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Theorem ucnextcn 20014
Description: Extension by continuity. Theorem 2 of [BourbakiTop1] p. II.20. Given an uniform space on a set  X, a subset  A dense in  X, and a function  F uniformly continuous from  A to  Y, that function can be extended by continuity to the whole  X, and its extension is uniformly continuous. (Contributed by Thierry Arnoux, 25-Jan-2018.)
Hypotheses
Ref Expression
ucnextcn.x  |-  X  =  ( Base `  V
)
ucnextcn.y  |-  Y  =  ( Base `  W
)
ucnextcn.j  |-  J  =  ( TopOpen `  V )
ucnextcn.k  |-  K  =  ( TopOpen `  W )
ucnextcn.s  |-  S  =  (UnifSt `  V )
ucnextcn.t  |-  T  =  (UnifSt `  ( Vs  A
) )
ucnextcn.u  |-  U  =  (UnifSt `  W )
ucnextcn.v  |-  ( ph  ->  V  e.  TopSp )
ucnextcn.r  |-  ( ph  ->  V  e. UnifSp )
ucnextcn.w  |-  ( ph  ->  W  e.  TopSp )
ucnextcn.z  |-  ( ph  ->  W  e. CUnifSp )
ucnextcn.h  |-  ( ph  ->  K  e.  Haus )
ucnextcn.a  |-  ( ph  ->  A  C_  X )
ucnextcn.f  |-  ( ph  ->  F  e.  ( T Cnu U ) )
ucnextcn.c  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  X )
Assertion
Ref Expression
ucnextcn  |-  ( ph  ->  ( ( JCnExt K
) `  F )  e.  ( J  Cn  K
) )

Proof of Theorem ucnextcn
Dummy variables  a 
b  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ucnextcn.x . 2  |-  X  =  ( Base `  V
)
2 ucnextcn.y . 2  |-  Y  =  ( Base `  W
)
3 ucnextcn.j . 2  |-  J  =  ( TopOpen `  V )
4 ucnextcn.k . 2  |-  K  =  ( TopOpen `  W )
5 ucnextcn.u . 2  |-  U  =  (UnifSt `  W )
6 ucnextcn.v . 2  |-  ( ph  ->  V  e.  TopSp )
7 ucnextcn.w . 2  |-  ( ph  ->  W  e.  TopSp )
8 ucnextcn.z . 2  |-  ( ph  ->  W  e. CUnifSp )
9 ucnextcn.h . 2  |-  ( ph  ->  K  e.  Haus )
10 ucnextcn.a . 2  |-  ( ph  ->  A  C_  X )
11 ucnextcn.f . . . 4  |-  ( ph  ->  F  e.  ( T Cnu U ) )
12 ucnextcn.r . . . . . 6  |-  ( ph  ->  V  e. UnifSp )
13 ucnextcn.t . . . . . . 7  |-  T  =  (UnifSt `  ( Vs  A
) )
141, 13ressust 19974 . . . . . 6  |-  ( ( V  e. UnifSp  /\  A  C_  X )  ->  T  e.  (UnifOn `  A )
)
1512, 10, 14syl2anc 661 . . . . 5  |-  ( ph  ->  T  e.  (UnifOn `  A ) )
16 cuspusp 20010 . . . . . . . 8  |-  ( W  e. CUnifSp  ->  W  e. UnifSp )
178, 16syl 16 . . . . . . 7  |-  ( ph  ->  W  e. UnifSp )
182, 5, 4isusp 19971 . . . . . . 7  |-  ( W  e. UnifSp 
<->  ( U  e.  (UnifOn `  Y )  /\  K  =  (unifTop `  U )
) )
1917, 18sylib 196 . . . . . 6  |-  ( ph  ->  ( U  e.  (UnifOn `  Y )  /\  K  =  (unifTop `  U )
) )
2019simpld 459 . . . . 5  |-  ( ph  ->  U  e.  (UnifOn `  Y ) )
21 isucn 19988 . . . . 5  |-  ( ( T  e.  (UnifOn `  A )  /\  U  e.  (UnifOn `  Y )
)  ->  ( F  e.  ( T Cnu U )  <->  ( F : A --> Y  /\  A. w  e.  U  E. v  e.  T  A. y  e.  A  A. z  e.  A  (
y v z  -> 
( F `  y
) w ( F `
 z ) ) ) ) )
2215, 20, 21syl2anc 661 . . . 4  |-  ( ph  ->  ( F  e.  ( T Cnu U )  <->  ( F : A --> Y  /\  A. w  e.  U  E. v  e.  T  A. y  e.  A  A. z  e.  A  (
y v z  -> 
( F `  y
) w ( F `
 z ) ) ) ) )
2311, 22mpbid 210 . . 3  |-  ( ph  ->  ( F : A --> Y  /\  A. w  e.  U  E. v  e.  T  A. y  e.  A  A. z  e.  A  ( y v z  ->  ( F `  y ) w ( F `  z ) ) ) )
2423simpld 459 . 2  |-  ( ph  ->  F : A --> Y )
25 ucnextcn.c . 2  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  X )
2620adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  U  e.  (UnifOn `  Y )
)
2726elfvexd 5830 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  Y  e.  _V )
28 simpr 461 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
2925adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( cls `  J
) `  A )  =  X )
3028, 29eleqtrrd 2545 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  ( ( cls `  J
) `  A )
)
311, 3istps 18676 . . . . . . . . 9  |-  ( V  e.  TopSp 
<->  J  e.  (TopOn `  X ) )
326, 31sylib 196 . . . . . . . 8  |-  ( ph  ->  J  e.  (TopOn `  X ) )
3332adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  J  e.  (TopOn `  X )
)
3410adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  A  C_  X )
35 trnei 19600 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X  /\  x  e.  X )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) ) )
3633, 34, 28, 35syl3anc 1219 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) ) )
3730, 36mpbid 210 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) )
38 filfbas 19556 . . . . 5  |-  ( ( ( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  ->  ( ( ( nei `  J ) `
 { x }
)t 
A )  e.  (
fBas `  A )
)
3937, 38syl 16 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  (
fBas `  A )
)
4024adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  F : A --> Y )
41 fmval 19651 . . . 4  |-  ( ( Y  e.  _V  /\  ( ( ( nei `  J ) `  {
x } )t  A )  e.  ( fBas `  A
)  /\  F : A
--> Y )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  =  ( Y filGen ran  (
a  e.  ( ( ( nei `  J
) `  { x } )t  A )  |->  ( F
" a ) ) ) )
4227, 39, 40, 41syl3anc 1219 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  =  ( Y filGen ran  (
a  e.  ( ( ( nei `  J
) `  { x } )t  A )  |->  ( F
" a ) ) ) )
4315adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  T  e.  (UnifOn `  A )
)
4411adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F  e.  ( T Cnu U ) )
45 ucnextcn.s . . . . . . . . . . 11  |-  S  =  (UnifSt `  V )
461, 45, 3isusp 19971 . . . . . . . . . 10  |-  ( V  e. UnifSp 
<->  ( S  e.  (UnifOn `  X )  /\  J  =  (unifTop `  S )
) )
4712, 46sylib 196 . . . . . . . . 9  |-  ( ph  ->  ( S  e.  (UnifOn `  X )  /\  J  =  (unifTop `  S )
) )
4847simpld 459 . . . . . . . 8  |-  ( ph  ->  S  e.  (UnifOn `  X ) )
4948adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  S  e.  (UnifOn `  X )
)
5012adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  V  e. UnifSp )
516adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  V  e.  TopSp )
521, 3, 45neipcfilu 20006 . . . . . . . 8  |-  ( ( V  e. UnifSp  /\  V  e. 
TopSp  /\  x  e.  X
)  ->  ( ( nei `  J ) `  { x } )  e.  (CauFilu `  S ) )
5350, 51, 28, 52syl3anc 1219 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( nei `  J
) `  { x } )  e.  (CauFilu `  S ) )
54 0nelfb 19539 . . . . . . . 8  |-  ( ( ( ( nei `  J
) `  { x } )t  A )  e.  (
fBas `  A )  ->  -.  (/)  e.  ( ( ( nei `  J
) `  { x } )t  A ) )
5539, 54syl 16 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  -.  (/) 
e.  ( ( ( nei `  J ) `
 { x }
)t 
A ) )
56 trcfilu 20004 . . . . . . 7  |-  ( ( S  e.  (UnifOn `  X )  /\  (
( ( nei `  J
) `  { x } )  e.  (CauFilu `  S )  /\  -.  (/) 
e.  ( ( ( nei `  J ) `
 { x }
)t 
A ) )  /\  A  C_  X )  -> 
( ( ( nei `  J ) `  {
x } )t  A )  e.  (CauFilu `  ( St  ( A  X.  A ) ) ) )
5749, 53, 55, 34, 56syl121anc 1224 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  (CauFilu `  ( St  ( A  X.  A ) ) ) )
5843elfvexd 5830 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  _V )
59 ressuss 19973 . . . . . . . . 9  |-  ( A  e.  _V  ->  (UnifSt `  ( Vs  A ) )  =  ( (UnifSt `  V
)t  ( A  X.  A
) ) )
6045oveq1i 6213 . . . . . . . . 9  |-  ( St  ( A  X.  A ) )  =  ( (UnifSt `  V )t  ( A  X.  A ) )
6159, 13, 603eqtr4g 2520 . . . . . . . 8  |-  ( A  e.  _V  ->  T  =  ( St  ( A  X.  A ) ) )
6261fveq2d 5806 . . . . . . 7  |-  ( A  e.  _V  ->  (CauFilu `  T )  =  (CauFilu `  ( St  ( A  X.  A ) ) ) )
6358, 62syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (CauFilu `  T )  =  (CauFilu `  ( St  ( A  X.  A ) ) ) )
6457, 63eleqtrrd 2545 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  (CauFilu `  T ) )
65 imaeq2 5276 . . . . . . 7  |-  ( a  =  b  ->  ( F " a )  =  ( F " b
) )
6665cbvmptv 4494 . . . . . 6  |-  ( a  e.  ( ( ( nei `  J ) `
 { x }
)t 
A )  |->  ( F
" a ) )  =  ( b  e.  ( ( ( nei `  J ) `  {
x } )t  A ) 
|->  ( F " b
) )
6766rneqi 5177 . . . . 5  |-  ran  (
a  e.  ( ( ( nei `  J
) `  { x } )t  A )  |->  ( F
" a ) )  =  ran  ( b  e.  ( ( ( nei `  J ) `
 { x }
)t 
A )  |->  ( F
" b ) )
6843, 26, 44, 64, 67fmucnd 20002 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ran  ( a  e.  ( ( ( nei `  J
) `  { x } )t  A )  |->  ( F
" a ) )  e.  (CauFilu `  U ) )
69 cfilufg 20003 . . . 4  |-  ( ( U  e.  (UnifOn `  Y )  /\  ran  ( a  e.  ( ( ( nei `  J
) `  { x } )t  A )  |->  ( F
" a ) )  e.  (CauFilu `  U ) )  ->  ( Y filGen ran  ( a  e.  ( ( ( nei `  J
) `  { x } )t  A )  |->  ( F
" a ) ) )  e.  (CauFilu `  U
) )
7026, 68, 69syl2anc 661 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( Y filGen ran  ( a  e.  ( ( ( nei `  J ) `  {
x } )t  A ) 
|->  ( F " a
) ) )  e.  (CauFilu `  U ) )
7142, 70eqeltrd 2542 . 2  |-  ( (
ph  /\  x  e.  X )  ->  (
( Y  FilMap  F ) `
 ( ( ( nei `  J ) `
 { x }
)t 
A ) )  e.  (CauFilu `  U ) )
721, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 25, 71cnextucn 20013 1  |-  ( ph  ->  ( ( JCnExt K
) `  F )  e.  ( J  Cn  K
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   E.wrex 2800   _Vcvv 3078    C_ wss 3439   (/)c0 3748   {csn 3988   class class class wbr 4403    |-> cmpt 4461    X. cxp 4949   ran crn 4952   "cima 4954   -->wf 5525   ` cfv 5529  (class class class)co 6203   Basecbs 14295   ↾s cress 14296   ↾t crest 14481   TopOpenctopn 14482   fBascfbas 17932   filGencfg 17933  TopOnctopon 18634   TopSpctps 18636   clsccl 18757   neicnei 18836    Cn ccn 18963   Hauscha 19047   Filcfil 19553    FilMap cfm 19641  CnExtccnext 19766  UnifOncust 19909  unifTopcutop 19940  UnifStcuss 19963  UnifSpcusp 19964   Cnucucn 19985  CauFiluccfilu 19996  CUnifSpccusp 20007
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fi 7775  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-dec 10870  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-unif 14383  df-rest 14483  df-topgen 14504  df-fbas 17942  df-fg 17943  df-top 18638  df-bases 18640  df-topon 18641  df-topsp 18642  df-cld 18758  df-ntr 18759  df-cls 18760  df-nei 18837  df-cn 18966  df-cnp 18967  df-haus 19054  df-reg 19055  df-tx 19270  df-fil 19554  df-fm 19646  df-flim 19647  df-flf 19648  df-cnext 19767  df-ust 19910  df-utop 19941  df-uss 19966  df-usp 19967  df-ucn 19986  df-cfilu 19997  df-cusp 20008
This theorem is referenced by:  rrhcn  26591
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