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Theorem pcoval 21805
Description: The concatenation of two paths. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
pcoval.2  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
pcoval.3  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
Assertion
Ref Expression
pcoval  |-  ( ph  ->  ( F ( *p
`  J ) G )  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  x ) ) ,  ( G `
 ( ( 2  x.  x )  - 
1 ) ) ) ) )
Distinct variable groups:    x, F    x, G    ph, x    x, J

Proof of Theorem pcoval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pcoval.2 . 2  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
2 pcoval.3 . 2  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
3 fveq1 5850 . . . . . 6  |-  ( f  =  F  ->  (
f `  ( 2  x.  x ) )  =  ( F `  (
2  x.  x ) ) )
43adantr 465 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f `  (
2  x.  x ) )  =  ( F `
 ( 2  x.  x ) ) )
5 fveq1 5850 . . . . . 6  |-  ( g  =  G  ->  (
g `  ( (
2  x.  x )  -  1 ) )  =  ( G `  ( ( 2  x.  x )  -  1 ) ) )
65adantl 466 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( g `  (
( 2  x.  x
)  -  1 ) )  =  ( G `
 ( ( 2  x.  x )  - 
1 ) ) )
74, 6ifeq12d 3907 . . . 4  |-  ( ( f  =  F  /\  g  =  G )  ->  if ( x  <_ 
( 1  /  2
) ,  ( f `
 ( 2  x.  x ) ) ,  ( g `  (
( 2  x.  x
)  -  1 ) ) )  =  if ( x  <_  (
1  /  2 ) ,  ( F `  ( 2  x.  x
) ) ,  ( G `  ( ( 2  x.  x )  -  1 ) ) ) )
87mpteq2dv 4484 . . 3  |-  ( ( f  =  F  /\  g  =  G )  ->  ( 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 ) ) ) ) )
9 pcofval 21804 . . 3  |-  ( *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 ) ) ) ) )
10 ovex 6308 . . . 4  |-  ( 0 [,] 1 )  e. 
_V
1110mptex 6126 . . 3  |-  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  x ) ) ,  ( G `
 ( ( 2  x.  x )  - 
1 ) ) ) )  e.  _V
128, 9, 11ovmpt2a 6416 . 2  |-  ( ( F  e.  ( II 
Cn  J )  /\  G  e.  ( II  Cn  J ) )  -> 
( F ( *p
`  J ) G )  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  x ) ) ,  ( G `
 ( ( 2  x.  x )  - 
1 ) ) ) ) )
131, 2, 12syl2anc 661 1  |-  ( ph  ->  ( F ( *p
`  J ) G )  =  ( 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:    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844   ifcif 3887   class class class wbr 4397    |-> cmpt 4455   ` cfv 5571  (class class class)co 6280   0cc0 9524   1c1 9525    x. cmul 9529    <_ cle 9661    - cmin 9843    / cdiv 10249   2c2 10628   [,]cicc 11587    Cn ccn 20020   IIcii 21673   *pcpco 21794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-map 7461  df-top 19693  df-topon 19696  df-cn 20023  df-pco 21799
This theorem is referenced by:  pcovalg  21806  pco1  21809  pcocn  21811  copco  21812  pcopt  21816  pcopt2  21817  pcoass  21818
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