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Theorem cnextucn 19719
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 2433 . 2  |-  U. J  =  U. J
2 eqid 2433 . 2  |-  U. K  =  U. K
3 cnextucn.v . . 3  |-  ( ph  ->  V  e.  TopSp )
4 cnextucn.j . . . 4  |-  J  =  ( TopOpen `  V )
54tpstop 18385 . . 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 18384 . . . . 5  |-  ( W  e.  TopSp  ->  Y  =  U. K )
139, 12syl 16 . . . 4  |-  ( ph  ->  Y  =  U. K
)
14 feq3 5532 . . . 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 18384 . . . 4  |-  ( V  e.  TopSp  ->  X  =  U. J )
203, 19syl 16 . . 3  |-  ( ph  ->  X  =  U. J
)
2117, 20sseqtrd 3380 . 2  |-  ( ph  ->  A  C_  U. J )
22 cnextucn.c . . 3  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  X )
2322, 20eqtrd 2465 . 2  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  U. J )
2410, 11istps 18382 . . . . . 6  |-  ( W  e.  TopSp 
<->  K  e.  (TopOn `  Y ) )
259, 24sylib 196 . . . . 5  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
2625adantr 462 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  K  e.  (TopOn `  Y )
)
2720eleq2d 2500 . . . . . . 7  |-  ( ph  ->  ( x  e.  X  <->  x  e.  U. J ) )
2827biimpar 482 . . . . . 6  |-  ( (
ph  /\  x  e.  U. J )  ->  x  e.  X )
2922adantr 462 . . . . . 6  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( cls `  J
) `  A )  =  X )
3028, 29eleqtrrd 2510 . . . . 5  |-  ( (
ph  /\  x  e.  U. J )  ->  x  e.  ( ( cls `  J
) `  A )
)
311toptopon 18379 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
326, 31sylib 196 . . . . . . . 8  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
33 fveq2 5679 . . . . . . . . . 10  |-  ( X  =  U. J  -> 
(TopOn `  X )  =  (TopOn `  U. J ) )
3433eleq2d 2500 . . . . . . . . 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 462 . . . . . 6  |-  ( (
ph  /\  x  e.  U. J )  ->  J  e.  (TopOn `  X )
)
3817adantr 462 . . . . . 6  |-  ( (
ph  /\  x  e.  U. J )  ->  A  C_  X )
39 trnei 19306 . . . . . 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 1211 . . . . 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 462 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  F : A --> Y )
43 flfval 19404 . . . 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 1211 . . 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 462 . . . 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 467 . . . . 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 5682 . . . . 5  |-  (CauFilu `  U
)  =  (CauFilu `  (UnifSt `  W ) )
5148, 50syl6eleq 2523 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  (CauFilu `  (UnifSt `  W
) ) )
52 fvex 5689 . . . . . . 7  |-  ( Base `  W )  e.  _V
5310, 52eqeltri 2503 . . . . . 6  |-  Y  e. 
_V
5453a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  U. J )  ->  Y  e.  _V )
55 filfbas 19262 . . . . . 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 19358 . . . . 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 1211 . . . 4  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  ( Fil `  Y
) )
5910, 11cuspcvg 19717 . . . 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 1211 . . 3  |-  ( (
ph  /\  x  e.  U. J )  ->  ( K  fLim  ( ( Y 
FilMap  F ) `  (
( ( nei `  J
) `  { x } )t  A ) ) )  =/=  (/) )
6144, 60eqnetrd 2616 . 2  |-  ( (
ph  /\  x  e.  U. J )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/) )
62 cuspusp 19716 . . . 4  |-  ( W  e. CUnifSp  ->  W  e. UnifSp )
6345, 62syl 16 . . 3  |-  ( ph  ->  W  e. UnifSp )
6411uspreg 19690 . . 3  |-  ( ( W  e. UnifSp  /\  K  e. 
Haus )  ->  K  e.  Reg )
6563, 7, 64syl2anc 654 . 2  |-  ( ph  ->  K  e.  Reg )
661, 2, 6, 7, 16, 21, 23, 61, 65cnextcn 19480 1  |-  ( ph  ->  ( ( JCnExt K
) `  F )  e.  ( J  Cn  K
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362    e. wcel 1755    =/= wne 2596   _Vcvv 2962    C_ wss 3316   (/)c0 3625   {csn 3865   U.cuni 4079   -->wf 5402   ` cfv 5406  (class class class)co 6080   Basecbs 14156   ↾t crest 14341   TopOpenctopn 14342   fBascfbas 17647   Topctop 18339  TopOnctopon 18340   TopSpctps 18342   clsccl 18463   neicnei 18542    Cn ccn 18669   Hauscha 18753   Regcreg 18754   Filcfil 19259    FilMap cfm 19347    fLim cflim 19348    fLimf cflf 19349  CnExtccnext 19472  UnifStcuss 19669  UnifSpcusp 19670  CauFiluccfilu 19702  CUnifSpccusp 19713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-en 7299  df-fin 7302  df-fi 7649  df-rest 14343  df-topgen 14364  df-fbas 17657  df-fg 17658  df-top 18344  df-bases 18346  df-topon 18347  df-topsp 18348  df-cld 18464  df-ntr 18465  df-cls 18466  df-nei 18543  df-cn 18672  df-cnp 18673  df-haus 18760  df-reg 18761  df-tx 18976  df-fil 19260  df-fm 19352  df-flim 19353  df-flf 19354  df-cnext 19473  df-ust 19616  df-utop 19647  df-usp 19673  df-cusp 19714
This theorem is referenced by:  ucnextcn  19720
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