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Theorem xkofvcn 20636
Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 20608.) (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 20425 . . . 4  |-  ( R  e. 𝑛Locally 
Comp  ->  R  e.  Top )
3 eqid 2423 . . . . 5  |-  ( S  ^ko  R )  =  ( S  ^ko  R )
43xkotopon 20552 . . . 4  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( S  ^ko  R )  e.  (TopOn `  ( R  Cn  S
) ) )
52, 4sylan 473 . . 3  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( S  ^ko  R )  e.  (TopOn `  ( R  Cn  S
) ) )
62adantr 466 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  R  e. 
Top )
7 xkofvcn.1 . . . . 5  |-  X  = 
U. R
87toptopon 19885 . . . 4  |-  ( R  e.  Top  <->  R  e.  (TopOn `  X ) )
96, 8sylib 199 . . 3  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  R  e.  (TopOn `  X )
)
105, 9cnmpt1st 20620 . . . 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 20621 . . . . 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 7139 . . . . . . 7  |-  1o  e.  On
13 distopon 19949 . . . . . . 7  |-  ( 1o  e.  On  ->  ~P 1o  e.  (TopOn `  1o ) )
1412, 13mp1i 13 . . . . . 6  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ~P 1o  e.  (TopOn `  1o )
)
15 xkoccn 20571 . . . . . 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 665 . . . . 5  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( y  e.  X  |->  ( 1o 
X.  { y } ) )  e.  ( R  Cn  ( R  ^ko  ~P 1o ) ) )
17 sneq 3946 . . . . . 6  |-  ( y  =  x  ->  { y }  =  { x } )
1817xpeq2d 4815 . . . . 5  |-  ( y  =  x  ->  ( 1o  X.  { y } )  =  ( 1o 
X.  { x }
) )
195, 9, 11, 9, 16, 18cnmpt21 20623 . . . 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 19948 . . . . . 6  |-  ( 1o  e.  On  ->  ~P 1o  e.  Top )
2112, 20mp1i 13 . . . . 5  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ~P 1o  e.  Top )
22 eqid 2423 . . . . . 6  |-  ( R  ^ko  ~P 1o )  =  ( R  ^ko  ~P 1o )
2322xkotopon 20552 . . . . 5  |-  ( ( ~P 1o  e.  Top  /\  R  e.  Top )  ->  ( R  ^ko  ~P 1o )  e.  (TopOn `  ( ~P 1o  Cn  R ) ) )
2421, 6, 23syl2anc 665 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( R  ^ko  ~P 1o )  e.  (TopOn `  ( ~P 1o  Cn  R ) ) )
25 simpl 458 . . . . 5  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  R  e. 𝑛Locally  Comp )
26 simpr 462 . . . . 5  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  S  e. 
Top )
27 eqid 2423 . . . . . 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 20612 . . . . 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 1264 . . . 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 4949 . . . . 5  |-  ( g  =  f  ->  (
g  o.  h )  =  ( f  o.  h ) )
31 coeq2 4950 . . . . 5  |-  ( h  =  ( 1o  X.  { x } )  ->  ( f  o.  h )  =  ( f  o.  ( 1o 
X.  { x }
) ) )
3230, 31sylan9eq 2477 . . . 4  |-  ( ( g  =  f  /\  h  =  ( 1o  X.  { x } ) )  ->  ( g  o.  h )  =  ( f  o.  ( 1o 
X.  { x }
) ) )
335, 9, 10, 19, 5, 24, 29, 32cnmpt22 20626 . . 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 2423 . . . . 5  |-  ( S  ^ko  ~P 1o )  =  ( S  ^ko  ~P 1o )
3534xkotopon 20552 . . . 4  |-  ( ( ~P 1o  e.  Top  /\  S  e.  Top )  ->  ( S  ^ko  ~P 1o )  e.  (TopOn `  ( ~P 1o  Cn  S ) ) )
3621, 26, 35syl2anc 665 . . 3  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( S  ^ko  ~P 1o )  e.  (TopOn `  ( ~P 1o  Cn  S ) ) )
37 0lt1o 7156 . . . . 5  |-  (/)  e.  1o
3837a1i 11 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  (/)  e.  1o )
39 unipw 4609 . . . . . 6  |-  U. ~P 1o  =  1o
4039eqcomi 2432 . . . . 5  |-  1o  =  U. ~P 1o
4140xkopjcn 20608 . . . 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 1264 . . 3  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( g  e.  ( ~P 1o  Cn  S )  |->  ( g `
 (/) ) )  e.  ( ( S  ^ko  ~P 1o )  Cn  S ) )
43 fveq1 5819 . . . 4  |-  ( g  =  ( f  o.  ( 1o  X.  {
x } ) )  ->  ( g `  (/) )  =  ( ( f  o.  ( 1o 
X.  { x }
) ) `  (/) ) )
44 vex 3020 . . . . . . 7  |-  x  e. 
_V
4544fconst 5724 . . . . . 6  |-  ( 1o 
X.  { x }
) : 1o --> { x }
46 fvco3 5897 . . . . . 6  |-  ( ( ( 1o  X.  {
x } ) : 1o --> { x }  /\  (/)  e.  1o )  ->  ( ( f  o.  ( 1o  X.  { x } ) ) `  (/) )  =  ( f `  (
( 1o  X.  {
x } ) `  (/) ) ) )
4745, 37, 46mp2an 676 . . . . 5  |-  ( ( f  o.  ( 1o 
X.  { x }
) ) `  (/) )  =  ( f `  (
( 1o  X.  {
x } ) `  (/) ) )
4844fvconst2 6074 . . . . . . 7  |-  ( (/)  e.  1o  ->  ( ( 1o  X.  { x }
) `  (/) )  =  x )
4937, 48ax-mp 5 . . . . . 6  |-  ( ( 1o  X.  { x } ) `  (/) )  =  x
5049fveq2i 5823 . . . . 5  |-  ( f `
 ( ( 1o 
X.  { x }
) `  (/) ) )  =  ( f `  x )
5147, 50eqtri 2445 . . . 4  |-  ( ( f  o.  ( 1o 
X.  { x }
) ) `  (/) )  =  ( f `  x
)
5243, 51syl6eq 2473 . . 3  |-  ( g  =  ( f  o.  ( 1o  X.  {
x } ) )  ->  ( g `  (/) )  =  ( f `
 x ) )
535, 9, 33, 36, 42, 52cnmpt21 20623 . 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 2505 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 370    = wceq 1437    e. wcel 1872   (/)c0 3699   ~Pcpw 3919   {csn 3936   U.cuni 4157    |-> cmpt 4420    X. cxp 4789    o. ccom 4795   Oncon0 5380   -->wf 5535   ` cfv 5539  (class class class)co 6244    |-> cmpt2 6246   1oc1o 7125   Topctop 19854  TopOnctopon 19855    Cn ccn 20177   Compccmp 20338  𝑛Locally cnlly 20417    tX ctx 20512    ^ko cxko 20513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-int 4194  df-iun 4239  df-iin 4240  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-om 6646  df-1st 6746  df-2nd 6747  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7473  df-en 7520  df-dom 7521  df-sdom 7522  df-fin 7523  df-fi 7873  df-rest 15259  df-topgen 15280  df-pt 15281  df-top 19858  df-bases 19859  df-topon 19860  df-ntr 19972  df-nei 20051  df-cn 20180  df-cnp 20181  df-cmp 20339  df-nlly 20419  df-tx 20514  df-xko 20515
This theorem is referenced by:  cnmptk1p  20637  cnmptk2  20638
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