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Theorem pcoval2 18994
Description: Evaluate the concatenation of two paths on the second half. (Contributed by Jeff Madsen, 15-Jun-2010.) (Proof shortened 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 ) )
pcoval2.4  |-  ( ph  ->  ( F `  1
)  =  ( G `
 0 ) )
Assertion
Ref Expression
pcoval2  |-  ( (
ph  /\  X  e.  ( ( 1  / 
2 ) [,] 1
) )  ->  (
( F ( *p
`  J ) G ) `  X )  =  ( G `  ( ( 2  x.  X )  -  1 ) ) )

Proof of Theorem pcoval2
StepHypRef Expression
1 0re 9047 . . . . 5  |-  0  e.  RR
2 1re 9046 . . . . 5  |-  1  e.  RR
32rehalfcli 10172 . . . . . 6  |-  ( 1  /  2 )  e.  RR
4 halfgt0 10144 . . . . . 6  |-  0  <  ( 1  /  2
)
51, 3, 4ltleii 9152 . . . . 5  |-  0  <_  ( 1  /  2
)
6 1le1 9606 . . . . 5  |-  1  <_  1
7 iccss 10934 . . . . 5  |-  ( ( ( 0  e.  RR  /\  1  e.  RR )  /\  ( 0  <_ 
( 1  /  2
)  /\  1  <_  1 ) )  ->  (
( 1  /  2
) [,] 1 ) 
C_  ( 0 [,] 1 ) )
81, 2, 5, 6, 7mp4an 655 . . . 4  |-  ( ( 1  /  2 ) [,] 1 )  C_  ( 0 [,] 1
)
98sseli 3304 . . 3  |-  ( X  e.  ( ( 1  /  2 ) [,] 1 )  ->  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 18990 . . 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 461 . 2  |-  ( (
ph  /\  X  e.  ( ( 1  / 
2 ) [,] 1
) )  ->  (
( F ( *p
`  J ) G ) `  X )  =  if ( X  <_  ( 1  / 
2 ) ,  ( F `  ( 2  x.  X ) ) ,  ( G `  ( ( 2  x.  X )  -  1 ) ) ) )
14 pcoval2.4 . . . . . . . 8  |-  ( ph  ->  ( F `  1
)  =  ( G `
 0 ) )
1514adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( F `  1 )  =  ( G ` 
0 ) )
16 simprr 734 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  X  <_  ( 1  /  2
) )
173, 2elicc2i 10932 . . . . . . . . . . . . 13  |-  ( X  e.  ( ( 1  /  2 ) [,] 1 )  <->  ( X  e.  RR  /\  ( 1  /  2 )  <_  X  /\  X  <_  1
) )
1817simp2bi 973 . . . . . . . . . . . 12  |-  ( X  e.  ( ( 1  /  2 ) [,] 1 )  ->  (
1  /  2 )  <_  X )
1918ad2antrl 709 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
1  /  2 )  <_  X )
2017simp1bi 972 . . . . . . . . . . . . 13  |-  ( X  e.  ( ( 1  /  2 ) [,] 1 )  ->  X  e.  RR )
2120ad2antrl 709 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  X  e.  RR )
22 letri3 9116 . . . . . . . . . . . 12  |-  ( ( X  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( X  =  ( 1  /  2
)  <->  ( X  <_ 
( 1  /  2
)  /\  ( 1  /  2 )  <_  X ) ) )
2321, 3, 22sylancl 644 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( X  =  ( 1  /  2 )  <->  ( X  <_  ( 1  /  2
)  /\  ( 1  /  2 )  <_  X ) ) )
2416, 19, 23mpbir2and 889 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  X  =  ( 1  / 
2 ) )
2524oveq2d 6056 . . . . . . . . 9  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
2  x.  X )  =  ( 2  x.  ( 1  /  2
) ) )
26 2cn 10026 . . . . . . . . . 10  |-  2  e.  CC
27 2ne0 10039 . . . . . . . . . 10  |-  2  =/=  0
2826, 27recidi 9701 . . . . . . . . 9  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
2925, 28syl6eq 2452 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
2  x.  X )  =  1 )
3029fveq2d 5691 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( F `  ( 2  x.  X ) )  =  ( F `  1
) )
3129oveq1d 6055 . . . . . . . . 9  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
( 2  x.  X
)  -  1 )  =  ( 1  -  1 ) )
32 1m1e0 10024 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
3331, 32syl6eq 2452 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
( 2  x.  X
)  -  1 )  =  0 )
3433fveq2d 5691 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( G `  ( (
2  x.  X )  -  1 ) )  =  ( G ` 
0 ) )
3515, 30, 343eqtr4d 2446 . . . . . 6  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( F `  ( 2  x.  X ) )  =  ( G `  (
( 2  x.  X
)  -  1 ) ) )
3635ifeq1d 3713 . . . . 5  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  if ( X  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  X ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  =  if ( X  <_  ( 1  / 
2 ) ,  ( G `  ( ( 2  x.  X )  -  1 ) ) ,  ( G `  ( ( 2  x.  X )  -  1 ) ) ) )
37 ifid 3731 . . . . 5  |-  if ( X  <_  ( 1  /  2 ) ,  ( G `  (
( 2  x.  X
)  -  1 ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  =  ( G `  ( ( 2  x.  X )  -  1 ) )
3836, 37syl6eq 2452 . . . 4  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  if ( X  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  X ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  =  ( G `  ( ( 2  x.  X )  -  1 ) ) )
3938expr 599 . . 3  |-  ( (
ph  /\  X  e.  ( ( 1  / 
2 ) [,] 1
) )  ->  ( X  <_  ( 1  / 
2 )  ->  if ( X  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  X ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  =  ( G `  ( ( 2  x.  X )  -  1 ) ) ) )
40 iffalse 3706 . . 3  |-  ( -.  X  <_  ( 1  /  2 )  ->  if ( X  <_  (
1  /  2 ) ,  ( F `  ( 2  x.  X
) ) ,  ( G `  ( ( 2  x.  X )  -  1 ) ) )  =  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )
4139, 40pm2.61d1 153 . 2  |-  ( (
ph  /\  X  e.  ( ( 1  / 
2 ) [,] 1
) )  ->  if ( X  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  X ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  =  ( G `  ( ( 2  x.  X )  -  1 ) ) )
4213, 41eqtrd 2436 1  |-  ( (
ph  /\  X  e.  ( ( 1  / 
2 ) [,] 1
) )  ->  (
( F ( *p
`  J ) G ) `  X )  =  ( G `  ( ( 2  x.  X )  -  1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3280   ifcif 3699   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    <_ cle 9077    - cmin 9247    / cdiv 9633   2c2 10005   [,]cicc 10875    Cn ccn 17242   IIcii 18858   *pcpco 18978
This theorem is referenced by:  pcoass  19002  pcorevlem  19004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-icc 10879  df-top 16918  df-topon 16921  df-cn 17245  df-pco 18983
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