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Theorem pcoval1 18615
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 8928 . . . . 5  |-  0  e.  RR
2 1re 8927 . . . . 5  |-  1  e.  RR
3 0le0 9917 . . . . 5  |-  0  <_  0
4 rehalfcl 10030 . . . . . . 7  |-  ( 1  e.  RR  ->  (
1  /  2 )  e.  RR )
52, 4ax-mp 8 . . . . . 6  |-  ( 1  /  2 )  e.  RR
6 halflt1 10025 . . . . . 6  |-  ( 1  /  2 )  <  1
75, 2, 6ltleii 9031 . . . . 5  |-  ( 1  /  2 )  <_ 
1
8 iccss 10810 . . . . 5  |-  ( ( ( 0  e.  RR  /\  1  e.  RR )  /\  ( 0  <_ 
0  /\  ( 1  /  2 )  <_ 
1 ) )  -> 
( 0 [,] (
1  /  2 ) )  C_  ( 0 [,] 1 ) )
91, 2, 3, 7, 8mp4an 654 . . . 4  |-  ( 0 [,] ( 1  / 
2 ) )  C_  ( 0 [,] 1
)
109sseli 3252 . . 3  |-  ( X  e.  ( 0 [,] ( 1  /  2
) )  ->  X  e.  ( 0 [,] 1
) )
11 pcoval.2 . . . 4  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
12 pcoval.3 . . . 4  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
1311, 12pcovalg 18614 . . 3  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
( F ( *p
`  J ) G ) `  X )  =  if ( X  <_  ( 1  / 
2 ) ,  ( F `  ( 2  x.  X ) ) ,  ( G `  ( ( 2  x.  X )  -  1 ) ) ) )
1410, 13sylan2 460 . 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 ) ) ) )
15 elii1 18537 . . . . 5  |-  ( X  e.  ( 0 [,] ( 1  /  2
) )  <->  ( X  e.  ( 0 [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )
1615simprbi 450 . . . 4  |-  ( X  e.  ( 0 [,] ( 1  /  2
) )  ->  X  <_  ( 1  /  2
) )
17 iftrue 3647 . . . 4  |-  ( X  <_  ( 1  / 
2 )  ->  if ( X  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  X ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  =  ( F `  ( 2  x.  X
) ) )
1816, 17syl 15 . . 3  |-  ( X  e.  ( 0 [,] ( 1  /  2
) )  ->  if ( X  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  X ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  =  ( F `  ( 2  x.  X
) ) )
1918adantl 452 . 2  |-  ( (
ph  /\  X  e.  ( 0 [,] (
1  /  2 ) ) )  ->  if ( X  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  X ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  =  ( F `  ( 2  x.  X
) ) )
2014, 19eqtrd 2390 1  |-  ( (
ph  /\  X  e.  ( 0 [,] (
1  /  2 ) ) )  ->  (
( F ( *p
`  J ) G ) `  X )  =  ( F `  ( 2  x.  X
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    C_ wss 3228   ifcif 3641   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   RRcr 8826   0cc0 8827   1c1 8828    x. cmul 8832    <_ cle 8958    - cmin 9127    / cdiv 9513   2c2 9885   [,]cicc 10751    Cn ccn 17060   IIcii 18482   *pcpco 18602
This theorem is referenced by:  pco0  18616  pcoass  18626  pcorevlem  18628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-2 9894  df-icc 10755  df-top 16742  df-topon 16745  df-cn 17063  df-pco 18607
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