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Theorem pcoval1 20560
Description: Evaluate the concatenation of two paths on the first half. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised 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
pcoval1  |-  ( (
ph  /\  X  e.  ( 0 [,] (
1  /  2 ) ) )  ->  (
( F ( *p
`  J ) G ) `  X )  =  ( F `  ( 2  x.  X
) ) )

Proof of Theorem pcoval1
StepHypRef Expression
1 0re 9378 . . . . 5  |-  0  e.  RR
2 1re 9377 . . . . 5  |-  1  e.  RR
3 0le0 10403 . . . . 5  |-  0  <_  0
4 halfre 10532 . . . . . 6  |-  ( 1  /  2 )  e.  RR
5 halflt1 10535 . . . . . 6  |-  ( 1  /  2 )  <  1
64, 2, 5ltleii 9489 . . . . 5  |-  ( 1  /  2 )  <_ 
1
7 iccss 11355 . . . . 5  |-  ( ( ( 0  e.  RR  /\  1  e.  RR )  /\  ( 0  <_ 
0  /\  ( 1  /  2 )  <_ 
1 ) )  -> 
( 0 [,] (
1  /  2 ) )  C_  ( 0 [,] 1 ) )
81, 2, 3, 6, 7mp4an 673 . . . 4  |-  ( 0 [,] ( 1  / 
2 ) )  C_  ( 0 [,] 1
)
98sseli 3347 . . 3  |-  ( X  e.  ( 0 [,] ( 1  /  2
) )  ->  X  e.  ( 0 [,] 1
) )
10 pcoval.2 . . . 4  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
11 pcoval.3 . . . 4  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
1210, 11pcovalg 20559 . . 3  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
( F ( *p
`  J ) G ) `  X )  =  if ( X  <_  ( 1  / 
2 ) ,  ( F `  ( 2  x.  X ) ) ,  ( G `  ( ( 2  x.  X )  -  1 ) ) ) )
139, 12sylan2 474 . 2  |-  ( (
ph  /\  X  e.  ( 0 [,] (
1  /  2 ) ) )  ->  (
( F ( *p
`  J ) G ) `  X )  =  if ( X  <_  ( 1  / 
2 ) ,  ( F `  ( 2  x.  X ) ) ,  ( G `  ( ( 2  x.  X )  -  1 ) ) ) )
14 elii1 20482 . . . . 5  |-  ( X  e.  ( 0 [,] ( 1  /  2
) )  <->  ( X  e.  ( 0 [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )
1514simprbi 464 . . . 4  |-  ( X  e.  ( 0 [,] ( 1  /  2
) )  ->  X  <_  ( 1  /  2
) )
16 iftrue 3792 . . . 4  |-  ( X  <_  ( 1  / 
2 )  ->  if ( X  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  X ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  =  ( F `  ( 2  x.  X
) ) )
1715, 16syl 16 . . 3  |-  ( X  e.  ( 0 [,] ( 1  /  2
) )  ->  if ( X  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  X ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  =  ( F `  ( 2  x.  X
) ) )
1817adantl 466 . 2  |-  ( (
ph  /\  X  e.  ( 0 [,] (
1  /  2 ) ) )  ->  if ( X  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  X ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  =  ( F `  ( 2  x.  X
) ) )
1913, 18eqtrd 2470 1  |-  ( (
ph  /\  X  e.  ( 0 [,] (
1  /  2 ) ) )  ->  (
( F ( *p
`  J ) G ) `  X )  =  ( F `  ( 2  x.  X
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3323   ifcif 3786   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   RRcr 9273   0cc0 9274   1c1 9275    x. cmul 9279    <_ cle 9411    - cmin 9587    / cdiv 9985   2c2 10363   [,]cicc 11295    Cn ccn 18803   IIcii 20426   *pcpco 20547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-2 10372  df-icc 11299  df-top 18478  df-topon 18481  df-cn 18806  df-pco 20552
This theorem is referenced by:  pco0  20561  pcoass  20571  pcorevlem  20573
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