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Theorem pcoval2 19041
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 9091 . . . . 5  |-  0  e.  RR
2 1re 9090 . . . . 5  |-  1  e.  RR
32rehalfcli 10216 . . . . . 6  |-  ( 1  /  2 )  e.  RR
4 halfgt0 10188 . . . . . 6  |-  0  <  ( 1  /  2
)
51, 3, 4ltleii 9196 . . . . 5  |-  0  <_  ( 1  /  2
)
6 1le1 9650 . . . . 5  |-  1  <_  1
7 iccss 10978 . . . . 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 3344 . . 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 19037 . . 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 10976 . . . . . . . . . . . . 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 9160 . . . . . . . . . . . 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 6097 . . . . . . . . 9  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
2  x.  X )  =  ( 2  x.  ( 1  /  2
) ) )
26 2cn 10070 . . . . . . . . . 10  |-  2  e.  CC
27 2ne0 10083 . . . . . . . . . 10  |-  2  =/=  0
2826, 27recidi 9745 . . . . . . . . 9  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
2925, 28syl6eq 2484 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
2  x.  X )  =  1 )
3029fveq2d 5732 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( F `  ( 2  x.  X ) )  =  ( F `  1
) )
3129oveq1d 6096 . . . . . . . . 9  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
( 2  x.  X
)  -  1 )  =  ( 1  -  1 ) )
32 1m1e0 10068 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
3331, 32syl6eq 2484 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
( 2  x.  X
)  -  1 )  =  0 )
3433fveq2d 5732 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( G `  ( (
2  x.  X )  -  1 ) )  =  ( G ` 
0 ) )
3515, 30, 343eqtr4d 2478 . . . . . 6  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( F `  ( 2  x.  X ) )  =  ( G `  (
( 2  x.  X
)  -  1 ) ) )
3635ifeq1d 3753 . . . . 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 3771 . . . . 5  |-  if ( X  <_  ( 1  /  2 ) ,  ( G `  (
( 2  x.  X
)  -  1 ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  =  ( G `  ( ( 2  x.  X )  -  1 ) )
3836, 37syl6eq 2484 . . . 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 3746 . . 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 2468 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 1652    e. wcel 1725    C_ wss 3320   ifcif 3739   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   RRcr 8989   0cc0 8990   1c1 8991    x. cmul 8995    <_ cle 9121    - cmin 9291    / cdiv 9677   2c2 10049   [,]cicc 10919    Cn ccn 17288   IIcii 18905   *pcpco 19025
This theorem is referenced by:  pcoass  19049  pcorevlem  19051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-2 10058  df-icc 10923  df-top 16963  df-topon 16966  df-cn 17291  df-pco 19030
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