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Theorem pcovalg 18510
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 18509 . . 3  |-  ( ph  ->  ( F ( *p
`  J ) G )  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  x ) ) ,  ( G `
 ( ( 2  x.  x )  - 
1 ) ) ) ) )
43fveq1d 5527 . 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 4026 . . . 4  |-  ( x  =  X  ->  (
x  <_  ( 1  /  2 )  <->  X  <_  ( 1  /  2 ) ) )
6 oveq2 5866 . . . . 5  |-  ( x  =  X  ->  (
2  x.  x )  =  ( 2  x.  X ) )
76fveq2d 5529 . . . 4  |-  ( x  =  X  ->  ( F `  ( 2  x.  x ) )  =  ( F `  (
2  x.  X ) ) )
86oveq1d 5873 . . . . 5  |-  ( x  =  X  ->  (
( 2  x.  x
)  -  1 )  =  ( ( 2  x.  X )  - 
1 ) )
98fveq2d 5529 . . . 4  |-  ( x  =  X  ->  ( G `  ( (
2  x.  x )  -  1 ) )  =  ( G `  ( ( 2  x.  X )  -  1 ) ) )
105, 7, 9ifbieq12d 3587 . . 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 2283 . . 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 5539 . . . 4  |-  ( F `
 ( 2  x.  X ) )  e. 
_V
13 fvex 5539 . . . 4  |-  ( G `
 ( ( 2  x.  X )  - 
1 ) )  e. 
_V
1412, 13ifex 3623 . . 3  |-  if ( X  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  X ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  e.  _V
1510, 11, 14fvmpt 5602 . 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 2335 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 358    = wceq 1623    e. wcel 1684   ifcif 3565   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738    x. cmul 8742    <_ cle 8868    - cmin 9037    / cdiv 9423   2c2 9795   [,]cicc 10659    Cn ccn 16954   IIcii 18379   *pcpco 18498
This theorem is referenced by:  pcoval1  18511  pcoval2  18514  pcohtpylem  18517
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957  df-pco 18503
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