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Theorem pcoval 19037
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 5728 . . . . . 6  |-  ( f  =  F  ->  (
f `  ( 2  x.  x ) )  =  ( F `  (
2  x.  x ) ) )
43adantr 453 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f `  (
2  x.  x ) )  =  ( F `
 ( 2  x.  x ) ) )
5 fveq1 5728 . . . . . 6  |-  ( g  =  G  ->  (
g `  ( (
2  x.  x )  -  1 ) )  =  ( G `  ( ( 2  x.  x )  -  1 ) ) )
65adantl 454 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( g `  (
( 2  x.  x
)  -  1 ) )  =  ( G `
 ( ( 2  x.  x )  - 
1 ) ) )
74, 6ifeq12d 3756 . . . 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 4297 . . 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 19036 . . 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 6107 . . . 4  |-  ( 0 [,] 1 )  e. 
_V
1110mptex 5967 . . 3  |-  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  x ) ) ,  ( G `
 ( ( 2  x.  x )  - 
1 ) ) ) )  e.  _V
128, 9, 11ovmpt2a 6205 . 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 644 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 set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   ifcif 3740   class class class wbr 4213    e. cmpt 4267   ` cfv 5455  (class class class)co 6082   0cc0 8991   1c1 8992    x. cmul 8996    <_ cle 9122    - cmin 9292    / cdiv 9678   2c2 10050   [,]cicc 10920    Cn ccn 17289   IIcii 18906   *pcpco 19026
This theorem is referenced by:  pcovalg  19038  pco1  19041  pcocn  19043  copco  19044  pcopt  19048  pcopt2  19049  pcoass  19050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-map 7021  df-top 16964  df-topon 16967  df-cn 17292  df-pco 19031
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