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Theorem pcofval 19028
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 6082 . . . 4  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
2 eqidd 2437 . . . 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 6129 . . 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 19023 . . 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 6099 . . . 4  |-  ( II 
Cn  J )  e. 
_V
65, 5mpt2ex 6418 . . 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 5799 . 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 5359 . . . . . 6  |-  dom  *p  C_ 
Top
98sseli 3337 . . . . 5  |-  ( J  e.  dom  *p  ->  J  e.  Top )
109con3i 129 . . . 4  |-  ( -.  J  e.  Top  ->  -.  J  e.  dom  *p )
11 ndmfv 5748 . . . 4  |-  ( -.  J  e.  dom  *p  ->  ( *p `  J
)  =  (/) )
1210, 11syl 16 . . 3  |-  ( -.  J  e.  Top  ->  ( *p `  J )  =  (/) )
13 cntop2 17298 . . . . . . 7  |-  ( f  e.  ( II  Cn  J )  ->  J  e.  Top )
1413con3i 129 . . . . . 6  |-  ( -.  J  e.  Top  ->  -.  f  e.  ( II 
Cn  J ) )
1514eq0rdv 3655 . . . . 5  |-  ( -.  J  e.  Top  ->  ( II  Cn  J )  =  (/) )
16 mpt2eq12 6127 . . . . 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 6420 . . . 4  |-  ( f  e.  (/) ,  g  e.  (/)  |->  ( x  e.  ( 0 [,] 1
)  |->  if ( x  <_  ( 1  / 
2 ) ,  ( f `  ( 2  x.  x ) ) ,  ( g `  ( ( 2  x.  x )  -  1 ) ) ) ) )  =  (/)
1917, 18syl6eq 2484 . . 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 2471 . 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 1652    e. wcel 1725   (/)c0 3621   ifcif 3732   class class class wbr 4205    e. cmpt 4259   dom cdm 4871   ` cfv 5447  (class class class)co 6074    e. cmpt2 6076   0cc0 8983   1c1 8984    x. cmul 8988    <_ cle 9114    - cmin 9284    / cdiv 9670   2c2 10042   [,]cicc 10912   Topctop 16951    Cn ccn 17281   IIcii 18898   *pcpco 19018
This theorem is referenced by:  pcoval  19029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-map 7013  df-top 16956  df-topon 16959  df-cn 17284  df-pco 19023
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