Theorem pcoval2 18994

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 9047	. . . . 5 |- 0 e. RR
2		1re 9046	. . . . 5 |- 1 e. RR
3	2	rehalfcli 10172	. . . . . 6 |- ( 1 / 2 ) e. RR
4		halfgt0 10144	. . . . . 6 |- 0 < ( 1 / 2 )
5	1, 3, 4	ltleii 9152	. . . . 5 |- 0 <_ ( 1 / 2 )
6		1le1 9606	. . . . 5 |- 1 <_ 1
7		iccss 10934	. . . . 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 655	. . . 4 |- ( ( 1 / 2 ) [,] 1 ) C_ ( 0 [,] 1 )
9	8	sseli 3304	. . 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 18990	. . 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 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 ) )
15	14	adantr 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 ) )
17	3, 2	elicc2i 10932	. . . . . . . . . . . . 13 |- ( X e. ( ( 1 / 2 ) [,] 1 ) <-> ( X e. RR /\ ( 1 / 2 ) <_ X /\ X <_ 1 ) )
18	17	simp2bi 973	. . . . . . . . . . . 12 |- ( X e. ( ( 1 / 2 ) [,] 1 ) -> ( 1 / 2 ) <_ X )
19	18	ad2antrl 709	. . . . . . . . . . 11 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( 1 / 2 ) <_ X )
20	17	simp1bi 972	. . . . . . . . . . . . 13 |- ( X e. ( ( 1 / 2 ) [,] 1 ) -> X e. RR )
21	20	ad2antrl 709	. . . . . . . . . . . 12 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> X e. RR )
22		letri3 9116	. . . . . . . . . . . 12 |- ( ( X e. RR /\ ( 1 / 2 ) e. RR ) -> ( X = ( 1 / 2 ) <-> ( X <_ ( 1 / 2 ) /\ ( 1 / 2 ) <_ X ) ) )
23	21, 3, 22	sylancl 644	. . . . . . . . . . 11 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( X = ( 1 / 2 ) <-> ( X <_ ( 1 / 2 ) /\ ( 1 / 2 ) <_ X ) ) )
24	16, 19, 23	mpbir2and 889	. . . . . . . . . 10 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> X = ( 1 / 2 ) )
25	24	oveq2d 6056	. . . . . . . . 9 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( 2 x. X ) = ( 2 x. ( 1 / 2 ) ) )
26		2cn 10026	. . . . . . . . . 10 |- 2 e. CC
27		2ne0 10039	. . . . . . . . . 10 |- 2 =/= 0
28	26, 27	recidi 9701	. . . . . . . . 9 |- ( 2 x. ( 1 / 2 ) ) = 1
29	25, 28	syl6eq 2452	. . . . . . . 8 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( 2 x. X ) = 1 )
30	29	fveq2d 5691	. . . . . . 7 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( F ` ( 2 x. X ) ) = ( F ` 1 ) )
31	29	oveq1d 6055	. . . . . . . . 9 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( ( 2 x. X ) - 1 ) = ( 1 - 1 ) )
32		1m1e0 10024	. . . . . . . . 9 |- ( 1 - 1 ) = 0
33	31, 32	syl6eq 2452	. . . . . . . 8 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( ( 2 x. X ) - 1 ) = 0 )
34	33	fveq2d 5691	. . . . . . 7 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( G ` ( ( 2 x. X ) - 1 ) ) = ( G ` 0 ) )
35	15, 30, 34	3eqtr4d 2446	. . . . . 6 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( F ` ( 2 x. X ) ) = ( G ` ( ( 2 x. X ) - 1 ) ) )
36	35	ifeq1d 3713	. . . . 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 3731	. . . . 5 |- if ( X <_ ( 1 / 2 ) , ( G ` ( ( 2 x. X ) - 1 ) ) , ( G ` ( ( 2 x. X ) - 1 ) ) ) = ( G ` ( ( 2 x. X ) - 1 ) )
38	36, 37	syl6eq 2452	. . . 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 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 3706	. . 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 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 ) ) )
42	13, 41	eqtrd 2436	1 |- ( ( ph /\ X e. ( ( 1 / 2 ) [,] 1 ) ) -> ( ( F ( *p ` J ) G ) ` X ) = ( G ` ( ( 2 x. X ) - 1 ) ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   C_ wss 3280   ifcif 3699   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   RRcr 8945   0cc0 8946   1c1 8947   x. cmul 8951   <_ cle 9077   - cmin 9247   / cdiv 9633   2c2 10005   [,]cicc 10875   Cn ccn 17242   IIcii 18858   *pcpco 18978

This theorem is referenced by:  pcoass  19002  pcorevlem  19004

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-icc 10879  df-top 16918  df-topon 16921  df-cn 17245  df-pco 18983