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

Step	Hyp	Ref	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 )
5	1, 3, 4	ltleii 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 ) )
8	1, 2, 5, 6, 7	mp4an 677	. . . 4 |- ( ( 1 / 2 ) [,] 1 ) C_ ( 0 [,] 1 )
9	8	sseli 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 ) )
12	10, 11	pcovalg 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 ) ) ) )
13	9, 12	sylan2 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 ) )
15	14	adantr 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 ) )
17	3, 2	elicc2i 11700	. . . . . . . . . . . . 13 |- ( X e. ( ( 1 / 2 ) [,] 1 ) <-> ( X e. RR /\ ( 1 / 2 ) <_ X /\ X <_ 1 ) )
18	17	simp2bi 1021	. . . . . . . . . . . 12 |- ( X e. ( ( 1 / 2 ) [,] 1 ) -> ( 1 / 2 ) <_ X )
19	18	ad2antrl 732	. . . . . . . . . . 11 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( 1 / 2 ) <_ X )
20	17	simp1bi 1020	. . . . . . . . . . . . 13 |- ( X e. ( ( 1 / 2 ) [,] 1 ) -> X e. RR )
21	20	ad2antrl 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 ) ) )
23	21, 3, 22	sylancl 666	. . . . . . . . . . 11 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( X = ( 1 / 2 ) <-> ( X <_ ( 1 / 2 ) /\ ( 1 / 2 ) <_ X ) ) )
24	16, 19, 23	mpbir2and 930	. . . . . . . . . 10 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> X = ( 1 / 2 ) )
25	24	oveq2d 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
28	26, 27	recidi 10337	. . . . . . . . 9 |- ( 2 x. ( 1 / 2 ) ) = 1
29	25, 28	syl6eq 2486	. . . . . . . 8 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( 2 x. X ) = 1 )
30	29	fveq2d 5885	. . . . . . 7 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( F ` ( 2 x. X ) ) = ( F ` 1 ) )
31	29	oveq1d 6320	. . . . . . . . 9 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( ( 2 x. X ) - 1 ) = ( 1 - 1 ) )
32		1m1e0 10678	. . . . . . . . 9 |- ( 1 - 1 ) = 0
33	31, 32	syl6eq 2486	. . . . . . . 8 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( ( 2 x. X ) - 1 ) = 0 )
34	33	fveq2d 5885	. . . . . . 7 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( G ` ( ( 2 x. X ) - 1 ) ) = ( G ` 0 ) )
35	15, 30, 34	3eqtr4d 2480	. . . . . 6 |- ( ( ph /\ ( X e. ( ( 1 / 2 ) [,] 1 ) /\ X <_ ( 1 / 2 ) ) ) -> ( F ` ( 2 x. X ) ) = ( G ` ( ( 2 x. X ) - 1 ) ) )
36	35	ifeq1d 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 ) )
38	36, 37	syl6eq 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 ) ) )
39	38	expr 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 ) ) )
41	39, 40	pm2.61d1 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 ) ) )
42	13, 41	eqtrd 2470	1 |- ( ( ph /\ X e. ( ( 1 / 2 ) [,] 1 ) ) -> ( ( F ( *p ` J ) G ) ` X ) = ( G ` ( ( 2 x. X ) - 1 ) ) )

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