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Theorem pcoval2 21940
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 9642 . . . . 5  |-  0  e.  RR
2 1re 9641 . . . . 5  |-  1  e.  RR
3 halfre 10828 . . . . . 6  |-  ( 1  /  2 )  e.  RR
4 halfgt0 10830 . . . . . 6  |-  0  <  ( 1  /  2
)
51, 3, 4ltleii 9756 . . . . 5  |-  0  <_  ( 1  /  2
)
6 1le1 10239 . . . . 5  |-  1  <_  1
7 iccss 11702 . . . . 5  |-  ( ( ( 0  e.  RR  /\  1  e.  RR )  /\  ( 0  <_ 
( 1  /  2
)  /\  1  <_  1 ) )  ->  (
( 1  /  2
) [,] 1 ) 
C_  ( 0 [,] 1 ) )
81, 2, 5, 6, 7mp4an 677 . . . 4  |-  ( ( 1  /  2 ) [,] 1 )  C_  ( 0 [,] 1
)
98sseli 3466 . . 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 21936 . . 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 476 . 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 466 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( F `  1 )  =  ( G ` 
0 ) )
16 simprr 764 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  X  <_  ( 1  /  2
) )
173, 2elicc2i 11700 . . . . . . . . . . . . 13  |-  ( X  e.  ( ( 1  /  2 ) [,] 1 )  <->  ( X  e.  RR  /\  ( 1  /  2 )  <_  X  /\  X  <_  1
) )
1817simp2bi 1021 . . . . . . . . . . . 12  |-  ( X  e.  ( ( 1  /  2 ) [,] 1 )  ->  (
1  /  2 )  <_  X )
1918ad2antrl 732 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
1  /  2 )  <_  X )
2017simp1bi 1020 . . . . . . . . . . . . 13  |-  ( X  e.  ( ( 1  /  2 ) [,] 1 )  ->  X  e.  RR )
2120ad2antrl 732 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  X  e.  RR )
22 letri3 9718 . . . . . . . . . . . 12  |-  ( ( X  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( X  =  ( 1  /  2
)  <->  ( X  <_ 
( 1  /  2
)  /\  ( 1  /  2 )  <_  X ) ) )
2321, 3, 22sylancl 666 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( X  =  ( 1  /  2 )  <->  ( X  <_  ( 1  /  2
)  /\  ( 1  /  2 )  <_  X ) ) )
2416, 19, 23mpbir2and 930 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  X  =  ( 1  / 
2 ) )
2524oveq2d 6321 . . . . . . . . 9  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
2  x.  X )  =  ( 2  x.  ( 1  /  2
) ) )
26 2cn 10680 . . . . . . . . . 10  |-  2  e.  CC
27 2ne0 10702 . . . . . . . . . 10  |-  2  =/=  0
2826, 27recidi 10337 . . . . . . . . 9  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
2925, 28syl6eq 2486 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
2  x.  X )  =  1 )
3029fveq2d 5885 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( F `  ( 2  x.  X ) )  =  ( F `  1
) )
3129oveq1d 6320 . . . . . . . . 9  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
( 2  x.  X
)  -  1 )  =  ( 1  -  1 ) )
32 1m1e0 10678 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
3331, 32syl6eq 2486 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  (
( 2  x.  X
)  -  1 )  =  0 )
3433fveq2d 5885 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( G `  ( (
2  x.  X )  -  1 ) )  =  ( G ` 
0 ) )
3515, 30, 343eqtr4d 2480 . . . . . 6  |-  ( (
ph  /\  ( X  e.  ( ( 1  / 
2 ) [,] 1
)  /\  X  <_  ( 1  /  2 ) ) )  ->  ( F `  ( 2  x.  X ) )  =  ( G `  (
( 2  x.  X
)  -  1 ) ) )
3635ifeq1d 3933 . . . . 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 3952 . . . . 5  |-  if ( X  <_  ( 1  /  2 ) ,  ( G `  (
( 2  x.  X
)  -  1 ) ) ,  ( G `
 ( ( 2  x.  X )  - 
1 ) ) )  =  ( G `  ( ( 2  x.  X )  -  1 ) )
3836, 37syl6eq 2486 . . . 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 618 . . 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 3924 . . 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 162 . 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 2470 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 187    /\ wa 370    = wceq 1437    e. wcel 1870    C_ wss 3442   ifcif 3915   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   RRcr 9537   0cc0 9538   1c1 9539    x. cmul 9543    <_ cle 9675    - cmin 9859    / cdiv 10268   2c2 10659   [,]cicc 11638    Cn ccn 20171   IIcii 21803   *pcpco 21924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-2 10668  df-icc 11642  df-top 19852  df-topon 19854  df-cn 20174  df-pco 21929
This theorem is referenced by:  pcoass  21948  pcorevlem  21950
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