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Theorem cnextucn 20005
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 2452 . 2  |-  U. J  =  U. J
2 eqid 2452 . 2  |-  U. K  =  U. K
3 cnextucn.v . . 3  |-  ( ph  ->  V  e.  TopSp )
4 cnextucn.j . . . 4  |-  J  =  ( TopOpen `  V )
54tpstop 18671 . . 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 18670 . . . . 5  |-  ( W  e.  TopSp  ->  Y  =  U. K )
139, 12syl 16 . . . 4  |-  ( ph  ->  Y  =  U. K
)
14 feq3 5647 . . . 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 18670 . . . 4  |-  ( V  e.  TopSp  ->  X  =  U. J )
203, 19syl 16 . . 3  |-  ( ph  ->  X  =  U. J
)
2117, 20sseqtrd 3495 . 2  |-  ( ph  ->  A  C_  U. J )
22 cnextucn.c . . 3  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  X )
2322, 20eqtrd 2493 . 2  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  U. J )
2410, 11istps 18668 . . . . . 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 2522 . . . . . . 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 2543 . . . . 5  |-  ( (
ph  /\  x  e.  U. J )  ->  x  e.  ( ( cls `  J
) `  A )
)
311toptopon 18665 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
326, 31sylib 196 . . . . . . . 8  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
33 fveq2 5794 . . . . . . . . . 10  |-  ( X  =  U. J  -> 
(TopOn `  X )  =  (TopOn `  U. J ) )
3433eleq2d 2522 . . . . . . . . 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 19592 . . . . . 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 1219 . . . . 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 19690 . . . 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 1219 . . 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 5797 . . . . 5  |-  (CauFilu `  U
)  =  (CauFilu `  (UnifSt `  W ) )
5148, 50syl6eleq 2550 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  (CauFilu `  (UnifSt `  W
) ) )
52 fvex 5804 . . . . . . 7  |-  ( Base `  W )  e.  _V
5310, 52eqeltri 2536 . . . . . 6  |-  Y  e. 
_V
5453a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  U. J )  ->  Y  e.  _V )
55 filfbas 19548 . . . . . 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 19644 . . . . 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 1219 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  ( Fil `  Y
) )
5910, 11cuspcvg 20003 . . . 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 1219 . . 3  |-  ( (
ph  /\  x  e.  U. J )  ->  ( K  fLim  ( ( Y 
FilMap  F ) `  (
( ( nei `  J
) `  { x } )t  A ) ) )  =/=  (/) )
6144, 60eqnetrd 2742 . 2  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/) )
62 cuspusp 20002 . . . 4  |-  ( W  e. CUnifSp  ->  W  e. UnifSp )
6345, 62syl 16 . . 3  |-  ( ph  ->  W  e. UnifSp )
6411uspreg 19976 . . 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 19766 1  |-  ( ph  ->  ( ( JCnExt K
) `  F )  e.  ( J  Cn  K
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645   _Vcvv 3072    C_ wss 3431   (/)c0 3740   {csn 3980   U.cuni 4194   -->wf 5517   ` cfv 5521  (class class class)co 6195   Basecbs 14287   ↾t crest 14473   TopOpenctopn 14474   fBascfbas 17924   Topctop 18625  TopOnctopon 18626   TopSpctps 18628   clsccl 18749   neicnei 18828    Cn ccn 18955   Hauscha 19039   Regcreg 19040   Filcfil 19545    FilMap cfm 19633    fLim cflim 19634    fLimf cflf 19635  CnExtccnext 19758  UnifStcuss 19955  UnifSpcusp 19956  CauFiluccfilu 19988  CUnifSpccusp 19999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-map 7321  df-pm 7322  df-en 7416  df-fin 7419  df-fi 7767  df-rest 14475  df-topgen 14496  df-fbas 17934  df-fg 17935  df-top 18630  df-bases 18632  df-topon 18633  df-topsp 18634  df-cld 18750  df-ntr 18751  df-cls 18752  df-nei 18829  df-cn 18958  df-cnp 18959  df-haus 19046  df-reg 19047  df-tx 19262  df-fil 19546  df-fm 19638  df-flim 19639  df-flf 19640  df-cnext 19759  df-ust 19902  df-utop 19933  df-usp 19959  df-cusp 20000
This theorem is referenced by:  ucnextcn  20006
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