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Theorem xkofvcn 17704
Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 17676.) (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
xkofvcn.1  |-  X  = 
U. R
xkofvcn.2  |-  F  =  ( f  e.  ( R  Cn  S ) ,  x  e.  X  |->  ( f `  x
) )
Assertion
Ref Expression
xkofvcn  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  F  e.  ( ( ( S  ^ k o  R
)  tX  R )  Cn  S ) )
Distinct variable groups:    x, f, R    S, f, x    f, X, x
Allowed substitution hints:    F( x, f)

Proof of Theorem xkofvcn
Dummy variables  g  h  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xkofvcn.2 . 2  |-  F  =  ( f  e.  ( R  Cn  S ) ,  x  e.  X  |->  ( f `  x
) )
2 nllytop 17524 . . . 4  |-  ( R  e. 𝑛Locally 
Comp  ->  R  e.  Top )
3 eqid 2435 . . . . 5  |-  ( S  ^ k o  R
)  =  ( S  ^ k o  R
)
43xkotopon 17620 . . . 4  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( S  ^ k o  R )  e.  (TopOn `  ( R  Cn  S
) ) )
52, 4sylan 458 . . 3  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( S  ^ k o  R
)  e.  (TopOn `  ( R  Cn  S
) ) )
62adantr 452 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  R  e. 
Top )
7 xkofvcn.1 . . . . 5  |-  X  = 
U. R
87toptopon 16986 . . . 4  |-  ( R  e.  Top  <->  R  e.  (TopOn `  X ) )
96, 8sylib 189 . . 3  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  R  e.  (TopOn `  X )
)
105, 9cnmpt1st 17688 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( f  e.  ( R  Cn  S ) ,  x  e.  X  |->  f )  e.  ( ( ( S  ^ k o  R )  tX  R
)  Cn  ( S  ^ k o  R
) ) )
115, 9cnmpt2nd 17689 . . . . 5  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( f  e.  ( R  Cn  S ) ,  x  e.  X  |->  x )  e.  ( ( ( S  ^ k o  R )  tX  R
)  Cn  R ) )
12 1on 6722 . . . . . . 7  |-  1o  e.  On
13 distopon 17049 . . . . . . 7  |-  ( 1o  e.  On  ->  ~P 1o  e.  (TopOn `  1o ) )
1412, 13mp1i 12 . . . . . 6  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ~P 1o  e.  (TopOn `  1o )
)
15 xkoccn 17639 . . . . . 6  |-  ( ( ~P 1o  e.  (TopOn `  1o )  /\  R  e.  (TopOn `  X )
)  ->  ( y  e.  X  |->  ( 1o 
X.  { y } ) )  e.  ( R  Cn  ( R  ^ k o  ~P 1o ) ) )
1614, 9, 15syl2anc 643 . . . . 5  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( y  e.  X  |->  ( 1o 
X.  { y } ) )  e.  ( R  Cn  ( R  ^ k o  ~P 1o ) ) )
17 sneq 3817 . . . . . 6  |-  ( y  =  x  ->  { y }  =  { x } )
1817xpeq2d 4893 . . . . 5  |-  ( y  =  x  ->  ( 1o  X.  { y } )  =  ( 1o 
X.  { x }
) )
195, 9, 11, 9, 16, 18cnmpt21 17691 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( f  e.  ( R  Cn  S ) ,  x  e.  X  |->  ( 1o 
X.  { x }
) )  e.  ( ( ( S  ^ k o  R )  tX  R )  Cn  ( R  ^ k o  ~P 1o ) ) )
20 distop 17048 . . . . . 6  |-  ( 1o  e.  On  ->  ~P 1o  e.  Top )
2112, 20mp1i 12 . . . . 5  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ~P 1o  e.  Top )
22 eqid 2435 . . . . . 6  |-  ( R  ^ k o  ~P 1o )  =  ( R  ^ k o  ~P 1o )
2322xkotopon 17620 . . . . 5  |-  ( ( ~P 1o  e.  Top  /\  R  e.  Top )  ->  ( R  ^ k o  ~P 1o )  e.  (TopOn `  ( ~P 1o  Cn  R ) ) )
2421, 6, 23syl2anc 643 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( R  ^ k o  ~P 1o )  e.  (TopOn `  ( ~P 1o  Cn  R ) ) )
25 simpl 444 . . . . 5  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  R  e. 𝑛Locally  Comp )
26 simpr 448 . . . . 5  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  S  e. 
Top )
27 eqid 2435 . . . . . 6  |-  ( g  e.  ( R  Cn  S ) ,  h  e.  ( ~P 1o  Cn  R )  |->  ( g  o.  h ) )  =  ( g  e.  ( R  Cn  S
) ,  h  e.  ( ~P 1o  Cn  R )  |->  ( g  o.  h ) )
2827xkococn 17680 . . . . 5  |-  ( ( ~P 1o  e.  Top  /\  R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( g  e.  ( R  Cn  S ) ,  h  e.  ( ~P 1o  Cn  R )  |->  ( g  o.  h ) )  e.  ( ( ( S  ^ k o  R )  tX  ( R  ^ k o  ~P 1o ) )  Cn  ( S  ^ k o  ~P 1o ) ) )
2921, 25, 26, 28syl3anc 1184 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( g  e.  ( R  Cn  S ) ,  h  e.  ( ~P 1o  Cn  R )  |->  ( g  o.  h ) )  e.  ( ( ( S  ^ k o  R )  tX  ( R  ^ k o  ~P 1o ) )  Cn  ( S  ^ k o  ~P 1o ) ) )
30 coeq1 5021 . . . . 5  |-  ( g  =  f  ->  (
g  o.  h )  =  ( f  o.  h ) )
31 coeq2 5022 . . . . 5  |-  ( h  =  ( 1o  X.  { x } )  ->  ( f  o.  h )  =  ( f  o.  ( 1o 
X.  { x }
) ) )
3230, 31sylan9eq 2487 . . . 4  |-  ( ( g  =  f  /\  h  =  ( 1o  X.  { x } ) )  ->  ( g  o.  h )  =  ( f  o.  ( 1o 
X.  { x }
) ) )
335, 9, 10, 19, 5, 24, 29, 32cnmpt22 17694 . . 3  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( f  e.  ( R  Cn  S ) ,  x  e.  X  |->  ( f  o.  ( 1o  X.  { x } ) ) )  e.  ( ( ( S  ^ k o  R )  tX  R )  Cn  ( S  ^ k o  ~P 1o ) ) )
34 eqid 2435 . . . . 5  |-  ( S  ^ k o  ~P 1o )  =  ( S  ^ k o  ~P 1o )
3534xkotopon 17620 . . . 4  |-  ( ( ~P 1o  e.  Top  /\  S  e.  Top )  ->  ( S  ^ k o  ~P 1o )  e.  (TopOn `  ( ~P 1o  Cn  S ) ) )
3621, 26, 35syl2anc 643 . . 3  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( S  ^ k o  ~P 1o )  e.  (TopOn `  ( ~P 1o  Cn  S ) ) )
37 0lt1o 6739 . . . . 5  |-  (/)  e.  1o
3837a1i 11 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  (/)  e.  1o )
39 unipw 4406 . . . . . 6  |-  U. ~P 1o  =  1o
4039eqcomi 2439 . . . . 5  |-  1o  =  U. ~P 1o
4140xkopjcn 17676 . . . 4  |-  ( ( ~P 1o  e.  Top  /\  S  e.  Top  /\  (/) 
e.  1o )  -> 
( g  e.  ( ~P 1o  Cn  S
)  |->  ( g `  (/) ) )  e.  ( ( S  ^ k o  ~P 1o )  Cn  S ) )
4221, 26, 38, 41syl3anc 1184 . . 3  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( g  e.  ( ~P 1o  Cn  S )  |->  ( g `
 (/) ) )  e.  ( ( S  ^ k o  ~P 1o )  Cn  S ) )
43 fveq1 5718 . . . 4  |-  ( g  =  ( f  o.  ( 1o  X.  {
x } ) )  ->  ( g `  (/) )  =  ( ( f  o.  ( 1o 
X.  { x }
) ) `  (/) ) )
44 vex 2951 . . . . . . 7  |-  x  e. 
_V
4544fconst 5620 . . . . . 6  |-  ( 1o 
X.  { x }
) : 1o --> { x }
46 fvco3 5791 . . . . . 6  |-  ( ( ( 1o  X.  {
x } ) : 1o --> { x }  /\  (/)  e.  1o )  ->  ( ( f  o.  ( 1o  X.  { x } ) ) `  (/) )  =  ( f `  (
( 1o  X.  {
x } ) `  (/) ) ) )
4745, 37, 46mp2an 654 . . . . 5  |-  ( ( f  o.  ( 1o 
X.  { x }
) ) `  (/) )  =  ( f `  (
( 1o  X.  {
x } ) `  (/) ) )
4844fvconst2 5938 . . . . . . 7  |-  ( (/)  e.  1o  ->  ( ( 1o  X.  { x }
) `  (/) )  =  x )
4937, 48ax-mp 8 . . . . . 6  |-  ( ( 1o  X.  { x } ) `  (/) )  =  x
5049fveq2i 5722 . . . . 5  |-  ( f `
 ( ( 1o 
X.  { x }
) `  (/) ) )  =  ( f `  x )
5147, 50eqtri 2455 . . . 4  |-  ( ( f  o.  ( 1o 
X.  { x }
) ) `  (/) )  =  ( f `  x
)
5243, 51syl6eq 2483 . . 3  |-  ( g  =  ( f  o.  ( 1o  X.  {
x } ) )  ->  ( g `  (/) )  =  ( f `
 x ) )
535, 9, 33, 36, 42, 52cnmpt21 17691 . 2  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( f  e.  ( R  Cn  S ) ,  x  e.  X  |->  ( f `
 x ) )  e.  ( ( ( S  ^ k o  R )  tX  R
)  Cn  S ) )
541, 53syl5eqel 2519 1  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  F  e.  ( ( ( S  ^ k o  R
)  tX  R )  Cn  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   (/)c0 3620   ~Pcpw 3791   {csn 3806   U.cuni 4007    e. cmpt 4258   Oncon0 4573    X. cxp 4867    o. ccom 4873   -->wf 5441   ` cfv 5445  (class class class)co 6072    e. cmpt2 6074   1oc1o 6708   Topctop 16946  TopOnctopon 16947    Cn ccn 17276   Compccmp 17437  𝑛Locally cnlly 17516    tX ctx 17580    ^ k o cxko 17581
This theorem is referenced by:  cnmptk1p  17705  cnmptk2  17706
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-map 7011  df-ixp 7055  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-fi 7407  df-rest 13638  df-topgen 13655  df-pt 13656  df-top 16951  df-bases 16953  df-topon 16954  df-ntr 17072  df-nei 17150  df-cn 17279  df-cnp 17280  df-cmp 17438  df-nlly 17518  df-tx 17582  df-xko 17583
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