Theorem pcoval2 20593

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

Step	Hyp	Ref	Expression
1		0re 9391	. . . . 5 |- 0 e. RR
2		1re 9390	. . . . 5 |- 1 e. RR
3		halfre 10545	. . . . . 6 |- ( 1 / 2 ) e. RR
4		halfgt0 10547	. . . . . 6 |- 0 < ( 1 / 2 )
5	1, 3, 4	ltleii 9502	. . . . 5 |- 0 <_ ( 1 / 2 )
6		1le1 9969	. . . . 5 |- 1 <_ 1
7		iccss 11368	. . . . 5 |- ( ( ( 0 e. RR /\ 1 e. RR ) /\ ( 0 <_ ( 1 / 2 ) /\ 1 <_ 1 ) ) -> ( ( 1 / 2 ) [,] 1 ) C_ ( 0 [,] 1 ) )
8	1, 2, 5, 6, 7	mp4an 673	. . . 4 |- ( ( 1 / 2 ) [,] 1 ) C_ ( 0 [,] 1 )
9	8	sseli 3357	. . 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 ) )
12	10, 11	pcovalg 20589	. . 3 |- ( ( ph /\ X e. ( 0 [,] 1 ) ) -> ( ( F ( *p ` J ) G ) ` X ) = if ( X <_ ( 1 / 2 ) , ( F ` ( 2 x. X ) ) , ( G ` ( ( 2 x. X ) - 1 ) ) ) )
13	9, 12	sylan2 474	. 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 ) )
15	14	adantr 465	. . . . . . 7 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( F ` 1 ) = ( G ` 0 ) )
16		simprr 756	. . . . . . . . . . 11 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> X <_ ( 1 / 2 ) )
17	3, 2	elicc2i 11366	. . . . . . . . . . . . 13 |- ( X e. ( ( 1 / 2 ) [,] 1 ) <-> ( X e. RR /\ ( 1 / 2 ) <_ X /\ X <_ 1 ) )
18	17	simp2bi 1004	. . . . . . . . . . . 12 |- ( X e. ( ( 1 / 2 ) [,] 1 ) -> ( 1 / 2 ) <_ X )
19	18	ad2antrl 727	. . . . . . . . . . 11 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( 1 / 2 ) <_ X )
20	17	simp1bi 1003	. . . . . . . . . . . . 13 |- ( X e. ( ( 1 / 2 ) [,] 1 ) -> X e. RR )
21	20	ad2antrl 727	. . . . . . . . . . . 12 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> X e. RR )
22		letri3 9465	. . . . . . . . . . . 12 |- ( ( X e. RR /\ ( 1 / 2 ) e. RR ) -> ( X = ( 1 / 2 ) <-> ( X <_ ( 1 / 2 ) /\ ( 1 / 2 ) <_ X ) ) )
23	21, 3, 22	sylancl 662	. . . . . . . . . . 11 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( X = ( 1 / 2 ) <-> ( X <_ ( 1 / 2 ) /\ ( 1 / 2 ) <_ X ) ) )
24	16, 19, 23	mpbir2and 913	. . . . . . . . . 10 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> X = ( 1 / 2 ) )
25	24	oveq2d 6112	. . . . . . . . 9 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( 2 x. X ) = ( 2 x. ( 1 / 2 ) ) )
26		2cn 10397	. . . . . . . . . 10 |- 2 e. CC
27		2ne0 10419	. . . . . . . . . 10 |- 2 =/= 0
28	26, 27	recidi 10067	. . . . . . . . 9 |- ( 2 x. ( 1 / 2 ) ) = 1
29	25, 28	syl6eq 2491	. . . . . . . 8 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( 2 x. X ) = 1 )
30	29	fveq2d 5700	. . . . . . 7 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( F ` ( 2 x. X ) ) = ( F ` 1 ) )
31	29	oveq1d 6111	. . . . . . . . 9 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( ( 2 x. X ) - 1 ) = ( 1 - 1 ) )
32		1m1e0 10395	. . . . . . . . 9 |- ( 1 - 1 ) = 0
33	31, 32	syl6eq 2491	. . . . . . . 8 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( ( 2 x. X ) - 1 ) = 0 )
34	33	fveq2d 5700	. . . . . . 7 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( G ` ( ( 2 x. X ) - 1 ) ) = ( G ` 0 ) )
35	15, 30, 34	3eqtr4d 2485	. . . . . 6 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( F ` ( 2 x. X ) ) = ( G ` ( ( 2 x. X ) - 1 ) ) )
36	35	ifeq1d 3812	. . . . 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 3831	. . . . 5 |- if ( X <_ ( 1 / 2 ) , ( G ` ( ( 2 x. X ) - 1 ) ) , ( G ` ( ( 2 x. X ) - 1 ) ) ) = ( G ` ( ( 2 x. X ) - 1 ) )
38	36, 37	syl6eq 2491	. . . 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 ) ) )
39	38	expr 615	. . 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 3804	. . 3 |- ( -. X <_ ( 1 / 2 ) -> if ( X <_ ( 1 / 2 ) , ( F ` ( 2 x. X ) ) , ( G ` ( ( 2 x. X ) - 1 ) ) ) = ( G ` ( ( 2 x. X ) - 1 ) ) )
41	39, 40	pm2.61d1 159	. 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 ) ) )
42	13, 41	eqtrd 2475	1 |- ( ( ph /\ X e. ( ( 1 / 2 ) [,] 1 ) ) -> ( ( F ( *p ` J ) G ) ` X ) = ( G ` ( ( 2 x. X ) - 1 ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   C_ wss 3333   ifcif 3796   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   RRcr 9286   0cc0 9287   1c1 9288   x. cmul 9292   <_ cle 9424   - cmin 9600   / cdiv 9998   2c2 10376   [,]cicc 11308   Cn ccn 18833   IIcii 20456   *pcpco 20577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-2 10385  df-icc 11312  df-top 18508  df-topon 18511  df-cn 18836  df-pco 20582

This theorem is referenced by:  pcoass  20601  pcorevlem  20603