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Theorem pcoval1 22044
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 9643 . . . . 5  |-  0  e.  RR
2 1re 9642 . . . . 5  |-  1  e.  RR
3 0le0 10699 . . . . 5  |-  0  <_  0
4 halfre 10828 . . . . . 6  |-  ( 1  /  2 )  e.  RR
5 halflt1 10831 . . . . . 6  |-  ( 1  /  2 )  <  1
64, 2, 5ltleii 9757 . . . . 5  |-  ( 1  /  2 )  <_ 
1
7 iccss 11702 . . . . 5  |-  ( ( ( 0  e.  RR  /\  1  e.  RR )  /\  ( 0  <_ 
0  /\  ( 1  /  2 )  <_ 
1 ) )  -> 
( 0 [,] (
1  /  2 ) )  C_  ( 0 [,] 1 ) )
81, 2, 3, 6, 7mp4an 679 . . . 4  |-  ( 0 [,] ( 1  / 
2 ) )  C_  ( 0 [,] 1
)
98sseli 3428 . . 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 22043 . . 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 477 . 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 21963 . . . . 5  |-  ( X  e.  ( 0 [,] ( 1  /  2
) )  <->  ( X  e.  ( 0 [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )
1514simprbi 466 . . . 4  |-  ( X  e.  ( 0 [,] ( 1  /  2
) )  ->  X  <_  ( 1  /  2
) )
1615iftrued 3889 . . 3  |-  ( X  e.  ( 0 [,] ( 1  /  2
) )  ->  if ( X  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  X ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  =  ( F `  ( 2  x.  X
) ) )
1716adantl 468 . 2  |-  ( (
ph  /\  X  e.  ( 0 [,] (
1  /  2 ) ) )  ->  if ( X  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  X ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  =  ( F `  ( 2  x.  X
) ) )
1813, 17eqtrd 2485 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 371    = wceq 1444    e. wcel 1887    C_ wss 3404   ifcif 3881   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540    x. cmul 9544    <_ cle 9676    - cmin 9860    / cdiv 10269   2c2 10659   [,]cicc 11638    Cn ccn 20240   IIcii 21907   *pcpco 22031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-2 10668  df-icc 11642  df-top 19921  df-topon 19923  df-cn 20243  df-pco 22036
This theorem is referenced by:  pco0  22045  pcoass  22055  pcorevlem  22057
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