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Theorem pcofval 18988
Description: The value of the path concatenation function on a topological space. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)
Assertion
Ref Expression
pcofval  |-  ( *p
`  J )  =  ( f  e.  ( II  Cn  J ) ,  g  e.  ( II  Cn  J ) 
|->  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  <_ 
( 1  /  2
) ,  ( f `
 ( 2  x.  x ) ) ,  ( g `  (
( 2  x.  x
)  -  1 ) ) ) ) )
Distinct variable group:    f, g, x, J

Proof of Theorem pcofval
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 oveq2 6048 . . . 4  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
2 eqidd 2405 . . . 4  |-  ( j  =  J  ->  (
x  e.  ( 0 [,] 1 )  |->  if ( x  <_  (
1  /  2 ) ,  ( f `  ( 2  x.  x
) ) ,  ( g `  ( ( 2  x.  x )  -  1 ) ) ) )  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  (
1  /  2 ) ,  ( f `  ( 2  x.  x
) ) ,  ( g `  ( ( 2  x.  x )  -  1 ) ) ) ) )
31, 1, 2mpt2eq123dv 6095 . . 3  |-  ( j  =  J  ->  (
f  e.  ( II 
Cn  j ) ,  g  e.  ( II 
Cn  j )  |->  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  (
1  /  2 ) ,  ( f `  ( 2  x.  x
) ) ,  ( g `  ( ( 2  x.  x )  -  1 ) ) ) ) )  =  ( f  e.  ( II  Cn  J ) ,  g  e.  ( II  Cn  J ) 
|->  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  <_ 
( 1  /  2
) ,  ( f `
 ( 2  x.  x ) ) ,  ( g `  (
( 2  x.  x
)  -  1 ) ) ) ) ) )
4 df-pco 18983 . . 3  |-  *p  =  ( j  e.  Top  |->  ( f  e.  ( II  Cn  j ) ,  g  e.  ( II  Cn  j ) 
|->  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  <_ 
( 1  /  2
) ,  ( f `
 ( 2  x.  x ) ) ,  ( g `  (
( 2  x.  x
)  -  1 ) ) ) ) ) )
5 ovex 6065 . . . 4  |-  ( II 
Cn  J )  e. 
_V
65, 5mpt2ex 6384 . . 3  |-  ( f  e.  ( II  Cn  J ) ,  g  e.  ( II  Cn  J )  |->  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( f `  (
2  x.  x ) ) ,  ( g `
 ( ( 2  x.  x )  - 
1 ) ) ) ) )  e.  _V
73, 4, 6fvmpt 5765 . 2  |-  ( J  e.  Top  ->  ( *p `  J )  =  ( f  e.  ( II  Cn  J ) ,  g  e.  ( II  Cn  J ) 
|->  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  <_ 
( 1  /  2
) ,  ( f `
 ( 2  x.  x ) ) ,  ( g `  (
( 2  x.  x
)  -  1 ) ) ) ) ) )
84dmmptss 5325 . . . . . 6  |-  dom  *p  C_ 
Top
98sseli 3304 . . . . 5  |-  ( J  e.  dom  *p  ->  J  e.  Top )
109con3i 129 . . . 4  |-  ( -.  J  e.  Top  ->  -.  J  e.  dom  *p )
11 ndmfv 5714 . . . 4  |-  ( -.  J  e.  dom  *p  ->  ( *p `  J
)  =  (/) )
1210, 11syl 16 . . 3  |-  ( -.  J  e.  Top  ->  ( *p `  J )  =  (/) )
13 cntop2 17259 . . . . . . 7  |-  ( f  e.  ( II  Cn  J )  ->  J  e.  Top )
1413con3i 129 . . . . . 6  |-  ( -.  J  e.  Top  ->  -.  f  e.  ( II 
Cn  J ) )
1514eq0rdv 3622 . . . . 5  |-  ( -.  J  e.  Top  ->  ( II  Cn  J )  =  (/) )
16 mpt2eq12 6093 . . . . 5  |-  ( ( ( II  Cn  J
)  =  (/)  /\  (
II  Cn  J )  =  (/) )  ->  (
f  e.  ( II 
Cn  J ) ,  g  e.  ( II 
Cn  J )  |->  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  (
1  /  2 ) ,  ( f `  ( 2  x.  x
) ) ,  ( g `  ( ( 2  x.  x )  -  1 ) ) ) ) )  =  ( f  e.  (/) ,  g  e.  (/)  |->  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( f `  (
2  x.  x ) ) ,  ( g `
 ( ( 2  x.  x )  - 
1 ) ) ) ) ) )
1715, 15, 16syl2anc 643 . . . 4  |-  ( -.  J  e.  Top  ->  ( f  e.  ( II 
Cn  J ) ,  g  e.  ( II 
Cn  J )  |->  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  (
1  /  2 ) ,  ( f `  ( 2  x.  x
) ) ,  ( g `  ( ( 2  x.  x )  -  1 ) ) ) ) )  =  ( f  e.  (/) ,  g  e.  (/)  |->  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( f `  (
2  x.  x ) ) ,  ( g `
 ( ( 2  x.  x )  - 
1 ) ) ) ) ) )
18 mpt20 6386 . . . 4  |-  ( f  e.  (/) ,  g  e.  (/)  |->  ( x  e.  ( 0 [,] 1
)  |->  if ( x  <_  ( 1  / 
2 ) ,  ( f `  ( 2  x.  x ) ) ,  ( g `  ( ( 2  x.  x )  -  1 ) ) ) ) )  =  (/)
1917, 18syl6eq 2452 . . 3  |-  ( -.  J  e.  Top  ->  ( f  e.  ( II 
Cn  J ) ,  g  e.  ( II 
Cn  J )  |->  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  (
1  /  2 ) ,  ( f `  ( 2  x.  x
) ) ,  ( g `  ( ( 2  x.  x )  -  1 ) ) ) ) )  =  (/) )
2012, 19eqtr4d 2439 . 2  |-  ( -.  J  e.  Top  ->  ( *p `  J )  =  ( f  e.  ( II  Cn  J
) ,  g  e.  ( II  Cn  J
)  |->  ( x  e.  ( 0 [,] 1
)  |->  if ( x  <_  ( 1  / 
2 ) ,  ( f `  ( 2  x.  x ) ) ,  ( g `  ( ( 2  x.  x )  -  1 ) ) ) ) ) )
217, 20pm2.61i 158 1  |-  ( *p
`  J )  =  ( f  e.  ( II  Cn  J ) ,  g  e.  ( II  Cn  J ) 
|->  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  <_ 
( 1  /  2
) ,  ( f `
 ( 2  x.  x ) ) ,  ( g `  (
( 2  x.  x
)  -  1 ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1721   (/)c0 3588   ifcif 3699   class class class wbr 4172    e. cmpt 4226   dom cdm 4837   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   0cc0 8946   1c1 8947    x. cmul 8951    <_ cle 9077    - cmin 9247    / cdiv 9633   2c2 10005   [,]cicc 10875   Topctop 16913    Cn ccn 17242   IIcii 18858   *pcpco 18978
This theorem is referenced by:  pcoval  18989
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-map 6979  df-top 16918  df-topon 16921  df-cn 17245  df-pco 18983
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