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Theorem pcoval1 20720
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 9500 . . . . 5  |-  0  e.  RR
2 1re 9499 . . . . 5  |-  1  e.  RR
3 0le0 10525 . . . . 5  |-  0  <_  0
4 halfre 10654 . . . . . 6  |-  ( 1  /  2 )  e.  RR
5 halflt1 10657 . . . . . 6  |-  ( 1  /  2 )  <  1
64, 2, 5ltleii 9611 . . . . 5  |-  ( 1  /  2 )  <_ 
1
7 iccss 11477 . . . . 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 3463 . . 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 20719 . . 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 20642 . . . . 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 3908 . . . 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 2495 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 1370    e. wcel 1758    C_ wss 3439   ifcif 3902   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   RRcr 9395   0cc0 9396   1c1 9397    x. cmul 9401    <_ cle 9533    - cmin 9709    / cdiv 10107   2c2 10485   [,]cicc 11417    Cn ccn 18963   IIcii 20586   *pcpco 20707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-2 10494  df-icc 11421  df-top 18638  df-topon 18641  df-cn 18966  df-pco 20712
This theorem is referenced by:  pco0  20721  pcoass  20731  pcorevlem  20733
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