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Theorem xkofvcn 19920
Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 19892.) (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  ^ko  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 19740 . . . 4  |-  ( R  e. 𝑛Locally 
Comp  ->  R  e.  Top )
3 eqid 2467 . . . . 5  |-  ( S  ^ko  R )  =  ( S  ^ko  R )
43xkotopon 19836 . . . 4  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( S  ^ko  R )  e.  (TopOn `  ( R  Cn  S
) ) )
52, 4sylan 471 . . 3  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( S  ^ko  R )  e.  (TopOn `  ( R  Cn  S
) ) )
62adantr 465 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  R  e. 
Top )
7 xkofvcn.1 . . . . 5  |-  X  = 
U. R
87toptopon 19201 . . . 4  |-  ( R  e.  Top  <->  R  e.  (TopOn `  X ) )
96, 8sylib 196 . . 3  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  R  e.  (TopOn `  X )
)
105, 9cnmpt1st 19904 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( f  e.  ( R  Cn  S ) ,  x  e.  X  |->  f )  e.  ( ( ( S  ^ko  R )  tX  R
)  Cn  ( S  ^ko  R ) ) )
115, 9cnmpt2nd 19905 . . . . 5  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( f  e.  ( R  Cn  S ) ,  x  e.  X  |->  x )  e.  ( ( ( S  ^ko  R )  tX  R
)  Cn  R ) )
12 1on 7134 . . . . . . 7  |-  1o  e.  On
13 distopon 19264 . . . . . . 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 19855 . . . . . 6  |-  ( ( ~P 1o  e.  (TopOn `  1o )  /\  R  e.  (TopOn `  X )
)  ->  ( y  e.  X  |->  ( 1o 
X.  { y } ) )  e.  ( R  Cn  ( R  ^ko  ~P 1o ) ) )
1614, 9, 15syl2anc 661 . . . . 5  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( y  e.  X  |->  ( 1o 
X.  { y } ) )  e.  ( R  Cn  ( R  ^ko  ~P 1o ) ) )
17 sneq 4037 . . . . . 6  |-  ( y  =  x  ->  { y }  =  { x } )
1817xpeq2d 5023 . . . . 5  |-  ( y  =  x  ->  ( 1o  X.  { y } )  =  ( 1o 
X.  { x }
) )
195, 9, 11, 9, 16, 18cnmpt21 19907 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( f  e.  ( R  Cn  S ) ,  x  e.  X  |->  ( 1o 
X.  { x }
) )  e.  ( ( ( S  ^ko  R ) 
tX  R )  Cn  ( R  ^ko  ~P 1o ) ) )
20 distop 19263 . . . . . 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 2467 . . . . . 6  |-  ( R  ^ko  ~P 1o )  =  ( R  ^ko  ~P 1o )
2322xkotopon 19836 . . . . 5  |-  ( ( ~P 1o  e.  Top  /\  R  e.  Top )  ->  ( R  ^ko  ~P 1o )  e.  (TopOn `  ( ~P 1o  Cn  R ) ) )
2421, 6, 23syl2anc 661 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( R  ^ko  ~P 1o )  e.  (TopOn `  ( ~P 1o  Cn  R ) ) )
25 simpl 457 . . . . 5  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  R  e. 𝑛Locally  Comp )
26 simpr 461 . . . . 5  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  S  e. 
Top )
27 eqid 2467 . . . . . 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 19896 . . . . 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  ^ko  R )  tX  ( R  ^ko  ~P 1o ) )  Cn  ( S  ^ko  ~P 1o ) ) )
2921, 25, 26, 28syl3anc 1228 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( g  e.  ( R  Cn  S ) ,  h  e.  ( ~P 1o  Cn  R )  |->  ( g  o.  h ) )  e.  ( ( ( S  ^ko  R )  tX  ( R  ^ko  ~P 1o ) )  Cn  ( S  ^ko  ~P 1o ) ) )
30 coeq1 5158 . . . . 5  |-  ( g  =  f  ->  (
g  o.  h )  =  ( f  o.  h ) )
31 coeq2 5159 . . . . 5  |-  ( h  =  ( 1o  X.  { x } )  ->  ( f  o.  h )  =  ( f  o.  ( 1o 
X.  { x }
) ) )
3230, 31sylan9eq 2528 . . . 4  |-  ( ( g  =  f  /\  h  =  ( 1o  X.  { x } ) )  ->  ( g  o.  h )  =  ( f  o.  ( 1o 
X.  { x }
) ) )
335, 9, 10, 19, 5, 24, 29, 32cnmpt22 19910 . . 3  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( f  e.  ( R  Cn  S ) ,  x  e.  X  |->  ( f  o.  ( 1o  X.  { x } ) ) )  e.  ( ( ( S  ^ko  R ) 
tX  R )  Cn  ( S  ^ko  ~P 1o ) ) )
34 eqid 2467 . . . . 5  |-  ( S  ^ko  ~P 1o )  =  ( S  ^ko  ~P 1o )
3534xkotopon 19836 . . . 4  |-  ( ( ~P 1o  e.  Top  /\  S  e.  Top )  ->  ( S  ^ko  ~P 1o )  e.  (TopOn `  ( ~P 1o  Cn  S ) ) )
3621, 26, 35syl2anc 661 . . 3  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( S  ^ko  ~P 1o )  e.  (TopOn `  ( ~P 1o  Cn  S ) ) )
37 0lt1o 7151 . . . . 5  |-  (/)  e.  1o
3837a1i 11 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  (/)  e.  1o )
39 unipw 4697 . . . . . 6  |-  U. ~P 1o  =  1o
4039eqcomi 2480 . . . . 5  |-  1o  =  U. ~P 1o
4140xkopjcn 19892 . . . 4  |-  ( ( ~P 1o  e.  Top  /\  S  e.  Top  /\  (/) 
e.  1o )  -> 
( g  e.  ( ~P 1o  Cn  S
)  |->  ( g `  (/) ) )  e.  ( ( S  ^ko  ~P 1o )  Cn  S ) )
4221, 26, 38, 41syl3anc 1228 . . 3  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( g  e.  ( ~P 1o  Cn  S )  |->  ( g `
 (/) ) )  e.  ( ( S  ^ko  ~P 1o )  Cn  S ) )
43 fveq1 5863 . . . 4  |-  ( g  =  ( f  o.  ( 1o  X.  {
x } ) )  ->  ( g `  (/) )  =  ( ( f  o.  ( 1o 
X.  { x }
) ) `  (/) ) )
44 vex 3116 . . . . . . 7  |-  x  e. 
_V
4544fconst 5769 . . . . . 6  |-  ( 1o 
X.  { x }
) : 1o --> { x }
46 fvco3 5942 . . . . . 6  |-  ( ( ( 1o  X.  {
x } ) : 1o --> { x }  /\  (/)  e.  1o )  ->  ( ( f  o.  ( 1o  X.  { x } ) ) `  (/) )  =  ( f `  (
( 1o  X.  {
x } ) `  (/) ) ) )
4745, 37, 46mp2an 672 . . . . 5  |-  ( ( f  o.  ( 1o 
X.  { x }
) ) `  (/) )  =  ( f `  (
( 1o  X.  {
x } ) `  (/) ) )
4844fvconst2 6114 . . . . . . 7  |-  ( (/)  e.  1o  ->  ( ( 1o  X.  { x }
) `  (/) )  =  x )
4937, 48ax-mp 5 . . . . . 6  |-  ( ( 1o  X.  { x } ) `  (/) )  =  x
5049fveq2i 5867 . . . . 5  |-  ( f `
 ( ( 1o 
X.  { x }
) `  (/) ) )  =  ( f `  x )
5147, 50eqtri 2496 . . . 4  |-  ( ( f  o.  ( 1o 
X.  { x }
) ) `  (/) )  =  ( f `  x
)
5243, 51syl6eq 2524 . . 3  |-  ( g  =  ( f  o.  ( 1o  X.  {
x } ) )  ->  ( g `  (/) )  =  ( f `
 x ) )
535, 9, 33, 36, 42, 52cnmpt21 19907 . 2  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( f  e.  ( R  Cn  S ) ,  x  e.  X  |->  ( f `
 x ) )  e.  ( ( ( S  ^ko  R )  tX  R
)  Cn  S ) )
541, 53syl5eqel 2559 1  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  F  e.  ( ( ( S  ^ko  R )  tX  R )  Cn  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   (/)c0 3785   ~Pcpw 4010   {csn 4027   U.cuni 4245    |-> cmpt 4505   Oncon0 4878    X. cxp 4997    o. ccom 5003   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1oc1o 7120   Topctop 19161  TopOnctopon 19162    Cn ccn 19491   Compccmp 19652  𝑛Locally cnlly 19732    tX ctx 19796    ^ko cxko 19797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fi 7867  df-rest 14674  df-topgen 14695  df-pt 14696  df-top 19166  df-bases 19168  df-topon 19169  df-ntr 19287  df-nei 19365  df-cn 19494  df-cnp 19495  df-cmp 19653  df-nlly 19734  df-tx 19798  df-xko 19799
This theorem is referenced by:  cnmptk1p  19921  cnmptk2  19922
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