Theorem pcovalg 22812

Description: Evaluate the concatenation of two paths. (Contributed 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 ) )

Assertion

Ref	Expression
pcovalg	|- ( ( ph /\ X e. ( 0 [,] 1 ) ) -> ( ( F ( *p ` J ) G ) ` X ) = if ( X <_ ( 1 / 2 ) , ( F ` ( 2 x. X ) ) , ( G ` ( ( 2 x. X ) - 1 ) ) ) )

Proof of Theorem pcovalg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pcoval.2	. . . 4 |- ( ph -> F e. ( II Cn J ) )
2		pcoval.3	. . . 4 |- ( ph -> G e. ( II Cn J ) )
3	1, 2	pcoval 22811	. . 3 |- ( ph -> ( F ( *p ` J ) G ) = ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( G ` ( ( 2 x. x ) - 1 ) ) ) ) )
4	3	fveq1d 6193	. 2 |- ( ph -> ( ( F ( *p ` J ) G ) ` X ) = ( ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( G ` ( ( 2 x. x ) - 1 ) ) ) ) ` X ) )
5		breq1 4656	. . . 4 |- ( x = X -> ( x <_ ( 1 / 2 ) <-> X <_ ( 1 / 2 ) ) )
6		oveq2 6658	. . . . 5 |- ( x = X -> ( 2 x. x ) = ( 2 x. X ) )
7	6	fveq2d 6195	. . . 4 |- ( x = X -> ( F ` ( 2 x. x ) ) = ( F ` ( 2 x. X ) ) )
8	6	oveq1d 6665	. . . . 5 |- ( x = X -> ( ( 2 x. x ) - 1 ) = ( ( 2 x. X ) - 1 ) )
9	8	fveq2d 6195	. . . 4 |- ( x = X -> ( G ` ( ( 2 x. x ) - 1 ) ) = ( G ` ( ( 2 x. X ) - 1 ) ) )
10	5, 7, 9	ifbieq12d 4113	. . 3 |- ( x = X -> if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( G ` ( ( 2 x. x ) - 1 ) ) ) = if ( X <_ ( 1 / 2 ) , ( F ` ( 2 x. X ) ) , ( G ` ( ( 2 x. X ) - 1 ) ) ) )
11		eqid 2622	. . 3 |- ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( G ` ( ( 2 x. x ) - 1 ) ) ) ) = ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( G ` ( ( 2 x. x ) - 1 ) ) ) )
12		fvex 6201	. . . 4 |- ( F ` ( 2 x. X ) ) e. _V
13		fvex 6201	. . . 4 |- ( G ` ( ( 2 x. X ) - 1 ) ) e. _V
14	12, 13	ifex 4156	. . 3 |- if ( X <_ ( 1 / 2 ) , ( F ` ( 2 x. X ) ) , ( G ` ( ( 2 x. X ) - 1 ) ) ) e. _V
15	10, 11, 14	fvmpt 6282	. 2 |- ( X e. ( 0 [,] 1 ) -> ( ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( G ` ( ( 2 x. x ) - 1 ) ) ) ) ` X ) = if ( X <_ ( 1 / 2 ) , ( F ` ( 2 x. X ) ) , ( G ` ( ( 2 x. X ) - 1 ) ) ) )
16	4, 15	sylan9eq 2676	1 |- ( ( ph /\ X e. ( 0 [,] 1 ) ) -> ( ( F ( *p ` J ) G ) ` X ) = if ( X <_ ( 1 / 2 ) , ( F ` ( 2 x. X ) ) , ( G ` ( ( 2 x. X ) - 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   x. cmul 9941   <_ cle 10075   - cmin 10266   / cdiv 10684   2c2 11070   [,]cicc 12178   Cn ccn 21028   IIcii 22678   *pcpco 22800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-top 20699  df-topon 20716  df-cn 21031  df-pco 22805

This theorem is referenced by:  pcoval1  22813  pcoval2  22816  pcohtpylem  22819