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Theorem pcovalg 19037
Description: Evaluate the concatenation of two paths. (Contributed by Mario Carneiro, 7-Jun-2014.)
Hypotheses
Ref Expression
pcoval.2  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
pcoval.3  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
Assertion
Ref Expression
pcovalg  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
( F ( *p
`  J ) G ) `  X )  =  if ( X  <_  ( 1  / 
2 ) ,  ( F `  ( 2  x.  X ) ) ,  ( G `  ( ( 2  x.  X )  -  1 ) ) ) )

Proof of Theorem pcovalg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pcoval.2 . . . 4  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
2 pcoval.3 . . . 4  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
31, 2pcoval 19036 . . 3  |-  ( ph  ->  ( F ( *p
`  J ) G )  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  x ) ) ,  ( G `
 ( ( 2  x.  x )  - 
1 ) ) ) ) )
43fveq1d 5730 . 2  |-  ( ph  ->  ( ( F ( *p `  J ) G ) `  X
)  =  ( ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  (
1  /  2 ) ,  ( F `  ( 2  x.  x
) ) ,  ( G `  ( ( 2  x.  x )  -  1 ) ) ) ) `  X
) )
5 breq1 4215 . . . 4  |-  ( x  =  X  ->  (
x  <_  ( 1  /  2 )  <->  X  <_  ( 1  /  2 ) ) )
6 oveq2 6089 . . . . 5  |-  ( x  =  X  ->  (
2  x.  x )  =  ( 2  x.  X ) )
76fveq2d 5732 . . . 4  |-  ( x  =  X  ->  ( F `  ( 2  x.  x ) )  =  ( F `  (
2  x.  X ) ) )
86oveq1d 6096 . . . . 5  |-  ( x  =  X  ->  (
( 2  x.  x
)  -  1 )  =  ( ( 2  x.  X )  - 
1 ) )
98fveq2d 5732 . . . 4  |-  ( x  =  X  ->  ( G `  ( (
2  x.  x )  -  1 ) )  =  ( G `  ( ( 2  x.  X )  -  1 ) ) )
105, 7, 9ifbieq12d 3761 . . 3  |-  ( x  =  X  ->  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 ) ) ) )
11 eqid 2436 . . 3  |-  ( 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 ) ) ) )
12 fvex 5742 . . . 4  |-  ( F `
 ( 2  x.  X ) )  e. 
_V
13 fvex 5742 . . . 4  |-  ( G `
 ( ( 2  x.  X )  - 
1 ) )  e. 
_V
1412, 13ifex 3797 . . 3  |-  if ( X  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  X ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  e.  _V
1510, 11, 14fvmpt 5806 . 2  |-  ( X  e.  ( 0 [,] 1 )  ->  (
( x  e.  ( 0 [,] 1 ) 
|->  if ( x  <_ 
( 1  /  2
) ,  ( F `
 ( 2  x.  x ) ) ,  ( G `  (
( 2  x.  x
)  -  1 ) ) ) ) `  X )  =  if ( X  <_  (
1  /  2 ) ,  ( F `  ( 2  x.  X
) ) ,  ( G `  ( ( 2  x.  X )  -  1 ) ) ) )
164, 15sylan9eq 2488 1  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
( F ( *p
`  J ) G ) `  X )  =  if ( X  <_  ( 1  / 
2 ) ,  ( F `  ( 2  x.  X ) ) ,  ( G `  ( ( 2  x.  X )  -  1 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ifcif 3739   class class class wbr 4212    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   0cc0 8990   1c1 8991    x. cmul 8995    <_ cle 9121    - cmin 9291    / cdiv 9677   2c2 10049   [,]cicc 10919    Cn ccn 17288   IIcii 18905   *pcpco 19025
This theorem is referenced by:  pcoval1  19038  pcoval2  19041  pcohtpylem  19044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-map 7020  df-top 16963  df-topon 16966  df-cn 17291  df-pco 19030
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