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Theorem cnextucn 21098
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 2402 . 2  |-  U. J  =  U. J
2 eqid 2402 . 2  |-  U. K  =  U. K
3 cnextucn.v . . 3  |-  ( ph  ->  V  e.  TopSp )
4 cnextucn.j . . . 4  |-  J  =  ( TopOpen `  V )
54tpstop 19732 . . 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 19731 . . . . 5  |-  ( W  e.  TopSp  ->  Y  =  U. K )
139, 12syl 17 . . . 4  |-  ( ph  ->  Y  =  U. K
)
1413feq3d 5702 . . 3  |-  ( ph  ->  ( F : A --> Y 
<->  F : A --> U. K
) )
158, 14mpbid 210 . 2  |-  ( ph  ->  F : A --> U. K
)
16 cnextucn.a . . 3  |-  ( ph  ->  A  C_  X )
17 cnextucn.x . . . . 5  |-  X  =  ( Base `  V
)
1817, 4tpsuni 19731 . . . 4  |-  ( V  e.  TopSp  ->  X  =  U. J )
193, 18syl 17 . . 3  |-  ( ph  ->  X  =  U. J
)
2016, 19sseqtrd 3478 . 2  |-  ( ph  ->  A  C_  U. J )
21 cnextucn.c . . 3  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  X )
2221, 19eqtrd 2443 . 2  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  U. J )
2310, 11istps 19729 . . . . . 6  |-  ( W  e.  TopSp 
<->  K  e.  (TopOn `  Y ) )
249, 23sylib 196 . . . . 5  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
2524adantr 463 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  K  e.  (TopOn `  Y )
)
2619eleq2d 2472 . . . . . . 7  |-  ( ph  ->  ( x  e.  X  <->  x  e.  U. J ) )
2726biimpar 483 . . . . . 6  |-  ( (
ph  /\  x  e.  U. J )  ->  x  e.  X )
2821adantr 463 . . . . . 6  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( cls `  J
) `  A )  =  X )
2927, 28eleqtrrd 2493 . . . . 5  |-  ( (
ph  /\  x  e.  U. J )  ->  x  e.  ( ( cls `  J
) `  A )
)
301toptopon 19726 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
316, 30sylib 196 . . . . . . . 8  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
32 fveq2 5849 . . . . . . . . . 10  |-  ( X  =  U. J  -> 
(TopOn `  X )  =  (TopOn `  U. J ) )
3332eleq2d 2472 . . . . . . . . 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 232 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  X ) )
3635adantr 463 . . . . . 6  |-  ( (
ph  /\  x  e.  U. J )  ->  J  e.  (TopOn `  X )
)
3716adantr 463 . . . . . 6  |-  ( (
ph  /\  x  e.  U. J )  ->  A  C_  X )
38 trnei 20685 . . . . . 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 1230 . . . . 5  |-  ( (
ph  /\  x  e.  U. J )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) ) )
4029, 39mpbid 210 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) )
418adantr 463 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  F : A --> Y )
42 flfval 20783 . . . 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 1230 . . 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 463 . . . 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 468 . . . . 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 5852 . . . . 5  |-  (CauFilu `  U
)  =  (CauFilu `  (UnifSt `  W ) )
5047, 49syl6eleq 2500 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  (CauFilu `  (UnifSt `  W
) ) )
51 fvex 5859 . . . . . . 7  |-  ( Base `  W )  e.  _V
5210, 51eqeltri 2486 . . . . . 6  |-  Y  e. 
_V
5352a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  U. J )  ->  Y  e.  _V )
54 filfbas 20641 . . . . . 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 20737 . . . . 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 1230 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  ( Fil `  Y
) )
5810, 11cuspcvg 21096 . . . 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 1230 . . 3  |-  ( (
ph  /\  x  e.  U. J )  ->  ( K  fLim  ( ( Y 
FilMap  F ) `  (
( ( nei `  J
) `  { x } )t  A ) ) )  =/=  (/) )
6043, 59eqnetrd 2696 . 2  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/) )
61 cuspusp 21095 . . . 4  |-  ( W  e. CUnifSp  ->  W  e. UnifSp )
6244, 61syl 17 . . 3  |-  ( ph  ->  W  e. UnifSp )
6311uspreg 21069 . . 3  |-  ( ( W  e. UnifSp  /\  K  e. 
Haus )  ->  K  e.  Reg )
6462, 7, 63syl2anc 659 . 2  |-  ( ph  ->  K  e.  Reg )
651, 2, 6, 7, 15, 20, 22, 60, 64cnextcn 20859 1  |-  ( ph  ->  ( ( JCnExt K
) `  F )  e.  ( J  Cn  K
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   _Vcvv 3059    C_ wss 3414   (/)c0 3738   {csn 3972   U.cuni 4191   -->wf 5565   ` cfv 5569  (class class class)co 6278   Basecbs 14841   ↾t crest 15035   TopOpenctopn 15036   fBascfbas 18726   Topctop 19686  TopOnctopon 19687   TopSpctps 19689   clsccl 19811   neicnei 19891    Cn ccn 20018   Hauscha 20102   Regcreg 20103   Filcfil 20638    FilMap cfm 20726    fLim cflim 20727    fLimf cflf 20728  CnExtccnext 20851  UnifStcuss 21048  UnifSpcusp 21049  CauFiluccfilu 21081  CUnifSpccusp 21092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-fin 7558  df-fi 7905  df-rest 15037  df-topgen 15058  df-fbas 18736  df-fg 18737  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-cn 20021  df-cnp 20022  df-haus 20109  df-reg 20110  df-tx 20355  df-fil 20639  df-fm 20731  df-flim 20732  df-flf 20733  df-cnext 20852  df-ust 20995  df-utop 21026  df-usp 21052  df-cusp 21093
This theorem is referenced by:  ucnextcn  21099
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