp 21391
This theorem is referenced by:  ucnextcn  21397
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