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Theorem pcoval2 18530
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 8854 . . . . 5  |-  0  e.  RR
2 1re 8853 . . . . 5  |-  1  e.  RR
3 rehalfcl 9954 . . . . . . 7  |-  ( 1  e.  RR  ->  (
1  /  2 )  e.  RR )
42, 3ax-mp 8 . . . . . 6  |-  ( 1  /  2 )  e.  RR
5 halfgt0 9948 . . . . . 6  |-  0  <  ( 1  /  2
)
61, 4, 5ltleii 8957 . . . . 5  |-  0  <_  ( 1  /  2
)
7 1le1 9412 . . . . 5  |-  1  <_  1
8 iccss 10734 . . . . 5  |-  ( ( ( 0  e.  RR  /\  1  e.  RR )  /\  ( 0  <_ 
( 1  /  2
)  /\  1  <_  1 ) )  ->  (
( 1  /  2
) [,] 1 ) 
C_  ( 0 [,] 1 ) )
91, 2, 6, 7, 8mp4an 654 . . . 4  |-  ( ( 1  /  2 ) [,] 1 )  C_  ( 0 [,] 1
)
109sseli 3189 . . 3  |-  ( X  e.  ( ( 1  /  2 ) [,] 1 )  ->  X  e.  ( 0 [,] 1
) )
11 pcoval.2 . . . 4  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
12 pcoval.3 . . . 4  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
1311, 12pcovalg 18526 . . 3  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
( F ( *p
`  J ) G ) `  X )  =  if ( X  <_  ( 1  / 
2 ) ,  ( F `  ( 2  x.  X ) ) ,  ( G `  ( ( 2  x.  X )  -  1 ) ) ) )
1410, 13sylan2 460 . 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 ) ) ) )
15 pcoval2.4 . . . . . . . 8  |-  ( ph  ->  ( F `  1
)  =  ( G `
 0 ) )
1615adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( F `  1 )  =  ( G ` 
0 ) )
17 simprr 733 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  X  <_  ( 1  /  2
) )
184, 2elicc2i 10732 . . . . . . . . . . . . 13  |-  ( X  e.  ( ( 1  /  2 ) [,] 1 )  <->  ( X  e.  RR  /\  ( 1  /  2 )  <_  X  /\  X  <_  1
) )
1918simp2bi 971 . . . . . . . . . . . 12  |-  ( X  e.  ( ( 1  /  2 ) [,] 1 )  ->  (
1  /  2 )  <_  X )
2019ad2antrl 708 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
1  /  2 )  <_  X )
2118simp1bi 970 . . . . . . . . . . . . 13  |-  ( X  e.  ( ( 1  /  2 ) [,] 1 )  ->  X  e.  RR )
2221ad2antrl 708 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  X  e.  RR )
23 letri3 8923 . . . . . . . . . . . 12  |-  ( ( X  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( X  =  ( 1  /  2
)  <->  ( X  <_ 
( 1  /  2
)  /\  ( 1  /  2 )  <_  X ) ) )
2422, 4, 23sylancl 643 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( X  =  ( 1  /  2 )  <->  ( X  <_  ( 1  /  2
)  /\  ( 1  /  2 )  <_  X ) ) )
2517, 20, 24mpbir2and 888 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  X  =  ( 1  / 
2 ) )
2625oveq2d 5890 . . . . . . . . 9  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
2  x.  X )  =  ( 2  x.  ( 1  /  2
) ) )
27 2cn 9832 . . . . . . . . . 10  |-  2  e.  CC
28 2ne0 9845 . . . . . . . . . 10  |-  2  =/=  0
2927, 28recidi 9507 . . . . . . . . 9  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
3026, 29syl6eq 2344 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
2  x.  X )  =  1 )
3130fveq2d 5545 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( F `  ( 2  x.  X ) )  =  ( F `  1
) )
3230oveq1d 5889 . . . . . . . . 9  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
( 2  x.  X
)  -  1 )  =  ( 1  -  1 ) )
33 1m1e0 9830 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
3432, 33syl6eq 2344 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
( 2  x.  X
)  -  1 )  =  0 )
3534fveq2d 5545 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( G `  ( (
2  x.  X )  -  1 ) )  =  ( G ` 
0 ) )
3616, 31, 353eqtr4d 2338 . . . . . 6  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( F `  ( 2  x.  X ) )  =  ( G `  (
( 2  x.  X
)  -  1 ) ) )
3736ifeq1d 3592 . . . . 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 ) ) ) )
38 ifid 3610 . . . . 5  |-  if ( X  <_  ( 1  /  2 ) ,  ( G `  (
( 2  x.  X
)  -  1 ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  =  ( G `  ( ( 2  x.  X )  -  1 ) )
3937, 38syl6eq 2344 . . . 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 ) ) )
4039expr 598 . . 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 ) ) ) )
41 iffalse 3585 . . 3  |-  ( -.  X  <_  ( 1  /  2 )  ->  if ( X  <_  (
1  /  2 ) ,  ( F `  ( 2  x.  X
) ) ,  ( G `  ( ( 2  x.  X )  -  1 ) ) )  =  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )
4240, 41pm2.61d1 151 . 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 ) ) )
4314, 42eqtrd 2328 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   ifcif 3578   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    <_ cle 8884    - cmin 9053    / cdiv 9439   2c2 9811   [,]cicc 10675    Cn ccn 16970   IIcii 18395   *pcpco 18514
This theorem is referenced by:  pcoass  18538  pcorevlem  18540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-icc 10679  df-top 16652  df-topon 16655  df-cn 16973  df-pco 18519
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