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Theorem pcoval 20483
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 5687 . . . . . 6  |-  ( f  =  F  ->  (
f `  ( 2  x.  x ) )  =  ( F `  (
2  x.  x ) ) )
43adantr 462 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f `  (
2  x.  x ) )  =  ( F `
 ( 2  x.  x ) ) )
5 fveq1 5687 . . . . . 6  |-  ( g  =  G  ->  (
g `  ( (
2  x.  x )  -  1 ) )  =  ( G `  ( ( 2  x.  x )  -  1 ) ) )
65adantl 463 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( g `  (
( 2  x.  x
)  -  1 ) )  =  ( G `
 ( ( 2  x.  x )  - 
1 ) ) )
74, 6ifeq12d 3806 . . . 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 4376 . . 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 20482 . . 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 6115 . . . 4  |-  ( 0 [,] 1 )  e. 
_V
1110mptex 5945 . . 3  |-  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  x ) ) ,  ( G `
 ( ( 2  x.  x )  - 
1 ) ) ) )  e.  _V
128, 9, 11ovmpt2a 6220 . 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 656 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 1364    e. wcel 1761   ifcif 3788   class class class wbr 4289    e. cmpt 4347   ` cfv 5415  (class class class)co 6090   0cc0 9278   1c1 9279    x. cmul 9283    <_ cle 9415    - cmin 9591    / cdiv 9989   2c2 10367   [,]cicc 11299    Cn ccn 18728   IIcii 20351   *pcpco 20472
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-map 7212  df-top 18403  df-topon 18406  df-cn 18731  df-pco 20477
This theorem is referenced by:  pcovalg  20484  pco1  20487  pcocn  20489  copco  20490  pcopt  20494  pcopt2  20495  pcoass  20496
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