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Theorem xkofvcn 17721
Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 17693.) (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 17541 . . . 4  |-  ( R  e. 𝑛Locally 
Comp  ->  R  e.  Top )
3 eqid 2438 . . . . 5  |-  ( S  ^ k o  R
)  =  ( S  ^ k o  R
)
43xkotopon 17637 . . . 4  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( S  ^ k o  R )  e.  (TopOn `  ( R  Cn  S
) ) )
52, 4sylan 459 . . 3  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( S  ^ k o  R
)  e.  (TopOn `  ( R  Cn  S
) ) )
62adantr 453 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  R  e. 
Top )
7 xkofvcn.1 . . . . 5  |-  X  = 
U. R
87toptopon 17003 . . . 4  |-  ( R  e.  Top  <->  R  e.  (TopOn `  X ) )
96, 8sylib 190 . . 3  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  R  e.  (TopOn `  X )
)
105, 9cnmpt1st 17705 . . . 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 17706 . . . . 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 6734 . . . . . . 7  |-  1o  e.  On
13 distopon 17066 . . . . . . 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 17656 . . . . . 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 644 . . . . 5  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( y  e.  X  |->  ( 1o 
X.  { y } ) )  e.  ( R  Cn  ( R  ^ k o  ~P 1o ) ) )
17 sneq 3827 . . . . . 6  |-  ( y  =  x  ->  { y }  =  { x } )
1817xpeq2d 4905 . . . . 5  |-  ( y  =  x  ->  ( 1o  X.  { y } )  =  ( 1o 
X.  { x }
) )
195, 9, 11, 9, 16, 18cnmpt21 17708 . . . 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 17065 . . . . . 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 2438 . . . . . 6  |-  ( R  ^ k o  ~P 1o )  =  ( R  ^ k o  ~P 1o )
2322xkotopon 17637 . . . . 5  |-  ( ( ~P 1o  e.  Top  /\  R  e.  Top )  ->  ( R  ^ k o  ~P 1o )  e.  (TopOn `  ( ~P 1o  Cn  R ) ) )
2421, 6, 23syl2anc 644 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( R  ^ k o  ~P 1o )  e.  (TopOn `  ( ~P 1o  Cn  R ) ) )
25 simpl 445 . . . . 5  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  R  e. 𝑛Locally  Comp )
26 simpr 449 . . . . 5  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  S  e. 
Top )
27 eqid 2438 . . . . . 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 17697 . . . . 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 1185 . . . 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 5033 . . . . 5  |-  ( g  =  f  ->  (
g  o.  h )  =  ( f  o.  h ) )
31 coeq2 5034 . . . . 5  |-  ( h  =  ( 1o  X.  { x } )  ->  ( f  o.  h )  =  ( f  o.  ( 1o 
X.  { x }
) ) )
3230, 31sylan9eq 2490 . . . 4  |-  ( ( g  =  f  /\  h  =  ( 1o  X.  { x } ) )  ->  ( g  o.  h )  =  ( f  o.  ( 1o 
X.  { x }
) ) )
335, 9, 10, 19, 5, 24, 29, 32cnmpt22 17711 . . 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 2438 . . . . 5  |-  ( S  ^ k o  ~P 1o )  =  ( S  ^ k o  ~P 1o )
3534xkotopon 17637 . . . 4  |-  ( ( ~P 1o  e.  Top  /\  S  e.  Top )  ->  ( S  ^ k o  ~P 1o )  e.  (TopOn `  ( ~P 1o  Cn  S ) ) )
3621, 26, 35syl2anc 644 . . 3  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  ( S  ^ k o  ~P 1o )  e.  (TopOn `  ( ~P 1o  Cn  S ) ) )
37 0lt1o 6751 . . . . 5  |-  (/)  e.  1o
3837a1i 11 . . . 4  |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  (/)  e.  1o )
39 unipw 4417 . . . . . 6  |-  U. ~P 1o  =  1o
4039eqcomi 2442 . . . . 5  |-  1o  =  U. ~P 1o
4140xkopjcn 17693 . . . 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 1185 . . 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 5730 . . . 4  |-  ( g  =  ( f  o.  ( 1o  X.  {
x } ) )  ->  ( g `  (/) )  =  ( ( f  o.  ( 1o 
X.  { x }
) ) `  (/) ) )
44 vex 2961 . . . . . . 7  |-  x  e. 
_V
4544fconst 5632 . . . . . 6  |-  ( 1o 
X.  { x }
) : 1o --> { x }
46 fvco3 5803 . . . . . 6  |-  ( ( ( 1o  X.  {
x } ) : 1o --> { x }  /\  (/)  e.  1o )  ->  ( ( f  o.  ( 1o  X.  { x } ) ) `  (/) )  =  ( f `  (
( 1o  X.  {
x } ) `  (/) ) ) )
4745, 37, 46mp2an 655 . . . . 5  |-  ( ( f  o.  ( 1o 
X.  { x }
) ) `  (/) )  =  ( f `  (
( 1o  X.  {
x } ) `  (/) ) )
4844fvconst2 5950 . . . . . . 7  |-  ( (/)  e.  1o  ->  ( ( 1o  X.  { x }
) `  (/) )  =  x )
4937, 48ax-mp 5 . . . . . 6  |-  ( ( 1o  X.  { x } ) `  (/) )  =  x
5049fveq2i 5734 . . . . 5  |-  ( f `
 ( ( 1o 
X.  { x }
) `  (/) ) )  =  ( f `  x )
5147, 50eqtri 2458 . . . 4  |-  ( ( f  o.  ( 1o 
X.  { x }
) ) `  (/) )  =  ( f `  x
)
5243, 51syl6eq 2486 . . 3  |-  ( g  =  ( f  o.  ( 1o  X.  {
x } ) )  ->  ( g `  (/) )  =  ( f `
 x ) )
535, 9, 33, 36, 42, 52cnmpt21 17708 . 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 2522 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 360    = wceq 1653    e. wcel 1726   (/)c0 3630   ~Pcpw 3801   {csn 3816   U.cuni 4017    e. cmpt 4269   Oncon0 4584    X. cxp 4879    o. ccom 4885   -->wf 5453   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1oc1o 6720   Topctop 16963  TopOnctopon 16964    Cn ccn 17293   Compccmp 17454  𝑛Locally cnlly 17533    tX ctx 17597    ^ k o cxko 17598
This theorem is referenced by:  cnmptk1p  17722  cnmptk2  17723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-rest 13655  df-topgen 13672  df-pt 13673  df-top 16968  df-bases 16970  df-topon 16971  df-ntr 17089  df-nei 17167  df-cn 17296  df-cnp 17297  df-cmp 17455  df-nlly 17535  df-tx 17599  df-xko 17600
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