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Theorem xkofvcn 17639
Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 17611.) (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 17459 . . . 4  |-  ( R  e. 𝑛Locally 
Comp  ->  R  e.  Top )
3 eqid 2389 . . . . 5  |-  ( S  ^ k o  R
)  =  ( S  ^ k o  R
)
43xkotopon 17555 . . . 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 16923 . . . 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 17623 . . . 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 17624 . . . . 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 6669 . . . . . . 7  |-  1o  e.  On
13 distopon 16986 . . . . . . 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 17574 . . . . . 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 3770 . . . . . 6  |-  ( y  =  x  ->  { y }  =  { x } )
1817xpeq2d 4844 . . . . 5  |-  ( y  =  x  ->  ( 1o  X.  { y } )  =  ( 1o 
X.  { x }
) )
195, 9, 11, 9, 16, 18cnmpt21 17626 . . . 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 16985 . . . . . 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 2389 . . . . . 6  |-  ( R  ^ k o  ~P 1o )  =  ( R  ^ k o  ~P 1o )
2322xkotopon 17555 . . . . 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 2389 . . . . . 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 17615 . . . . 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 4972 . . . . 5  |-  ( g  =  f  ->  (
g  o.  h )  =  ( f  o.  h ) )
31 coeq2 4973 . . . . 5  |-  ( h  =  ( 1o  X.  { x } )  ->  ( f  o.  h )  =  ( f  o.  ( 1o 
X.  { x }
) ) )
3230, 31sylan9eq 2441 . . . 4  |-  ( ( g  =  f  /\  h  =  ( 1o  X.  { x } ) )  ->  ( g  o.  h )  =  ( f  o.  ( 1o 
X.  { x }
) ) )
335, 9, 10, 19, 5, 24, 29, 32cnmpt22 17629 . . 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 2389 . . . . 5  |-  ( S  ^ k o  ~P 1o )  =  ( S  ^ k o  ~P 1o )
3534xkotopon 17555 . . . 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 6686 . . . . 5  |-  (/)  e.  1o
3837a1i 11 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  (/)  e.  1o )
39 unipw 4357 . . . . . 6  |-  U. ~P 1o  =  1o
4039eqcomi 2393 . . . . 5  |-  1o  =  U. ~P 1o
4140xkopjcn 17611 . . . 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 5669 . . . 4  |-  ( g  =  ( f  o.  ( 1o  X.  {
x } ) )  ->  ( g `  (/) )  =  ( ( f  o.  ( 1o 
X.  { x }
) ) `  (/) ) )
44 vex 2904 . . . . . . 7  |-  x  e. 
_V
4544fconst 5571 . . . . . 6  |-  ( 1o 
X.  { x }
) : 1o --> { x }
46 fvco3 5741 . . . . . 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 5888 . . . . . . 7  |-  ( (/)  e.  1o  ->  ( ( 1o  X.  { x }
) `  (/) )  =  x )
4937, 48ax-mp 8 . . . . . 6  |-  ( ( 1o  X.  { x } ) `  (/) )  =  x
5049fveq2i 5673 . . . . 5  |-  ( f `
 ( ( 1o 
X.  { x }
) `  (/) ) )  =  ( f `  x )
5147, 50eqtri 2409 . . . 4  |-  ( ( f  o.  ( 1o 
X.  { x }
) ) `  (/) )  =  ( f `  x
)
5243, 51syl6eq 2437 . . 3  |-  ( g  =  ( f  o.  ( 1o  X.  {
x } ) )  ->  ( g `  (/) )  =  ( f `
 x ) )
535, 9, 33, 36, 42, 52cnmpt21 17626 . 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 2473 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 1649    e. wcel 1717   (/)c0 3573   ~Pcpw 3744   {csn 3759   U.cuni 3959    e. cmpt 4209   Oncon0 4524    X. cxp 4818    o. ccom 4824   -->wf 5392   ` cfv 5396  (class class class)co 6022    e. cmpt2 6024   1oc1o 6655   Topctop 16883  TopOnctopon 16884    Cn ccn 17212   Compccmp 17373  𝑛Locally cnlly 17451    tX ctx 17515    ^ k o cxko 17516
This theorem is referenced by:  cnmptk1p  17640  cnmptk2  17641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-map 6958  df-ixp 7002  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-fi 7353  df-rest 13579  df-topgen 13596  df-pt 13597  df-top 16888  df-bases 16890  df-topon 16891  df-ntr 17009  df-nei 17087  df-cn 17215  df-cnp 17216  df-cmp 17374  df-nlly 17453  df-tx 17517  df-xko 17518
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