Theorem pcoval 16073

Description: The concatenation of two paths.

Assertion

Ref	Expression
pcoval	|- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ (F` 1) = (G` 0))) -> (F(*p` J)G) = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))})

Distinct variable groups:   x,J,y   x,F,y   x,G,y

Proof of Theorem pcoval

Step	Hyp	Ref	Expression
1		pcofval 16072	. . 3 |- (J e. Top -> (*p` J) = {<.<.f, g>., h>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))))})})
2	1	opreqd 4899	. 2 |- (J e. Top -> (F(*p` J)G) = (F{<.<.f, g>., h>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))))})}G))
3		oprex 4907	. . . . . 6 |- (0[,]1) e. _V
4	3	opabex2 4539	. . . . 5 |- {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))} e. _V
5		id 73	. . . . 5 |- ((F e. (II Cn J) /\ G e. (II Cn J) /\ {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))} e. _V) -> (F e. (II Cn J) /\ G e. (II Cn J) /\ {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))} e. _V))
6	4, 5	mp3an3 1180	. . . 4 |- ((F e. (II Cn J) /\ G e. (II Cn J)) -> (F e. (II Cn J) /\ G e. (II Cn J) /\ {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))} e. _V))
7	6	3adant3 896	. . 3 |- ((F e. (II Cn J) /\ G e. (II Cn J) /\ (F` 1) = (G` 0)) -> (F e. (II Cn J) /\ G e. (II Cn J) /\ {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))} e. _V))
8		eleq1 1957	. . . . . . 7 |- (f = F -> (f e. (II Cn J) <-> F e. (II Cn J)))
9	8	anbi1d 679	. . . . . 6 |- (f = F -> ((f e. (II Cn J) /\ g e. (II Cn J)) <-> (F e. (II Cn J) /\ g e. (II Cn J))))
10		fveq1 4680	. . . . . . 7 |- (f = F -> (f` 1) = (F` 1))
11	10	eqeq1d 1892	. . . . . 6 |- (f = F -> ((f` 1) = (g` 0) <-> (F` 1) = (g` 0)))
12	9, 11	anbi12d 690	. . . . 5 |- (f = F -> (((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0)) <-> ((F e. (II Cn J) /\ g e. (II Cn J)) /\ (F` 1) = (g` 0))))
13		fveq1 4680	. . . . . . . . . 10 |- (f = F -> (f` (2 x. x)) = (F` (2 x. x)))
14	13	ifeq1d 2993	. . . . . . . . 9 |- (f = F -> 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))))
15	14	eqeq2d 1895	. . . . . . . 8 |- (f = F -> (y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))) <-> y = if(x <_ (1 / 2), (F` (2 x. x)), (g` ((2 x. x) - 1)))))
16	15	anbi2d 678	. . . . . . 7 |- (f = F -> ((x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1)))) <-> (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (g` ((2 x. x) - 1))))))
17	16	opabbidv 3401	. . . . . 6 |- (f = F -> {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))))} = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (g` ((2 x. x) - 1))))})
18	17	eqeq2d 1895	. . . . 5 |- (f = F -> (h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))))} <-> h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (g` ((2 x. x) - 1))))}))
19	12, 18	anbi12d 690	. . . 4 |- (f = F -> ((((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))))}) <-> (((F e. (II Cn J) /\ g e. (II Cn J)) /\ (F` 1) = (g` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (g` ((2 x. x) - 1))))})))
20		eleq1 1957	. . . . . . 7 |- (g = G -> (g e. (II Cn J) <-> G e. (II Cn J)))
21	20	anbi2d 678	. . . . . 6 |- (g = G -> ((F e. (II Cn J) /\ g e. (II Cn J)) <-> (F e. (II Cn J) /\ G e. (II Cn J))))
22		fveq1 4680	. . . . . . 7 |- (g = G -> (g` 0) = (G` 0))
23	22	eqeq2d 1895	. . . . . 6 |- (g = G -> ((F` 1) = (g` 0) <-> (F` 1) = (G` 0)))
24	21, 23	anbi12d 690	. . . . 5 |- (g = G -> (((F e. (II Cn J) /\ g e. (II Cn J)) /\ (F` 1) = (g` 0)) <-> ((F e. (II Cn J) /\ G e. (II Cn J)) /\ (F` 1) = (G` 0))))
25		fveq1 4680	. . . . . . . . . 10 |- (g = G -> (g` ((2 x. x) - 1)) = (G` ((2 x. x) - 1)))
26	25	ifeq2d 2994	. . . . . . . . 9 |- (g = G -> 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))))
27	26	eqeq2d 1895	. . . . . . . 8 |- (g = G -> (y = if(x <_ (1 / 2), (F` (2 x. x)), (g` ((2 x. x) - 1))) <-> y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1)))))
28	27	anbi2d 678	. . . . . . 7 |- (g = G -> ((x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (g` ((2 x. x) - 1)))) <-> (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))))
29	28	opabbidv 3401	. . . . . 6 |- (g = G -> {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (g` ((2 x. x) - 1))))} = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))})
30	29	eqeq2d 1895	. . . . 5 |- (g = G -> (h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (g` ((2 x. x) - 1))))} <-> h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))}))
31	24, 30	anbi12d 690	. . . 4 |- (g = G -> ((((F e. (II Cn J) /\ g e. (II Cn J)) /\ (F` 1) = (g` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (g` ((2 x. x) - 1))))}) <-> (((F e. (II Cn J) /\ G e. (II Cn J)) /\ (F` 1) = (G` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))})))
32		simpl 346	. . . . . 6 |- (((F e. (II Cn J) /\ G e. (II Cn J) /\ (F` 1) = (G` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))}) -> (F e. (II Cn J) /\ G e. (II Cn J) /\ (F` 1) = (G` 0)))
33		pm3.21 306	. . . . . 6 |- (h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))} -> ((F e. (II Cn J) /\ G e. (II Cn J) /\ (F` 1) = (G` 0)) -> ((F e. (II Cn J) /\ G e. (II Cn J) /\ (F` 1) = (G` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))})))
34	32, 33	impbid2 576	. . . . 5 |- (h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))} -> (((F e. (II Cn J) /\ G e. (II Cn J) /\ (F` 1) = (G` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))}) <-> (F e. (II Cn J) /\ G e. (II Cn J) /\ (F` 1) = (G` 0))))
35		df-3an 860	. . . . . 6 |- ((F e. (II Cn J) /\ G e. (II Cn J) /\ (F` 1) = (G` 0)) <-> ((F e. (II Cn J) /\ G e. (II Cn J)) /\ (F` 1) = (G` 0)))
36	35	anbi1i 539	. . . . 5 |- (((F e. (II Cn J) /\ G e. (II Cn J) /\ (F` 1) = (G` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))}) <-> (((F e. (II Cn J) /\ G e. (II Cn J)) /\ (F` 1) = (G` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))}))
37	34, 36	syl5bbr 593	. . . 4 |- (h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))} -> ((((F e. (II Cn J) /\ G e. (II Cn J)) /\ (F` 1) = (G` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))}) <-> (F e. (II Cn J) /\ G e. (II Cn J) /\ (F` 1) = (G` 0))))
38		moeq 2431	. . . . 5 |- E*h h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))))}
39	38	moani 1820	. . . 4 |- E*h(((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))))})
40		eqid 1884	. . . 4 |- {<.<.f, g>., h>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))))})} = {<.<.f, g>., h>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))))})}
41	19, 31, 37, 39, 40	oprabvaligg 10154	. . 3 |- ((F e. (II Cn J) /\ G e. (II Cn J) /\ {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))} e. _V) -> ((F e. (II Cn J) /\ G e. (II Cn J) /\ (F` 1) = (G` 0)) -> (F{<.<.f, g>., h>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))))})}G) = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))}))
42	7, 41	mpcom 60	. 2 |- ((F e. (II Cn J) /\ G e. (II Cn J) /\ (F` 1) = (G` 0)) -> (F{<.<.f, g>., h>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))))})}G) = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))})
43	2, 42	sylan9eq 1948	1 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ (F` 1) = (G` 0))) -> (F(*p` J)G) = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (F` (2 x. x)), (G` ((2 x. x) - 1))))})

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292  ifcif 2982   class class class wbr 3338  {copab 3395  ` cfv 3998  (class class class)co 4884  {copab2 4885  0cc0 6386  1c1 6387   x. cmul 6391   - cmin 6445   / cdiv 6447   <_ cle 6448  2c2 7145  [,]cicc 7527  Topctop 8857   Cn ccn 9028  IIcii 15865  *pcpco 16067

This theorem is referenced by:  pcoval1 16074  pcoval2 16075  pcocn 16076  pcopt 16084  pcoass 16085  pcorev 16087

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-pco 16069

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