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Theorem pcofval 18900
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 6022 . . . 4  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
2 eqidd 2382 . . . 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 6069 . . 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 18895 . . 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 6039 . . . 4  |-  ( II 
Cn  J )  e. 
_V
65, 5mpt2ex 6358 . . 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 5739 . 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 5300 . . . . . 6  |-  dom  *p  C_ 
Top
98sseli 3281 . . . . 5  |-  ( J  e.  dom  *p  ->  J  e.  Top )
109con3i 129 . . . 4  |-  ( -.  J  e.  Top  ->  -.  J  e.  dom  *p )
11 ndmfv 5689 . . . 4  |-  ( -.  J  e.  dom  *p  ->  ( *p `  J
)  =  (/) )
1210, 11syl 16 . . 3  |-  ( -.  J  e.  Top  ->  ( *p `  J )  =  (/) )
13 cntop2 17221 . . . . . . 7  |-  ( f  e.  ( II  Cn  J )  ->  J  e.  Top )
1413con3i 129 . . . . . 6  |-  ( -.  J  e.  Top  ->  -.  f  e.  ( II 
Cn  J ) )
1514eq0rdv 3599 . . . . 5  |-  ( -.  J  e.  Top  ->  ( II  Cn  J )  =  (/) )
16 mpt2eq12 6067 . . . . 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 6360 . . . 4  |-  ( f  e.  (/) ,  g  e.  (/)  |->  ( x  e.  ( 0 [,] 1
)  |->  if ( x  <_  ( 1  / 
2 ) ,  ( f `  ( 2  x.  x ) ) ,  ( g `  ( ( 2  x.  x )  -  1 ) ) ) ) )  =  (/)
1917, 18syl6eq 2429 . . 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 2416 . 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 1717   (/)c0 3565   ifcif 3676   class class class wbr 4147    e. cmpt 4201   dom cdm 4812   ` cfv 5388  (class class class)co 6014    e. cmpt2 6016   0cc0 8917   1c1 8918    x. cmul 8922    <_ cle 9048    - cmin 9217    / cdiv 9603   2c2 9975   [,]cicc 10845   Topctop 16875    Cn ccn 17204   IIcii 18770   *pcpco 18890
This theorem is referenced by:  pcoval  18901
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635
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 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-reu 2650  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-id 4433  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-1st 6282  df-2nd 6283  df-map 6950  df-top 16880  df-topon 16883  df-cn 17207  df-pco 18895
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