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Theorem pcoval2 22125
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 9661 . . . . 5  |-  0  e.  RR
2 1re 9660 . . . . 5  |-  1  e.  RR
3 halfre 10851 . . . . . 6  |-  ( 1  /  2 )  e.  RR
4 halfgt0 10853 . . . . . 6  |-  0  <  ( 1  /  2
)
51, 3, 4ltleii 9775 . . . . 5  |-  0  <_  ( 1  /  2
)
6 1le1 10262 . . . . 5  |-  1  <_  1
7 iccss 11727 . . . . 5  |-  ( ( ( 0  e.  RR  /\  1  e.  RR )  /\  ( 0  <_ 
( 1  /  2
)  /\  1  <_  1 ) )  ->  (
( 1  /  2
) [,] 1 ) 
C_  ( 0 [,] 1 ) )
81, 2, 5, 6, 7mp4an 687 . . . 4  |-  ( ( 1  /  2 ) [,] 1 )  C_  ( 0 [,] 1
)
98sseli 3414 . . 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 22121 . . 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 482 . 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 472 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( F `  1 )  =  ( G ` 
0 ) )
16 simprr 774 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  X  <_  ( 1  /  2
) )
173, 2elicc2i 11725 . . . . . . . . . . . . 13  |-  ( X  e.  ( ( 1  /  2 ) [,] 1 )  <->  ( X  e.  RR  /\  ( 1  /  2 )  <_  X  /\  X  <_  1
) )
1817simp2bi 1046 . . . . . . . . . . . 12  |-  ( X  e.  ( ( 1  /  2 ) [,] 1 )  ->  (
1  /  2 )  <_  X )
1918ad2antrl 742 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
1  /  2 )  <_  X )
2017simp1bi 1045 . . . . . . . . . . . . 13  |-  ( X  e.  ( ( 1  /  2 ) [,] 1 )  ->  X  e.  RR )
2120ad2antrl 742 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  X  e.  RR )
22 letri3 9737 . . . . . . . . . . . 12  |-  ( ( X  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( X  =  ( 1  /  2
)  <->  ( X  <_ 
( 1  /  2
)  /\  ( 1  /  2 )  <_  X ) ) )
2321, 3, 22sylancl 675 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( X  =  ( 1  /  2 )  <->  ( X  <_  ( 1  /  2
)  /\  ( 1  /  2 )  <_  X ) ) )
2416, 19, 23mpbir2and 936 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  X  =  ( 1  / 
2 ) )
2524oveq2d 6324 . . . . . . . . 9  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
2  x.  X )  =  ( 2  x.  ( 1  /  2
) ) )
26 2cn 10702 . . . . . . . . . 10  |-  2  e.  CC
27 2ne0 10724 . . . . . . . . . 10  |-  2  =/=  0
2826, 27recidi 10360 . . . . . . . . 9  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
2925, 28syl6eq 2521 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
2  x.  X )  =  1 )
3029fveq2d 5883 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( F `  ( 2  x.  X ) )  =  ( F `  1
) )
3129oveq1d 6323 . . . . . . . . 9  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
( 2  x.  X
)  -  1 )  =  ( 1  -  1 ) )
32 1m1e0 10700 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
3331, 32syl6eq 2521 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
( 2  x.  X
)  -  1 )  =  0 )
3433fveq2d 5883 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( G `  ( (
2  x.  X )  -  1 ) )  =  ( G ` 
0 ) )
3515, 30, 343eqtr4d 2515 . . . . . 6  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( F `  ( 2  x.  X ) )  =  ( G `  (
( 2  x.  X
)  -  1 ) ) )
3635ifeq1d 3890 . . . . 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 3909 . . . . 5  |-  if ( X  <_  ( 1  /  2 ) ,  ( G `  (
( 2  x.  X
)  -  1 ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  =  ( G `  ( ( 2  x.  X )  -  1 ) )
3836, 37syl6eq 2521 . . . 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 626 . . 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 3881 . . 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 164 . 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 2505 1  |-  ( (
ph  /\  X  e.  ( ( 1  / 
2 ) [,] 1
) )  ->  (
( F ( *p
`  J ) G ) `  X )  =  ( G `  ( ( 2  x.  X )  -  1 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    C_ wss 3390   ifcif 3872   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562    <_ cle 9694    - cmin 9880    / cdiv 10291   2c2 10681   [,]cicc 11663    Cn ccn 20317   IIcii 21985   *pcpco 22109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-2 10690  df-icc 11667  df-top 19998  df-topon 20000  df-cn 20320  df-pco 22114
This theorem is referenced by:  pcoass  22133  pcorevlem  22135
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