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Theorem cnextucn 20538
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 2467 . 2  |-  U. J  =  U. J
2 eqid 2467 . 2  |-  U. K  =  U. K
3 cnextucn.v . . 3  |-  ( ph  ->  V  e.  TopSp )
4 cnextucn.j . . . 4  |-  J  =  ( TopOpen `  V )
54tpstop 19204 . . 3  |-  ( V  e.  TopSp  ->  J  e.  Top )
63, 5syl 16 . 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 19203 . . . . 5  |-  ( W  e.  TopSp  ->  Y  =  U. K )
139, 12syl 16 . . . 4  |-  ( ph  ->  Y  =  U. K
)
14 feq3 5713 . . . 4  |-  ( Y  =  U. K  -> 
( F : A --> Y 
<->  F : A --> U. K
) )
1513, 14syl 16 . . 3  |-  ( ph  ->  ( F : A --> Y 
<->  F : A --> U. K
) )
168, 15mpbid 210 . 2  |-  ( ph  ->  F : A --> U. K
)
17 cnextucn.a . . 3  |-  ( ph  ->  A  C_  X )
18 cnextucn.x . . . . 5  |-  X  =  ( Base `  V
)
1918, 4tpsuni 19203 . . . 4  |-  ( V  e.  TopSp  ->  X  =  U. J )
203, 19syl 16 . . 3  |-  ( ph  ->  X  =  U. J
)
2117, 20sseqtrd 3540 . 2  |-  ( ph  ->  A  C_  U. J )
22 cnextucn.c . . 3  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  X )
2322, 20eqtrd 2508 . 2  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  U. J )
2410, 11istps 19201 . . . . . 6  |-  ( W  e.  TopSp 
<->  K  e.  (TopOn `  Y ) )
259, 24sylib 196 . . . . 5  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
2625adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  K  e.  (TopOn `  Y )
)
2720eleq2d 2537 . . . . . . 7  |-  ( ph  ->  ( x  e.  X  <->  x  e.  U. J ) )
2827biimpar 485 . . . . . 6  |-  ( (
ph  /\  x  e.  U. J )  ->  x  e.  X )
2922adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( cls `  J
) `  A )  =  X )
3028, 29eleqtrrd 2558 . . . . 5  |-  ( (
ph  /\  x  e.  U. J )  ->  x  e.  ( ( cls `  J
) `  A )
)
311toptopon 19198 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
326, 31sylib 196 . . . . . . . 8  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
33 fveq2 5864 . . . . . . . . . 10  |-  ( X  =  U. J  -> 
(TopOn `  X )  =  (TopOn `  U. J ) )
3433eleq2d 2537 . . . . . . . . 9  |-  ( X  =  U. J  -> 
( J  e.  (TopOn `  X )  <->  J  e.  (TopOn `  U. J ) ) )
3520, 34syl 16 . . . . . . . 8  |-  ( ph  ->  ( J  e.  (TopOn `  X )  <->  J  e.  (TopOn `  U. J ) ) )
3632, 35mpbird 232 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  X ) )
3736adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  U. J )  ->  J  e.  (TopOn `  X )
)
3817adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  U. J )  ->  A  C_  X )
39 trnei 20125 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X  /\  x  e.  X )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) ) )
4037, 38, 28, 39syl3anc 1228 . . . . 5  |-  ( (
ph  /\  x  e.  U. J )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) ) )
4130, 40mpbid 210 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) )
428adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  F : A --> Y )
43 flfval 20223 . . . 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 ) ) ) )
4426, 41, 42, 43syl3anc 1228 . . 3  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =  ( K  fLim  ( ( Y  FilMap  F ) `  ( ( ( nei `  J ) `  {
x } )t  A ) ) ) )
45 cnextucn.w . . . . 5  |-  ( ph  ->  W  e. CUnifSp )
4645adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  W  e. CUnifSp )
47 cnextucn.l . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  (CauFilu `  U ) )
4828, 47syldan 470 . . . . 5  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  (CauFilu `  U ) )
49 cnextucn.u . . . . . 6  |-  U  =  (UnifSt `  W )
5049fveq2i 5867 . . . . 5  |-  (CauFilu `  U
)  =  (CauFilu `  (UnifSt `  W ) )
5148, 50syl6eleq 2565 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  (CauFilu `  (UnifSt `  W
) ) )
52 fvex 5874 . . . . . . 7  |-  ( Base `  W )  e.  _V
5310, 52eqeltri 2551 . . . . . 6  |-  Y  e. 
_V
5453a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  U. J )  ->  Y  e.  _V )
55 filfbas 20081 . . . . . 6  |-  ( ( ( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  ->  ( ( ( nei `  J ) `
 { x }
)t 
A )  e.  (
fBas `  A )
)
5641, 55syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  (
fBas `  A )
)
57 fmfil 20177 . . . . 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
) )
5854, 56, 42, 57syl3anc 1228 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  ( Fil `  Y
) )
5910, 11cuspcvg 20536 . . . 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 ) ) )  =/=  (/) )
6046, 51, 58, 59syl3anc 1228 . . 3  |-  ( (
ph  /\  x  e.  U. J )  ->  ( K  fLim  ( ( Y 
FilMap  F ) `  (
( ( nei `  J
) `  { x } )t  A ) ) )  =/=  (/) )
6144, 60eqnetrd 2760 . 2  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/) )
62 cuspusp 20535 . . . 4  |-  ( W  e. CUnifSp  ->  W  e. UnifSp )
6345, 62syl 16 . . 3  |-  ( ph  ->  W  e. UnifSp )
6411uspreg 20509 . . 3  |-  ( ( W  e. UnifSp  /\  K  e. 
Haus )  ->  K  e.  Reg )
6563, 7, 64syl2anc 661 . 2  |-  ( ph  ->  K  e.  Reg )
661, 2, 6, 7, 16, 21, 23, 61, 65cnextcn 20299 1  |-  ( ph  ->  ( ( JCnExt K
) `  F )  e.  ( J  Cn  K
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    C_ wss 3476   (/)c0 3785   {csn 4027   U.cuni 4245   -->wf 5582   ` cfv 5586  (class class class)co 6282   Basecbs 14483   ↾t crest 14669   TopOpenctopn 14670   fBascfbas 18174   Topctop 19158  TopOnctopon 19159   TopSpctps 19161   clsccl 19282   neicnei 19361    Cn ccn 19488   Hauscha 19572   Regcreg 19573   Filcfil 20078    FilMap cfm 20166    fLim cflim 20167    fLimf cflf 20168  CnExtccnext 20291  UnifStcuss 20488  UnifSpcusp 20489  CauFiluccfilu 20521  CUnifSpccusp 20532
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-fin 7517  df-fi 7867  df-rest 14671  df-topgen 14692  df-fbas 18184  df-fg 18185  df-top 19163  df-bases 19165  df-topon 19166  df-topsp 19167  df-cld 19283  df-ntr 19284  df-cls 19285  df-nei 19362  df-cn 19491  df-cnp 19492  df-haus 19579  df-reg 19580  df-tx 19795  df-fil 20079  df-fm 20171  df-flim 20172  df-flf 20173  df-cnext 20292  df-ust 20435  df-utop 20466  df-usp 20492  df-cusp 20533
This theorem is referenced by:  ucnextcn  20539
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