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Theorem pcovalg 21380
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 21379 . . 3  |-  ( ph  ->  ( F ( *p
`  J ) G )  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  x ) ) ,  ( G `
 ( ( 2  x.  x )  - 
1 ) ) ) ) )
43fveq1d 5874 . 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 4456 . . . 4  |-  ( x  =  X  ->  (
x  <_  ( 1  /  2 )  <->  X  <_  ( 1  /  2 ) ) )
6 oveq2 6303 . . . . 5  |-  ( x  =  X  ->  (
2  x.  x )  =  ( 2  x.  X ) )
76fveq2d 5876 . . . 4  |-  ( x  =  X  ->  ( F `  ( 2  x.  x ) )  =  ( F `  (
2  x.  X ) ) )
86oveq1d 6310 . . . . 5  |-  ( x  =  X  ->  (
( 2  x.  x
)  -  1 )  =  ( ( 2  x.  X )  - 
1 ) )
98fveq2d 5876 . . . 4  |-  ( x  =  X  ->  ( G `  ( (
2  x.  x )  -  1 ) )  =  ( G `  ( ( 2  x.  X )  -  1 ) ) )
105, 7, 9ifbieq12d 3972 . . 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 2467 . . 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 5882 . . . 4  |-  ( F `
 ( 2  x.  X ) )  e. 
_V
13 fvex 5882 . . . 4  |-  ( G `
 ( ( 2  x.  X )  - 
1 ) )  e. 
_V
1412, 13ifex 4014 . . 3  |-  if ( X  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  X ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  e.  _V
1510, 11, 14fvmpt 5957 . 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 2528 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ifcif 3945   class class class wbr 4453    |-> cmpt 4511   ` cfv 5594  (class class class)co 6295   0cc0 9504   1c1 9505    x. cmul 9509    <_ cle 9641    - cmin 9817    / cdiv 10218   2c2 10597   [,]cicc 11544    Cn ccn 19593   IIcii 21247   *pcpco 21368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-map 7434  df-top 19268  df-topon 19271  df-cn 19596  df-pco 21373
This theorem is referenced by:  pcoval1  21381  pcoval2  21384  pcohtpylem  21387
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