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Theorem pcofval 21238
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 6283 . . . 4  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
2 eqidd 2461 . . . 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 6334 . . 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 21233 . . 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 6300 . . . 4  |-  ( II 
Cn  J )  e. 
_V
65, 5mpt2ex 6850 . . 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 5941 . 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 5494 . . . . . 6  |-  dom  *p  C_ 
Top
98sseli 3493 . . . . 5  |-  ( J  e.  dom  *p  ->  J  e.  Top )
109con3i 135 . . . 4  |-  ( -.  J  e.  Top  ->  -.  J  e.  dom  *p )
11 ndmfv 5881 . . . 4  |-  ( -.  J  e.  dom  *p  ->  ( *p `  J
)  =  (/) )
1210, 11syl 16 . . 3  |-  ( -.  J  e.  Top  ->  ( *p `  J )  =  (/) )
13 cntop2 19501 . . . . . . 7  |-  ( f  e.  ( II  Cn  J )  ->  J  e.  Top )
1413con3i 135 . . . . . 6  |-  ( -.  J  e.  Top  ->  -.  f  e.  ( II 
Cn  J ) )
1514eq0rdv 3813 . . . . 5  |-  ( -.  J  e.  Top  ->  ( II  Cn  J )  =  (/) )
16 mpt2eq12 6332 . . . . 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 661 . . . 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 6342 . . . 4  |-  ( f  e.  (/) ,  g  e.  (/)  |->  ( x  e.  ( 0 [,] 1
)  |->  if ( x  <_  ( 1  / 
2 ) ,  ( f `  ( 2  x.  x ) ) ,  ( g `  ( ( 2  x.  x )  -  1 ) ) ) ) )  =  (/)
1917, 18syl6eq 2517 . . 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 2504 . 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 164 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 setvar class
Syntax hints:   -. wn 3    = wceq 1374    e. wcel 1762   (/)c0 3778   ifcif 3932   class class class wbr 4440    |-> cmpt 4498   dom cdm 4992   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   0cc0 9481   1c1 9482    x. cmul 9486    <_ cle 9618    - cmin 9794    / cdiv 10195   2c2 10574   [,]cicc 11521   Topctop 19154    Cn ccn 19484   IIcii 21107   *pcpco 21228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-map 7412  df-top 19159  df-topon 19162  df-cn 19487  df-pco 21233
This theorem is referenced by:  pcoval  21239
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