Theorem pcoass 16085

Description: Order of concatenation does not affect homotopy class.

Assertion

Ref	Expression
pcoass	|- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> ((F(*p` J)G)(*p` J)H)(~=ph` J)(F(*p` J)(G(*p` J)H)))

Proof of Theorem pcoass

Step	Hyp	Ref	Expression
1		simp1 876	. . . . . . . . . . . . . . 15 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> J e. Top)
2		simp1 876	. . . . . . . . . . . . . . . 16 |- ((F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) -> F e. (II Cn J))
3	2	3ad2ant2 898	. . . . . . . . . . . . . . 15 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> F e. (II Cn J))
4		simp2 877	. . . . . . . . . . . . . . . 16 |- ((F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) -> G e. (II Cn J))
5	4	3ad2ant2 898	. . . . . . . . . . . . . . 15 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> G e. (II Cn J))
6		simp3l 904	. . . . . . . . . . . . . . 15 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> (F` 1) = (G` 0))
7		pcoval 16073	. . . . . . . . . . . . . . 15 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ (F` 1) = (G` 0))) -> (F(*p` J)G) = {<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), (G` ((2 x. v) - 1))))})
8	1, 3, 5, 6, 7	syl13anc 1102	. . . . . . . . . . . . . 14 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> (F(*p` J)G) = {<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), (G` ((2 x. v) - 1))))})
9	8	ad2antrr 440	. . . . . . . . . . . . 13 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) -> (F(*p` J)G) = {<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), (G` ((2 x. v) - 1))))})
10	9	fveq1d 4683	. . . . . . . . . . . 12 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) -> ((F(*p` J)G)` (2 x. x)) = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), (G` ((2 x. v) - 1))))}` (2 x. x)))
11		0re 6603	. . . . . . . . . . . . . . . . . . 19 |- 0 e. RR
12		2re 7163	. . . . . . . . . . . . . . . . . . . 20 |- 2 e. RR
13		2ne0 7174	. . . . . . . . . . . . . . . . . . . 20 |- 2 =/= 0
14	12, 13	rereccli 6979	. . . . . . . . . . . . . . . . . . 19 |- (1 / 2) e. RR
15		elicc2 7560	. . . . . . . . . . . . . . . . . . 19 |- ((0 e. RR /\ (1 / 2) e. RR) -> (x e. (0[,](1 / 2)) <-> (x e. RR /\ 0 <_ x /\ x <_ (1 / 2))))
16	11, 14, 15	mp2an 761	. . . . . . . . . . . . . . . . . 18 |- (x e. (0[,](1 / 2)) <-> (x e. RR /\ 0 <_ x /\ x <_ (1 / 2)))
17	16	biimpri 169	. . . . . . . . . . . . . . . . 17 |- ((x e. RR /\ 0 <_ x /\ x <_ (1 / 2)) -> x e. (0[,](1 / 2)))
18	17	3expa 1067	. . . . . . . . . . . . . . . 16 |- (((x e. RR /\ 0 <_ x) /\ x <_ (1 / 2)) -> x e. (0[,](1 / 2)))
19	18	3adantl3 1034	. . . . . . . . . . . . . . 15 |- (((x e. RR /\ 0 <_ x /\ x <_ 1) /\ x <_ (1 / 2)) -> x e. (0[,](1 / 2)))
20		1re 6598	. . . . . . . . . . . . . . . 16 |- 1 e. RR
21		elicc2 7560	. . . . . . . . . . . . . . . 16 |- ((0 e. RR /\ 1 e. RR) -> (x e. (0[,]1) <-> (x e. RR /\ 0 <_ x /\ x <_ 1)))
22	11, 20, 21	mp2an 761	. . . . . . . . . . . . . . 15 |- (x e. (0[,]1) <-> (x e. RR /\ 0 <_ x /\ x <_ 1))
23	19, 22	sylanb 498	. . . . . . . . . . . . . 14 |- ((x e. (0[,]1) /\ x <_ (1 / 2)) -> x e. (0[,](1 / 2)))
24		iihalf1 15872	. . . . . . . . . . . . . 14 |- (x e. (0[,](1 / 2)) -> (2 x. x) e. (0[,]1))
25		breq1 3341	. . . . . . . . . . . . . . . 16 |- (v = (2 x. x) -> (v <_ (1 / 2) <-> (2 x. x) <_ (1 / 2)))
26		opreq2 4890	. . . . . . . . . . . . . . . . 17 |- (v = (2 x. x) -> (2 x. v) = (2 x. (2 x. x)))
27	26	fveq2d 4685	. . . . . . . . . . . . . . . 16 |- (v = (2 x. x) -> (F` (2 x. v)) = (F` (2 x. (2 x. x))))
28	26	opreq1d 4897	. . . . . . . . . . . . . . . . 17 |- (v = (2 x. x) -> ((2 x. v) - 1) = ((2 x. (2 x. x)) - 1))
29	28	fveq2d 4685	. . . . . . . . . . . . . . . 16 |- (v = (2 x. x) -> (G` ((2 x. v) - 1)) = (G` ((2 x. (2 x. x)) - 1)))
30	25, 27, 29	ifbieq12d 2998	. . . . . . . . . . . . . . 15 |- (v = (2 x. x) -> if(v <_ (1 / 2), (F` (2 x. v)), (G` ((2 x. v) - 1))) = if((2 x. x) <_ (1 / 2), (F` (2 x. (2 x. x))), (G` ((2 x. (2 x. x)) - 1))))
31		eqid 1884	. . . . . . . . . . . . . . 15 |- {<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), (G` ((2 x. v) - 1))))} = {<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), (G` ((2 x. v) - 1))))}
32		fvex 4689	. . . . . . . . . . . . . . . 16 |- (F` (2 x. (2 x. x))) e. _V
33		fvex 4689	. . . . . . . . . . . . . . . 16 |- (G` ((2 x. (2 x. x)) - 1)) e. _V
34	32, 33	ifex 3031	. . . . . . . . . . . . . . 15 |- if((2 x. x) <_ (1 / 2), (F` (2 x. (2 x. x))), (G` ((2 x. (2 x. x)) - 1))) e. _V
35	30, 31, 34	fvopab4 4743	. . . . . . . . . . . . . 14 |- ((2 x. x) e. (0[,]1) -> ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), (G` ((2 x. v) - 1))))}` (2 x. x)) = if((2 x. x) <_ (1 / 2), (F` (2 x. (2 x. x))), (G` ((2 x. (2 x. x)) - 1))))
36	23, 24, 35	3syl 24	. . . . . . . . . . . . 13 |- ((x e. (0[,]1) /\ x <_ (1 / 2)) -> ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), (G` ((2 x. v) - 1))))}` (2 x. x)) = if((2 x. x) <_ (1 / 2), (F` (2 x. (2 x. x))), (G` ((2 x. (2 x. x)) - 1))))
37	36	adantll 428	. . . . . . . . . . . 12 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) -> ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), (G` ((2 x. v) - 1))))}` (2 x. x)) = if((2 x. x) <_ (1 / 2), (F` (2 x. (2 x. x))), (G` ((2 x. (2 x. x)) - 1))))
38		iccssre 7565	. . . . . . . . . . . . . . . . . 18 |- ((0 e. RR /\ 1 e. RR) -> (0[,]1) C_ RR)
39	11, 20, 38	mp2an 761	. . . . . . . . . . . . . . . . 17 |- (0[,]1) C_ RR
40	39	sseli 2617	. . . . . . . . . . . . . . . 16 |- (x e. (0[,]1) -> x e. RR)
41		2pos 7173	. . . . . . . . . . . . . . . . . . 19 |- 0 < 2
42	12, 41	pm3.2i 307	. . . . . . . . . . . . . . . . . 18 |- (2 e. RR /\ 0 < 2)
43		lemuldiv2 7059	. . . . . . . . . . . . . . . . . 18 |- ((x e. RR /\ (1 / 2) e. RR /\ (2 e. RR /\ 0 < 2)) -> ((2 x. x) <_ (1 / 2) <-> x <_ ((1 / 2) / 2)))
44	14, 42, 43	mp3an23 1183	. . . . . . . . . . . . . . . . 17 |- (x e. RR -> ((2 x. x) <_ (1 / 2) <-> x <_ ((1 / 2) / 2)))
45		ax1cn 6422	. . . . . . . . . . . . . . . . . . . 20 |- 1 e. CC
46		2cn 7164	. . . . . . . . . . . . . . . . . . . . 21 |- 2 e. CC
47	46, 13	pm3.2i 307	. . . . . . . . . . . . . . . . . . . 20 |- (2 e. CC /\ 2 =/= 0)
48		divdiv1 6972	. . . . . . . . . . . . . . . . . . . 20 |- ((1 e. CC /\ (2 e. CC /\ 2 =/= 0) /\ (2 e. CC /\ 2 =/= 0)) -> ((1 / 2) / 2) = (1 / (2 x. 2)))
49	45, 47, 47, 48	mp3an 1191	. . . . . . . . . . . . . . . . . . 19 |- ((1 / 2) / 2) = (1 / (2 x. 2))
50		2t2e4 7206	. . . . . . . . . . . . . . . . . . . 20 |- (2 x. 2) = 4
51	50	opreq2i 4893	. . . . . . . . . . . . . . . . . . 19 |- (1 / (2 x. 2)) = (1 / 4)
52	49, 51	eqtri 1908	. . . . . . . . . . . . . . . . . 18 |- ((1 / 2) / 2) = (1 / 4)
53	52	breq2i 3346	. . . . . . . . . . . . . . . . 17 |- (x <_ ((1 / 2) / 2) <-> x <_ (1 / 4))
54	44, 53	syl6bb 595	. . . . . . . . . . . . . . . 16 |- (x e. RR -> ((2 x. x) <_ (1 / 2) <-> x <_ (1 / 4)))
55	40, 54	syl 12	. . . . . . . . . . . . . . 15 |- (x e. (0[,]1) -> ((2 x. x) <_ (1 / 2) <-> x <_ (1 / 4)))
56	55	ad2antlr 441	. . . . . . . . . . . . . 14 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) -> ((2 x. x) <_ (1 / 2) <-> x <_ (1 / 4)))
57	56	ifbid 2996	. . . . . . . . . . . . 13 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) -> if((2 x. x) <_ (1 / 2), (F` (2 x. (2 x. x))), (G` ((2 x. (2 x. x)) - 1))) = if(x <_ (1 / 4), (F` (2 x. (2 x. x))), (G` ((2 x. (2 x. x)) - 1))))
58	23, 24	syl 12	. . . . . . . . . . . . . . . . . 18 |- ((x e. (0[,]1) /\ x <_ (1 / 2)) -> (2 x. x) e. (0[,]1))
59	58	adantr 425	. . . . . . . . . . . . . . . . 17 |- (((x e. (0[,]1) /\ x <_ (1 / 2)) /\ x <_ (1 / 4)) -> (2 x. x) e. (0[,]1))
60	28	fveq2d 4685	. . . . . . . . . . . . . . . . . . 19 |- (v = (2 x. x) -> ((G(*p` J)H)` ((2 x. v) - 1)) = ((G(*p` J)H)` ((2 x. (2 x. x)) - 1)))
61	25, 27, 60	ifbieq12d 2998	. . . . . . . . . . . . . . . . . 18 |- (v = (2 x. x) -> if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))) = if((2 x. x) <_ (1 / 2), (F` (2 x. (2 x. x))), ((G(*p` J)H)` ((2 x. (2 x. x)) - 1))))
62		eqid 1884	. . . . . . . . . . . . . . . . . 18 |- {<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))} = {<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}
63		fvex 4689	. . . . . . . . . . . . . . . . . . 19 |- ((G(*p` J)H)` ((2 x. (2 x. x)) - 1)) e. _V
64	32, 63	ifex 3031	. . . . . . . . . . . . . . . . . 18 |- if((2 x. x) <_ (1 / 2), (F` (2 x. (2 x. x))), ((G(*p` J)H)` ((2 x. (2 x. x)) - 1))) e. _V
65	61, 62, 64	fvopab4 4743	. . . . . . . . . . . . . . . . 17 |- ((2 x. x) e. (0[,]1) -> ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` (2 x. x)) = if((2 x. x) <_ (1 / 2), (F` (2 x. (2 x. x))), ((G(*p` J)H)` ((2 x. (2 x. x)) - 1))))
66	59, 65	syl 12	. . . . . . . . . . . . . . . 16 |- (((x e. (0[,]1) /\ x <_ (1 / 2)) /\ x <_ (1 / 4)) -> ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` (2 x. x)) = if((2 x. x) <_ (1 / 2), (F` (2 x. (2 x. x))), ((G(*p` J)H)` ((2 x. (2 x. x)) - 1))))
67	54	biimpar 461	. . . . . . . . . . . . . . . . . . 19 |- ((x e. RR /\ x <_ (1 / 4)) -> (2 x. x) <_ (1 / 2))
68	67, 40	sylan 497	. . . . . . . . . . . . . . . . . 18 |- ((x e. (0[,]1) /\ x <_ (1 / 4)) -> (2 x. x) <_ (1 / 2))
69	68	adantlr 429	. . . . . . . . . . . . . . . . 17 |- (((x e. (0[,]1) /\ x <_ (1 / 2)) /\ x <_ (1 / 4)) -> (2 x. x) <_ (1 / 2))
70		iftrue 2989	. . . . . . . . . . . . . . . . 17 |- ((2 x. x) <_ (1 / 2) -> if((2 x. x) <_ (1 / 2), (F` (2 x. (2 x. x))), ((G(*p` J)H)` ((2 x. (2 x. x)) - 1))) = (F` (2 x. (2 x. x))))
71	69, 70	syl 12	. . . . . . . . . . . . . . . 16 |- (((x e. (0[,]1) /\ x <_ (1 / 2)) /\ x <_ (1 / 4)) -> if((2 x. x) <_ (1 / 2), (F` (2 x. (2 x. x))), ((G(*p` J)H)` ((2 x. (2 x. x)) - 1))) = (F` (2 x. (2 x. x))))
72	66, 71	eqtr2d 1926	. . . . . . . . . . . . . . 15 |- (((x e. (0[,]1) /\ x <_ (1 / 2)) /\ x <_ (1 / 4)) -> (F` (2 x. (2 x. x))) = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` (2 x. x)))
73	72	adantlll 432	. . . . . . . . . . . . . 14 |- (((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) /\ x <_ (1 / 4)) -> (F` (2 x. (2 x. x))) = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` (2 x. x)))
74	73	ifeq1da 15693	. . . . . . . . . . . . 13 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) -> if(x <_ (1 / 4), (F` (2 x. (2 x. x))), (G` ((2 x. (2 x. x)) - 1))) = if(x <_ (1 / 4), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` (2 x. x)), (G` ((2 x. (2 x. x)) - 1))))
75		4re 7166	. . . . . . . . . . . . . . . . . . . . 21 |- 4 e. RR
76		4pos 7176	. . . . . . . . . . . . . . . . . . . . . 22 |- 0 < 4
77	75, 76	gt0ne0ii 6799	. . . . . . . . . . . . . . . . . . . . 21 |- 4 =/= 0
78	75, 77	rereccli 6979	. . . . . . . . . . . . . . . . . . . 20 |- (1 / 4) e. RR
79	78	recni 6467	. . . . . . . . . . . . . . . . . . . . 21 |- (1 / 4) e. CC
80	79	addid2i 6484	. . . . . . . . . . . . . . . . . . . 20 |- (0 + (1 / 4)) = (1 / 4)
81	75	recni 6467	. . . . . . . . . . . . . . . . . . . . . 22 |- 4 e. CC
82	46, 45, 81, 77	divdiri 6930	. . . . . . . . . . . . . . . . . . . . 21 |- ((2 + 1) / 4) = ((2 / 4) + (1 / 4))
83		df-3 7155	. . . . . . . . . . . . . . . . . . . . . 22 |- 3 = (2 + 1)
84	83	opreq1i 4892	. . . . . . . . . . . . . . . . . . . . 21 |- (3 / 4) = ((2 + 1) / 4)
85		divcan5 6957	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((1 e. CC /\ (2 e. CC /\ 2 =/= 0) /\ (2 e. CC /\ 2 =/= 0)) -> ((2 x. 1) / (2 x. 2)) = (1 / 2))
86	45, 47, 47, 85	mp3an 1191	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((2 x. 1) / (2 x. 2)) = (1 / 2)
87	46	mulid1i 6485	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (2 x. 1) = 2
88	87, 50	opreq12i 4894	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((2 x. 1) / (2 x. 2)) = (2 / 4)
89	86, 88	eqtr3i 1910	. . . . . . . . . . . . . . . . . . . . . 22 |- (1 / 2) = (2 / 4)
90	89	opreq1i 4892	. . . . . . . . . . . . . . . . . . . . 21 |- ((1 / 2) + (1 / 4)) = ((2 / 4) + (1 / 4))
91	82, 84, 90	3eqtr4ri 1923	. . . . . . . . . . . . . . . . . . . 20 |- ((1 / 2) + (1 / 4)) = (3 / 4)
92	11, 14, 78, 80, 91	iccshftri 15858	. . . . . . . . . . . . . . . . . . 19 |- (x e. (0[,](1 / 2)) -> (x + (1 / 4)) e. ((1 / 4)[,](3 / 4)))
93	11, 20	pm3.2i 307	. . . . . . . . . . . . . . . . . . . . 21 |- (0 e. RR /\ 1 e. RR)
94		3re 7165	. . . . . . . . . . . . . . . . . . . . . . 23 |- 3 e. RR
95	94, 75, 77	redivcli 6976	. . . . . . . . . . . . . . . . . . . . . 22 |- (3 / 4) e. RR
96	78, 95	pm3.2i 307	. . . . . . . . . . . . . . . . . . . . 21 |- ((1 / 4) e. RR /\ (3 / 4) e. RR)
97	75, 76	recgt0ii 6992	. . . . . . . . . . . . . . . . . . . . . . 23 |- 0 < (1 / 4)
98	11, 78, 97	ltleii 6756	. . . . . . . . . . . . . . . . . . . . . 22 |- 0 <_ (1 / 4)
99	94	ltp1i 6991	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 3 < (3 + 1)
100		df-4 7156	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 4 = (3 + 1)
101	99, 100	breqtrri 3362	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- 3 < 4
102	94, 75, 101	ltleii 6756	. . . . . . . . . . . . . . . . . . . . . . . 24 |- 3 <_ 4
103	81	mulid1i 6485	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (4 x. 1) = 4
104	102, 103	breqtrri 3362	. . . . . . . . . . . . . . . . . . . . . . 23 |- 3 <_ (4 x. 1)
105	75, 76	pm3.2i 307	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (4 e. RR /\ 0 < 4)
106		ledivmul 7051	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((3 e. RR /\ 1 e. RR /\ (4 e. RR /\ 0 < 4)) -> ((3 / 4) <_ 1 <-> 3 <_ (4 x. 1)))
107	94, 20, 105, 106	mp3an 1191	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((3 / 4) <_ 1 <-> 3 <_ (4 x. 1))
108	104, 107	mpbir 207	. . . . . . . . . . . . . . . . . . . . . 22 |- (3 / 4) <_ 1
109	98, 108	pm3.2i 307	. . . . . . . . . . . . . . . . . . . . 21 |- (0 <_ (1 / 4) /\ (3 / 4) <_ 1)
110		iccss 15855	. . . . . . . . . . . . . . . . . . . . 21 |- (((0 e. RR /\ 1 e. RR) /\ ((1 / 4) e. RR /\ (3 / 4) e. RR) /\ (0 <_ (1 / 4) /\ (3 / 4) <_ 1)) -> ((1 / 4)[,](3 / 4)) C_ (0[,]1))
111	93, 96, 109, 110	mp3an 1191	. . . . . . . . . . . . . . . . . . . 20 |- ((1 / 4)[,](3 / 4)) C_ (0[,]1)
112	111	sseli 2617	. . . . . . . . . . . . . . . . . . 19 |- ((x + (1 / 4)) e. ((1 / 4)[,](3 / 4)) -> (x + (1 / 4)) e. (0[,]1))
113	23, 92, 112	3syl 24	. . . . . . . . . . . . . . . . . 18 |- ((x e. (0[,]1) /\ x <_ (1 / 2)) -> (x + (1 / 4)) e. (0[,]1))
114	113	adantr 425	. . . . . . . . . . . . . . . . 17 |- (((x e. (0[,]1) /\ x <_ (1 / 2)) /\ -. x <_ (1 / 4)) -> (x + (1 / 4)) e. (0[,]1))
115	114	adantlll 432	. . . . . . . . . . . . . . . 16 |- (((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) /\ -. x <_ (1 / 4)) -> (x + (1 / 4)) e. (0[,]1))
116		breq1 3341	. . . . . . . . . . . . . . . . . 18 |- (v = (x + (1 / 4)) -> (v <_ (1 / 2) <-> (x + (1 / 4)) <_ (1 / 2)))
117		opreq2 4890	. . . . . . . . . . . . . . . . . . 19 |- (v = (x + (1 / 4)) -> (2 x. v) = (2 x. (x + (1 / 4))))
118	117	fveq2d 4685	. . . . . . . . . . . . . . . . . 18 |- (v = (x + (1 / 4)) -> (F` (2 x. v)) = (F` (2 x. (x + (1 / 4)))))
119	117	opreq1d 4897	. . . . . . . . . . . . . . . . . . 19 |- (v = (x + (1 / 4)) -> ((2 x. v) - 1) = ((2 x. (x + (1 / 4))) - 1))
120	119	fveq2d 4685	. . . . . . . . . . . . . . . . . 18 |- (v = (x + (1 / 4)) -> ((G(*p` J)H)` ((2 x. v) - 1)) = ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1)))
121	116, 118, 120	ifbieq12d 2998	. . . . . . . . . . . . . . . . 17 |- (v = (x + (1 / 4)) -> if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))) = if((x + (1 / 4)) <_ (1 / 2), (F` (2 x. (x + (1 / 4)))), ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1))))
122		fvex 4689	. . . . . . . . . . . . . . . . . 18 |- (F` (2 x. (x + (1 / 4)))) e. _V
123		fvex 4689	. . . . . . . . . . . . . . . . . 18 |- ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1)) e. _V
124	122, 123	ifex 3031	. . . . . . . . . . . . . . . . 17 |- if((x + (1 / 4)) <_ (1 / 2), (F` (2 x. (x + (1 / 4)))), ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1))) e. _V
125	121, 62, 124	fvopab4 4743	. . . . . . . . . . . . . . . 16 |- ((x + (1 / 4)) e. (0[,]1) -> ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` (x + (1 / 4))) = if((x + (1 / 4)) <_ (1 / 2), (F` (2 x. (x + (1 / 4)))), ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1))))
126	115, 125	syl 12	. . . . . . . . . . . . . . 15 |- (((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) /\ -. x <_ (1 / 4)) -> ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` (x + (1 / 4))) = if((x + (1 / 4)) <_ (1 / 2), (F` (2 x. (x + (1 / 4)))), ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1))))
127		leadd1 6808	. . . . . . . . . . . . . . . . . . . . . 22 |- ((x e. RR /\ (1 / 4) e. RR /\ (1 / 4) e. RR) -> (x <_ (1 / 4) <-> (x + (1 / 4)) <_ ((1 / 4) + (1 / 4))))
128	78, 78, 127	mp3an23 1183	. . . . . . . . . . . . . . . . . . . . 21 |- (x e. RR -> (x <_ (1 / 4) <-> (x + (1 / 4)) <_ ((1 / 4) + (1 / 4))))
129	40, 128	syl 12	. . . . . . . . . . . . . . . . . . . 20 |- (x e. (0[,]1) -> (x <_ (1 / 4) <-> (x + (1 / 4)) <_ ((1 / 4) + (1 / 4))))
130		df-2 7154	. . . . . . . . . . . . . . . . . . . . . . 23 |- 2 = (1 + 1)
131	130	opreq1i 4892	. . . . . . . . . . . . . . . . . . . . . 22 |- (2 / 4) = ((1 + 1) / 4)
132	45, 45, 81, 77	divdiri 6930	. . . . . . . . . . . . . . . . . . . . . 22 |- ((1 + 1) / 4) = ((1 / 4) + (1 / 4))
133	89, 131, 132	3eqtrri 1913	. . . . . . . . . . . . . . . . . . . . 21 |- ((1 / 4) + (1 / 4)) = (1 / 2)
134	133	breq2i 3346	. . . . . . . . . . . . . . . . . . . 20 |- ((x + (1 / 4)) <_ ((1 / 4) + (1 / 4)) <-> (x + (1 / 4)) <_ (1 / 2))
135	129, 134	syl6rbb 596	. . . . . . . . . . . . . . . . . . 19 |- (x e. (0[,]1) -> ((x + (1 / 4)) <_ (1 / 2) <-> x <_ (1 / 4)))
136	135	ad2antrr 440	. . . . . . . . . . . . . . . . . 18 |- (((x e. (0[,]1) /\ x <_ (1 / 2)) /\ -. x <_ (1 / 4)) -> ((x + (1 / 4)) <_ (1 / 2) <-> x <_ (1 / 4)))
137	136	adantlll 432	. . . . . . . . . . . . . . . . 17 |- (((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) /\ -. x <_ (1 / 4)) -> ((x + (1 / 4)) <_ (1 / 2) <-> x <_ (1 / 4)))
138	137	ifbid 2996	. . . . . . . . . . . . . . . 16 |- (((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) /\ -. x <_ (1 / 4)) -> if((x + (1 / 4)) <_ (1 / 2), (F` (2 x. (x + (1 / 4)))), ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1))) = if(x <_ (1 / 4), (F` (2 x. (x + (1 / 4)))), ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1))))
139		iffalse 2991	. . . . . . . . . . . . . . . . 17 |- (-. x <_ (1 / 4) -> if(x <_ (1 / 4), (F` (2 x. (x + (1 / 4)))), ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1))) = ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1)))
140	139	adantl 424	. . . . . . . . . . . . . . . 16 |- (((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) /\ -. x <_ (1 / 4)) -> if(x <_ (1 / 4), (F` (2 x. (x + (1 / 4)))), ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1))) = ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1)))
141		pcoval1 16074	. . . . . . . . . . . . . . . . . . . 20 |- (((J e. Top /\ (G e. (II Cn J) /\ H e. (II Cn J) /\ (G` 1) = (H` 0))) /\ ((2 x. (x + (1 / 4))) - 1) e. (0[,](1 / 2))) -> ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1)) = (G` (2 x. ((2 x. (x + (1 / 4))) - 1))))
142		simp3 878	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) -> H e. (II Cn J))
143	142	3ad2ant2 898	. . . . . . . . . . . . . . . . . . . . . 22 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> H e. (II Cn J))
144		simp3r 905	. . . . . . . . . . . . . . . . . . . . . 22 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> (G` 1) = (H` 0))
145	5, 143, 144	3jca 1050	. . . . . . . . . . . . . . . . . . . . 21 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> (G e. (II Cn J) /\ H e. (II Cn J) /\ (G` 1) = (H` 0)))
146	1, 145	jca 310	. . . . . . . . . . . . . . . . . . . 20 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> (J e. Top /\ (G e. (II Cn J) /\ H e. (II Cn J) /\ (G` 1) = (H` 0))))
147		recn 6466	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (x e. RR -> x e. CC)
148		adddi 6462	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((2 e. CC /\ x e. CC /\ (1 / 4) e. CC) -> (2 x. (x + (1 / 4))) = ((2 x. x) + (2 x. (1 / 4))))
149	46, 79, 148	mp3an13 1182	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (x e. CC -> (2 x. (x + (1 / 4))) = ((2 x. x) + (2 x. (1 / 4))))
150	147, 149	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (x e. RR -> (2 x. (x + (1 / 4))) = ((2 x. x) + (2 x. (1 / 4))))
151	14	recni 6467	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (1 / 2) e. CC
152	151, 46, 79, 13	divmuli 6894	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (((1 / 2) / 2) = (1 / 4) <-> (2 x. (1 / 4)) = (1 / 2))
153	52, 152	mpbi 206	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (2 x. (1 / 4)) = (1 / 2)
154	153	a1i 8	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (x e. RR -> (2 x. (1 / 4)) = (1 / 2))
155	154	opreq2d 4898	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (x e. RR -> ((2 x. x) + (2 x. (1 / 4))) = ((2 x. x) + (1 / 2)))
156	150, 155	eqtrd 1925	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (x e. RR -> (2 x. (x + (1 / 4))) = ((2 x. x) + (1 / 2)))
157	156	opreq1d 4897	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (x e. RR -> ((2 x. (x + (1 / 4))) - 1) = (((2 x. x) + (1 / 2)) - 1))
158		mulcl 6456	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((2 e. CC /\ x e. CC) -> (2 x. x) e. CC)
159	46, 158	mpan 759	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (x e. CC -> (2 x. x) e. CC)
160		subsub3 6628	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (((2 x. x) e. CC /\ 1 e. CC /\ (1 / 2) e. CC) -> ((2 x. x) - (1 - (1 / 2))) = (((2 x. x) + (1 / 2)) - 1))
161	45, 151, 160	mp3an23 1183	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((2 x. x) e. CC -> ((2 x. x) - (1 - (1 / 2))) = (((2 x. x) + (1 / 2)) - 1))
162	159, 161	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (x e. CC -> ((2 x. x) - (1 - (1 / 2))) = (((2 x. x) + (1 / 2)) - 1))
163		2halves 7225	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (1 e. CC -> ((1 / 2) + (1 / 2)) = 1)
164	45, 163	ax-mp 7	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ((1 / 2) + (1 / 2)) = 1
165	45, 151, 151, 164	subaddrii 6529	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (1 - (1 / 2)) = (1 / 2)
166	165	a1i 8	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (x e. CC -> (1 - (1 / 2)) = (1 / 2))
167	166	opreq2d 4898	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (x e. CC -> ((2 x. x) - (1 - (1 / 2))) = ((2 x. x) - (1 / 2)))
168	162, 167	eqtr3d 1927	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (x e. CC -> (((2 x. x) + (1 / 2)) - 1) = ((2 x. x) - (1 / 2)))
169	147, 168	syl 12	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (x e. RR -> (((2 x. x) + (1 / 2)) - 1) = ((2 x. x) - (1 / 2)))
170	157, 169	eqtrd 1925	. . . . . . . . . . . . . . . . . . . . . . 23 |- (x e. RR -> ((2 x. (x + (1 / 4))) - 1) = ((2 x. x) - (1 / 2)))
171	40, 170	syl 12	. . . . . . . . . . . . . . . . . . . . . 22 |- (x e. (0[,]1) -> ((2 x. (x + (1 / 4))) - 1) = ((2 x. x) - (1 / 2)))
172	171	adantr 425	. . . . . . . . . . . . . . . . . . . . 21 |- ((x e. (0[,]1) /\ (x <_ (1 / 2) /\ -. x <_ (1 / 4))) -> ((2 x. (x + (1 / 4))) - 1) = ((2 x. x) - (1 / 2)))
173		ltnle 6680	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (((1 / 4) e. RR /\ x e. RR) -> ((1 / 4) < x <-> -. x <_ (1 / 4)))
174		ltle 6690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (((1 / 4) e. RR /\ x e. RR) -> ((1 / 4) < x -> (1 / 4) <_ x))
175	173, 174	sylbird 222	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (((1 / 4) e. RR /\ x e. RR) -> (-. x <_ (1 / 4) -> (1 / 4) <_ x))
176	78, 175	mpan 759	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (x e. RR -> (-. x <_ (1 / 4) -> (1 / 4) <_ x))
177	176	imp 377	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((x e. RR /\ -. x <_ (1 / 4)) -> (1 / 4) <_ x)
178	177	adantlr 429	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((x e. RR /\ x <_ (1 / 2)) /\ -. x <_ (1 / 4)) -> (1 / 4) <_ x)
179		elicc2 7560	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (((1 / 4) e. RR /\ (1 / 2) e. RR) -> (x e. ((1 / 4)[,](1 / 2)) <-> (x e. RR /\ (1 / 4) <_ x /\ x <_ (1 / 2))))
180	78, 14, 179	mp2an 761	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (x e. ((1 / 4)[,](1 / 2)) <-> (x e. RR /\ (1 / 4) <_ x /\ x <_ (1 / 2)))
181		mulcom 6459	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ((2 e. CC /\ x e. CC) -> (2 x. x) = (x x. 2))
182	181, 46, 147	sylancr 526	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (x e. RR -> (2 x. x) = (x x. 2))
183	182	3ad2ant1 897	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ((x e. RR /\ (1 / 4) <_ x /\ x <_ (1 / 2)) -> (2 x. x) = (x x. 2))
184	180, 183	sylbi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (x e. ((1 / 4)[,](1 / 2)) -> (2 x. x) = (x x. 2))
185	12, 41	elrpii 7234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- 2 e. RR+
186	79, 46	mulcomi 6476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ((1 / 4) x. 2) = (2 x. (1 / 4))
187	186, 153	eqtri 1908	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ((1 / 4) x. 2) = (1 / 2)
188	45, 46, 13	divcan1i 6906	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ((1 / 2) x. 2) = 1
189	78, 14, 185, 187, 188	iccdili 15862	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (x e. ((1 / 4)[,](1 / 2)) -> (x x. 2) e. ((1 / 2)[,]1))
190	184, 189	eqeltrd 1971	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (x e. ((1 / 4)[,](1 / 2)) -> (2 x. x) e. ((1 / 2)[,]1))
191	151	subidi 6551	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ((1 / 2) - (1 / 2)) = 0
192	14, 20, 14, 191, 165	iccshftli 15860	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ((2 x. x) e. ((1 / 2)[,]1) -> ((2 x. x) - (1 / 2)) e. (0[,](1 / 2)))
193	190, 192	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (x e. ((1 / 4)[,](1 / 2)) -> ((2 x. x) - (1 / 2)) e. (0[,](1 / 2)))
194	180, 193	sylbir 218	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((x e. RR /\ (1 / 4) <_ x /\ x <_ (1 / 2)) -> ((2 x. x) - (1 / 2)) e. (0[,](1 / 2)))
195	194	3expa 1067	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (((x e. RR /\ (1 / 4) <_ x) /\ x <_ (1 / 2)) -> ((2 x. x) - (1 / 2)) e. (0[,](1 / 2)))
196	195	an1rs 547	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((x e. RR /\ x <_ (1 / 2)) /\ (1 / 4) <_ x) -> ((2 x. x) - (1 / 2)) e. (0[,](1 / 2)))
197	178, 196	syldan 516	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((x e. RR /\ x <_ (1 / 2)) /\ -. x <_ (1 / 4)) -> ((2 x. x) - (1 / 2)) e. (0[,](1 / 2)))
198	197	anasss 488	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((x e. RR /\ (x <_ (1 / 2) /\ -. x <_ (1 / 4))) -> ((2 x. x) - (1 / 2)) e. (0[,](1 / 2)))
199	198	3ad2antl1 1038	. . . . . . . . . . . . . . . . . . . . . 22 |- (((x e. RR /\ 0 <_ x /\ x <_ 1) /\ (x <_ (1 / 2) /\ -. x <_ (1 / 4))) -> ((2 x. x) - (1 / 2)) e. (0[,](1 / 2)))
200	199, 22	sylanb 498	. . . . . . . . . . . . . . . . . . . . 21 |- ((x e. (0[,]1) /\ (x <_ (1 / 2) /\ -. x <_ (1 / 4))) -> ((2 x. x) - (1 / 2)) e. (0[,](1 / 2)))
201	172, 200	eqeltrd 1971	. . . . . . . . . . . . . . . . . . . 20 |- ((x e. (0[,]1) /\ (x <_ (1 / 2) /\ -. x <_ (1 / 4))) -> ((2 x. (x + (1 / 4))) - 1) e. (0[,](1 / 2)))
202	141, 146, 201	syl2an 503	. . . . . . . . . . . . . . . . . . 19 |- (((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ (x e. (0[,]1) /\ (x <_ (1 / 2) /\ -. x <_ (1 / 4)))) -> ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1)) = (G` (2 x. ((2 x. (x + (1 / 4))) - 1))))
203	202	anassrs 489	. . . . . . . . . . . . . . . . . 18 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ (x <_ (1 / 2) /\ -. x <_ (1 / 4))) -> ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1)) = (G` (2 x. ((2 x. (x + (1 / 4))) - 1))))
204	203	anassrs 489	. . . . . . . . . . . . . . . . 17 |- (((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) /\ -. x <_ (1 / 4)) -> ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1)) = (G` (2 x. ((2 x. (x + (1 / 4))) - 1))))
205	170	opreq2d 4898	. . . . . . . . . . . . . . . . . . . . . 22 |- (x e. RR -> (2 x. ((2 x. (x + (1 / 4))) - 1)) = (2 x. ((2 x. x) - (1 / 2))))
206		subdi 6590	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((2 e. CC /\ (2 x. x) e. CC /\ (1 / 2) e. CC) -> (2 x. ((2 x. x) - (1 / 2))) = ((2 x. (2 x. x)) - (2 x. (1 / 2))))
207	46, 151, 206	mp3an13 1182	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((2 x. x) e. CC -> (2 x. ((2 x. x) - (1 / 2))) = ((2 x. (2 x. x)) - (2 x. (1 / 2))))
208	147, 159, 207	3syl 24	. . . . . . . . . . . . . . . . . . . . . 22 |- (x e. RR -> (2 x. ((2 x. x) - (1 / 2))) = ((2 x. (2 x. x)) - (2 x. (1 / 2))))
209	46, 13	recidi 6916	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (2 x. (1 / 2)) = 1
210	209	a1i 8	. . . . . . . . . . . . . . . . . . . . . . 23 |- (x e. RR -> (2 x. (1 / 2)) = 1)
211	210	opreq2d 4898	. . . . . . . . . . . . . . . . . . . . . 22 |- (x e. RR -> ((2 x. (2 x. x)) - (2 x. (1 / 2))) = ((2 x. (2 x. x)) - 1))
212	205, 208, 211	3eqtrd 1929	. . . . . . . . . . . . . . . . . . . . 21 |- (x e. RR -> (2 x. ((2 x. (x + (1 / 4))) - 1)) = ((2 x. (2 x. x)) - 1))
213	40, 212	syl 12	. . . . . . . . . . . . . . . . . . . 20 |- (x e. (0[,]1) -> (2 x. ((2 x. (x + (1 / 4))) - 1)) = ((2 x. (2 x. x)) - 1))
214	213	ad2antrr 440	. . . . . . . . . . . . . . . . . . 19 |- (((x e. (0[,]1) /\ x <_ (1 / 2)) /\ -. x <_ (1 / 4)) -> (2 x. ((2 x. (x + (1 / 4))) - 1)) = ((2 x. (2 x. x)) - 1))
215	214	adantlll 432	. . . . . . . . . . . . . . . . . 18 |- (((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) /\ -. x <_ (1 / 4)) -> (2 x. ((2 x. (x + (1 / 4))) - 1)) = ((2 x. (2 x. x)) - 1))
216	215	fveq2d 4685	. . . . . . . . . . . . . . . . 17 |- (((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) /\ -. x <_ (1 / 4)) -> (G` (2 x. ((2 x. (x + (1 / 4))) - 1))) = (G` ((2 x. (2 x. x)) - 1)))
217	204, 216	eqtrd 1925	. . . . . . . . . . . . . . . 16 |- (((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) /\ -. x <_ (1 / 4)) -> ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1)) = (G` ((2 x. (2 x. x)) - 1)))
218	138, 140, 217	3eqtrd 1929	. . . . . . . . . . . . . . 15 |- (((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) /\ -. x <_ (1 / 4)) -> if((x + (1 / 4)) <_ (1 / 2), (F` (2 x. (x + (1 / 4)))), ((G(*p` J)H)` ((2 x. (x + (1 / 4))) - 1))) = (G` ((2 x. (2 x. x)) - 1)))
219	126, 218	eqtr2d 1926	. . . . . . . . . . . . . 14 |- (((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) /\ -. x <_ (1 / 4)) -> (G` ((2 x. (2 x. x)) - 1)) = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` (x + (1 / 4))))
220	219	ifeq2da 15694	. . . . . . . . . . . . 13 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) -> if(x <_ (1 / 4), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` (2 x. x)), (G` ((2 x. (2 x. x)) - 1))) = if(x <_ (1 / 4), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` (2 x. x)), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` (x + (1 / 4)))))
221	57, 74, 220	3eqtrd 1929	. . . . . . . . . . . 12 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) -> if((2 x. x) <_ (1 / 2), (F` (2 x. (2 x. x))), (G` ((2 x. (2 x. x)) - 1))) = if(x <_ (1 / 4), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` (2 x. x)), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` (x + (1 / 4)))))
222	10, 37, 221	3eqtrd 1929	. . . . . . . . . . 11 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) -> ((F(*p` J)G)` (2 x. x)) = if(x <_ (1 / 4), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` (2 x. x)), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` (x + (1 / 4)))))
223		fvif 15692	. . . . . . . . . . 11 |- ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` if(x <_ (1 / 4), (2 x. x), (x + (1 / 4)))) = if(x <_ (1 / 4), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` (2 x. x)), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` (x + (1 / 4))))
224	222, 223	syl6eqr 1946	. . . . . . . . . 10 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ x <_ (1 / 2)) -> ((F(*p` J)G)` (2 x. x)) = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` if(x <_ (1 / 4), (2 x. x), (x + (1 / 4)))))
225	224	ifeq1da 15693	. . . . . . . . 9 |- (((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) -> if(x <_ (1 / 2), ((F(*p` J)G)` (2 x. x)), (H` ((2 x. x) - 1))) = if(x <_ (1 / 2), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` if(x <_ (1 / 4), (2 x. x), (x + (1 / 4)))), (H` ((2 x. x) - 1))))
226		ltnle 6680	. . . . . . . . . . . . . . . . . . . 20 |- (((1 / 2) e. RR /\ x e. RR) -> ((1 / 2) < x <-> -. x <_ (1 / 2)))
227		ltle 6690	. . . . . . . . . . . . . . . . . . . 20 |- (((1 / 2) e. RR /\ x e. RR) -> ((1 / 2) < x -> (1 / 2) <_ x))
228	226, 227	sylbird 222	. . . . . . . . . . . . . . . . . . 19 |- (((1 / 2) e. RR /\ x e. RR) -> (-. x <_ (1 / 2) -> (1 / 2) <_ x))
229	14, 228	mpan 759	. . . . . . . . . . . . . . . . . 18 |- (x e. RR -> (-. x <_ (1 / 2) -> (1 / 2) <_ x))
230	229	imp 377	. . . . . . . . . . . . . . . . 17 |- ((x e. RR /\ -. x <_ (1 / 2)) -> (1 / 2) <_ x)
231	230	adantlr 429	. . . . . . . . . . . . . . . 16 |- (((x e. RR /\ x <_ 1) /\ -. x <_ (1 / 2)) -> (1 / 2) <_ x)
232		elicc2 7560	. . . . . . . . . . . . . . . . . . . 20 |- (((1 / 2) e. RR /\ 1 e. RR) -> (x e. ((1 / 2)[,]1) <-> (x e. RR /\ (1 / 2) <_ x /\ x <_ 1)))
233	14, 20, 232	mp2an 761	. . . . . . . . . . . . . . . . . . 19 |- (x e. ((1 / 2)[,]1) <-> (x e. RR /\ (1 / 2) <_ x /\ x <_ 1))
234		eqid 1884	. . . . . . . . . . . . . . . . . . . . 21 |- (1 / 2) = (1 / 2)
235	14, 20, 185, 52, 234	icccntri 15864	. . . . . . . . . . . . . . . . . . . 20 |- (x e. ((1 / 2)[,]1) -> (x / 2) e. ((1 / 4)[,](1 / 2)))
236		eqid 1884	. . . . . . . . . . . . . . . . . . . . 21 |- ((1 / 4) + (1 / 2)) = ((1 / 4) + (1 / 2))
237	78, 14, 14, 236, 164	iccshftri 15858	. . . . . . . . . . . . . . . . . . . 20 |- ((x / 2) e. ((1 / 4)[,](1 / 2)) -> ((x / 2) + (1 / 2)) e. (((1 / 4) + (1 / 2))[,]1))
238	78, 14	readdcli 6487	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((1 / 4) + (1 / 2)) e. RR
239	238, 20	pm3.2i 307	. . . . . . . . . . . . . . . . . . . . . 22 |- (((1 / 4) + (1 / 2)) e. RR /\ 1 e. RR)
240		halfgt0 7215	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- 0 < (1 / 2)
241	11, 14, 240	ltleii 6756	. . . . . . . . . . . . . . . . . . . . . . . 24 |- 0 <_ (1 / 2)
242	78, 14	addge0i 6777	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((0 <_ (1 / 4) /\ 0 <_ (1 / 2)) -> 0 <_ ((1 / 4) + (1 / 2)))
243	98, 241, 242	mp2an 761	. . . . . . . . . . . . . . . . . . . . . . 23 |- 0 <_ ((1 / 4) + (1 / 2))
244	20	leidi 6790	. . . . . . . . . . . . . . . . . . . . . . 23 |- 1 <_ 1
245	243, 244	pm3.2i 307	. . . . . . . . . . . . . . . . . . . . . 22 |- (0 <_ ((1 / 4) + (1 / 2)) /\ 1 <_ 1)
246		iccss 15855	. . . . . . . . . . . . . . . . . . . . . 22 |- (((0 e. RR /\ 1 e. RR) /\ (((1 / 4) + (1 / 2)) e. RR /\ 1 e. RR) /\ (0 <_ ((1 / 4) + (1 / 2)) /\ 1 <_ 1)) -> (((1 / 4) + (1 / 2))[,]1) C_ (0[,]1))
247	93, 239, 245, 246	mp3an 1191	. . . . . . . . . . . . . . . . . . . . 21 |- (((1 / 4) + (1 / 2))[,]1) C_ (0[,]1)
248	247	sseli 2617	. . . . . . . . . . . . . . . . . . . 20 |- (((x / 2) + (1 / 2)) e. (((1 / 4) + (1 / 2))[,]1) -> ((x / 2) + (1 / 2)) e. (0[,]1))
249	235, 237, 248	3syl 24	. . . . . . . . . . . . . . . . . . 19 |- (x e. ((1 / 2)[,]1) -> ((x / 2) + (1 / 2)) e. (0[,]1))
250	233, 249	sylbir 218	. . . . . . . . . . . . . . . . . 18 |- ((x e. RR /\ (1 / 2) <_ x /\ x <_ 1) -> ((x / 2) + (1 / 2)) e. (0[,]1))
251	250	3expa 1067	. . . . . . . . . . . . . . . . 17 |- (((x e. RR /\ (1 / 2) <_ x) /\ x <_ 1) -> ((x / 2) + (1 / 2)) e. (0[,]1))
252	251	an1rs 547	. . . . . . . . . . . . . . . 16 |- (((x e. RR /\ x <_ 1) /\ (1 / 2) <_ x) -> ((x / 2) + (1 / 2)) e. (0[,]1))
253	231, 252	syldan 516	. . . . . . . . . . . . . . 15 |- (((x e. RR /\ x <_ 1) /\ -. x <_ (1 / 2)) -> ((x / 2) + (1 / 2)) e. (0[,]1))
254	253	3adantl2 1033	. . . . . . . . . . . . . 14 |- (((x e. RR /\ 0 <_ x /\ x <_ 1) /\ -. x <_ (1 / 2)) -> ((x / 2) + (1 / 2)) e. (0[,]1))
255	254, 22	sylanb 498	. . . . . . . . . . . . 13 |- ((x e. (0[,]1) /\ -. x <_ (1 / 2)) -> ((x / 2) + (1 / 2)) e. (0[,]1))
256	255	adantll 428	. . . . . . . . . . . 12 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ -. x <_ (1 / 2)) -> ((x / 2) + (1 / 2)) e. (0[,]1))
257		breq1 3341	. . . . . . . . . . . . . 14 |- (v = ((x / 2) + (1 / 2)) -> (v <_ (1 / 2) <-> ((x / 2) + (1 / 2)) <_ (1 / 2)))
258		opreq2 4890	. . . . . . . . . . . . . . 15 |- (v = ((x / 2) + (1 / 2)) -> (2 x. v) = (2 x. ((x / 2) + (1 / 2))))
259	258	fveq2d 4685	. . . . . . . . . . . . . 14 |- (v = ((x / 2) + (1 / 2)) -> (F` (2 x. v)) = (F` (2 x. ((x / 2) + (1 / 2)))))
260	258	opreq1d 4897	. . . . . . . . . . . . . . 15 |- (v = ((x / 2) + (1 / 2)) -> ((2 x. v) - 1) = ((2 x. ((x / 2) + (1 / 2))) - 1))
261	260	fveq2d 4685	. . . . . . . . . . . . . 14 |- (v = ((x / 2) + (1 / 2)) -> ((G(*p` J)H)` ((2 x. v) - 1)) = ((G(*p` J)H)` ((2 x. ((x / 2) + (1 / 2))) - 1)))
262	257, 259, 261	ifbieq12d 2998	. . . . . . . . . . . . 13 |- (v = ((x / 2) + (1 / 2)) -> if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))) = if(((x / 2) + (1 / 2)) <_ (1 / 2), (F` (2 x. ((x / 2) + (1 / 2)))), ((G(*p` J)H)` ((2 x. ((x / 2) + (1 / 2))) - 1))))
263		fvex 4689	. . . . . . . . . . . . . 14 |- (F` (2 x. ((x / 2) + (1 / 2)))) e. _V
264		fvex 4689	. . . . . . . . . . . . . 14 |- ((G(*p` J)H)` ((2 x. ((x / 2) + (1 / 2))) - 1)) e. _V
265	263, 264	ifex 3031	. . . . . . . . . . . . 13 |- if(((x / 2) + (1 / 2)) <_ (1 / 2), (F` (2 x. ((x / 2) + (1 / 2)))), ((G(*p` J)H)` ((2 x. ((x / 2) + (1 / 2))) - 1))) e. _V
266	262, 62, 265	fvopab4 4743	. . . . . . . . . . . 12 |- (((x / 2) + (1 / 2)) e. (0[,]1) -> ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` ((x / 2) + (1 / 2))) = if(((x / 2) + (1 / 2)) <_ (1 / 2), (F` (2 x. ((x / 2) + (1 / 2)))), ((G(*p` J)H)` ((2 x. ((x / 2) + (1 / 2))) - 1))))
267	256, 266	syl 12	. . . . . . . . . . 11 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ -. x <_ (1 / 2)) -> ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` ((x / 2) + (1 / 2))) = if(((x / 2) + (1 / 2)) <_ (1 / 2), (F` (2 x. ((x / 2) + (1 / 2)))), ((G(*p` J)H)` ((2 x. ((x / 2) + (1 / 2))) - 1))))
268		lttr 6677	. . . . . . . . . . . . . . . . . . . 20 |- ((0 e. RR /\ (1 / 2) e. RR /\ x e. RR) -> ((0 < (1 / 2) /\ (1 / 2) < x) -> 0 < x))
269	11, 14, 268	mp3an12 1181	. . . . . . . . . . . . . . . . . . 19 |- (x e. RR -> ((0 < (1 / 2) /\ (1 / 2) < x) -> 0 < x))
270	240, 269	mpani 762	. . . . . . . . . . . . . . . . . 18 |- (x e. RR -> ((1 / 2) < x -> 0 < x))
271		halfpos2 7223	. . . . . . . . . . . . . . . . . . . 20 |- (x e. RR -> (0 < x <-> 0 < (x / 2)))
272	11	a1i 8	. . . . . . . . . . . . . . . . . . . . 21 |- (x e. RR -> 0 e. RR)
273		rehalfcl 7220	. . . . . . . . . . . . . . . . . . . . 21 |- (x e. RR -> (x / 2) e. RR)
274	14	a1i 8	. . . . . . . . . . . . . . . . . . . . 21 |- (x e. RR -> (1 / 2) e. RR)
275		ltadd1 6806	. . . . . . . . . . . . . . . . . . . . 21 |- ((0 e. RR /\ (x / 2) e. RR /\ (1 / 2) e. RR) -> (0 < (x / 2) <-> (0 + (1 / 2)) < ((x / 2) + (1 / 2))))
276	272, 273, 274, 275	syl111anc 1100	. . . . . . . . . . . . . . . . . . . 20 |- (x e. RR -> (0 < (x / 2) <-> (0 + (1 / 2)) < ((x / 2) + (1 / 2))))
277	271, 276	bitrd 587	. . . . . . . . . . . . . . . . . . 19 |- (x e. RR -> (0 < x <-> (0 + (1 / 2)) < ((x / 2) + (1 / 2))))
278	151	addid2i 6484	. . . . . . . . . . . . . . . . . . . 20 |- (0 + (1 / 2)) = (1 / 2)
279	278	breq1i 3345	. . . . . . . . . . . . . . . . . . 19 |- ((0 + (1 / 2)) < ((x / 2) + (1 / 2)) <-> (1 / 2) < ((x / 2) + (1 / 2)))
280	277, 279	syl6bb 595	. . . . . . . . . . . . . . . . . 18 |- (x e. RR -> (0 < x <-> (1 / 2) < ((x / 2) + (1 / 2))))
281	270, 280	sylibd 219	. . . . . . . . . . . . . . . . 17 |- (x e. RR -> ((1 / 2) < x -> (1 / 2) < ((x / 2) + (1 / 2))))
282	14, 226	mpan 759	. . . . . . . . . . . . . . . . 17 |- (x e. RR -> ((1 / 2) < x <-> -. x <_ (1 / 2)))
283		ltnle 6680	. . . . . . . . . . . . . . . . . 18 |- (((1 / 2) e. RR /\ ((x / 2) + (1 / 2)) e. RR) -> ((1 / 2) < ((x / 2) + (1 / 2)) <-> -. ((x / 2) + (1 / 2)) <_ (1 / 2)))
284		readdcl 6455	. . . . . . . . . . . . . . . . . . 19 |- (((x / 2) e. RR /\ (1 / 2) e. RR) -> ((x / 2) + (1 / 2)) e. RR)
285	284, 273, 14	sylancl 525	. . . . . . . . . . . . . . . . . 18 |- (x e. RR -> ((x / 2) + (1 / 2)) e. RR)
286	283, 14, 285	sylancr 526	. . . . . . . . . . . . . . . . 17 |- (x e. RR -> ((1 / 2) < ((x / 2) + (1 / 2)) <-> -. ((x / 2) + (1 / 2)) <_ (1 / 2)))
287	281, 282, 286	3imtr3d 601	. . . . . . . . . . . . . . . 16 |- (x e. RR -> (-. x <_ (1 / 2) -> -. ((x / 2) + (1 / 2)) <_ (1 / 2)))
288	40, 287	syl 12	. . . . . . . . . . . . . . 15 |- (x e. (0[,]1) -> (-. x <_ (1 / 2) -> -. ((x / 2) + (1 / 2)) <_ (1 / 2)))
289	288	imp 377	. . . . . . . . . . . . . 14 |- ((x e. (0[,]1) /\ -. x <_ (1 / 2)) -> -. ((x / 2) + (1 / 2)) <_ (1 / 2))
290	289	adantll 428	. . . . . . . . . . . . 13 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ -. x <_ (1 / 2)) -> -. ((x / 2) + (1 / 2)) <_ (1 / 2))
291		iffalse 2991	. . . . . . . . . . . . 13 |- (-. ((x / 2) + (1 / 2)) <_ (1 / 2) -> if(((x / 2) + (1 / 2)) <_ (1 / 2), (F` (2 x. ((x / 2) + (1 / 2)))), ((G(*p` J)H)` ((2 x. ((x / 2) + (1 / 2))) - 1))) = ((G(*p` J)H)` ((2 x. ((x / 2) + (1 / 2))) - 1)))
292	290, 291	syl 12	. . . . . . . . . . . 12 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ -. x <_ (1 / 2)) -> if(((x / 2) + (1 / 2)) <_ (1 / 2), (F` (2 x. ((x / 2) + (1 / 2)))), ((G(*p` J)H)` ((2 x. ((x / 2) + (1 / 2))) - 1))) = ((G(*p` J)H)` ((2 x. ((x / 2) + (1 / 2))) - 1)))
293		axresscn 6420	. . . . . . . . . . . . . . . . 17 |- RR C_ CC
294	39, 293	sstri 2626	. . . . . . . . . . . . . . . 16 |- (0[,]1) C_ CC
295	294	sseli 2617	. . . . . . . . . . . . . . 15 |- (x e. (0[,]1) -> x e. CC)
296		divdir 6933	. . . . . . . . . . . . . . . . . . . . 21 |- ((x e. CC /\ 1 e. CC /\ (2 e. CC /\ 2 =/= 0)) -> ((x + 1) / 2) = ((x / 2) + (1 / 2)))
297	45, 47, 296	mp3an23 1183	. . . . . . . . . . . . . . . . . . . 20 |- (x e. CC -> ((x + 1) / 2) = ((x / 2) + (1 / 2)))
298	297	eqcomd 1889	. . . . . . . . . . . . . . . . . . 19 |- (x e. CC -> ((x / 2) + (1 / 2)) = ((x + 1) / 2))
299	298	opreq2d 4898	. . . . . . . . . . . . . . . . . 18 |- (x e. CC -> (2 x. ((x / 2) + (1 / 2))) = (2 x. ((x + 1) / 2)))
300		peano2cn 6498	. . . . . . . . . . . . . . . . . . 19 |- (x e. CC -> (x + 1) e. CC)
301	46	a1i 8	. . . . . . . . . . . . . . . . . . 19 |- (x e. CC -> 2 e. CC)
302	13	a1i 8	. . . . . . . . . . . . . . . . . . 19 |- (x e. CC -> 2 =/= 0)
303		divcan2 6910	. . . . . . . . . . . . . . . . . . 19 |- (((x + 1) e. CC /\ 2 e. CC /\ 2 =/= 0) -> (2 x. ((x + 1) / 2)) = (x + 1))
304	300, 301, 302, 303	syl111anc 1100	. . . . . . . . . . . . . . . . . 18 |- (x e. CC -> (2 x. ((x + 1) / 2)) = (x + 1))
305	299, 304	eqtrd 1925	. . . . . . . . . . . . . . . . 17 |- (x e. CC -> (2 x. ((x / 2) + (1 / 2))) = (x + 1))
306	305	opreq1d 4897	. . . . . . . . . . . . . . . 16 |- (x e. CC -> ((2 x. ((x / 2) + (1 / 2))) - 1) = ((x + 1) - 1))
307		pncan 6557	. . . . . . . . . . . . . . . . 17 |- ((x e. CC /\ 1 e. CC) -> ((x + 1) - 1) = x)
308	45, 307	mpan2 760	. . . . . . . . . . . . . . . 16 |- (x e. CC -> ((x + 1) - 1) = x)
309	306, 308	eqtrd 1925	. . . . . . . . . . . . . . 15 |- (x e. CC -> ((2 x. ((x / 2) + (1 / 2))) - 1) = x)
310	295, 309	syl 12	. . . . . . . . . . . . . 14 |- (x e. (0[,]1) -> ((2 x. ((x / 2) + (1 / 2))) - 1) = x)
311	310	fveq2d 4685	. . . . . . . . . . . . 13 |- (x e. (0[,]1) -> ((G(*p` J)H)` ((2 x. ((x / 2) + (1 / 2))) - 1)) = ((G(*p` J)H)` x))
312	311	ad2antlr 441	. . . . . . . . . . . 12 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ -. x <_ (1 / 2)) -> ((G(*p` J)H)` ((2 x. ((x / 2) + (1 / 2))) - 1)) = ((G(*p` J)H)` x))
313		pcoval2 16075	. . . . . . . . . . . . . 14 |- (((J e. Top /\ (G e. (II Cn J) /\ H e. (II Cn J) /\ (G` 1) = (H` 0))) /\ x e. ((1 / 2)[,]1)) -> ((G(*p` J)H)` x) = (H` ((2 x. x) - 1)))
314	233	biimpri 169	. . . . . . . . . . . . . . . . . . 19 |- ((x e. RR /\ (1 / 2) <_ x /\ x <_ 1) -> x e. ((1 / 2)[,]1))
315	314	3expa 1067	. . . . . . . . . . . . . . . . . 18 |- (((x e. RR /\ (1 / 2) <_ x) /\ x <_ 1) -> x e. ((1 / 2)[,]1))
316	315	an1rs 547	. . . . . . . . . . . . . . . . 17 |- (((x e. RR /\ x <_ 1) /\ (1 / 2) <_ x) -> x e. ((1 / 2)[,]1))
317	231, 316	syldan 516	. . . . . . . . . . . . . . . 16 |- (((x e. RR /\ x <_ 1) /\ -. x <_ (1 / 2)) -> x e. ((1 / 2)[,]1))
318	317	3adantl2 1033	. . . . . . . . . . . . . . 15 |- (((x e. RR /\ 0 <_ x /\ x <_ 1) /\ -. x <_ (1 / 2)) -> x e. ((1 / 2)[,]1))
319	318, 22	sylanb 498	. . . . . . . . . . . . . 14 |- ((x e. (0[,]1) /\ -. x <_ (1 / 2)) -> x e. ((1 / 2)[,]1))
320	313, 146, 319	syl2an 503	. . . . . . . . . . . . 13 |- (((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ (x e. (0[,]1) /\ -. x <_ (1 / 2))) -> ((G(*p` J)H)` x) = (H` ((2 x. x) - 1)))
321	320	anassrs 489	. . . . . . . . . . . 12 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ -. x <_ (1 / 2)) -> ((G(*p` J)H)` x) = (H` ((2 x. x) - 1)))
322	292, 312, 321	3eqtrd 1929	. . . . . . . . . . 11 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ -. x <_ (1 / 2)) -> if(((x / 2) + (1 / 2)) <_ (1 / 2), (F` (2 x. ((x / 2) + (1 / 2)))), ((G(*p` J)H)` ((2 x. ((x / 2) + (1 / 2))) - 1))) = (H` ((2 x. x) - 1)))
323	267, 322	eqtr2d 1926	. . . . . . . . . 10 |- ((((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) /\ -. x <_ (1 / 2)) -> (H` ((2 x. x) - 1)) = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` ((x / 2) + (1 / 2))))
324	323	ifeq2da 15694	. . . . . . . . 9 |- (((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) -> if(x <_ (1 / 2), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` if(x <_ (1 / 4), (2 x. x), (x + (1 / 4)))), (H` ((2 x. x) - 1))) = if(x <_ (1 / 2), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` if(x <_ (1 / 4), (2 x. x), (x + (1 / 4)))), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` ((x / 2) + (1 / 2)))))
325	225, 324	eqtrd 1925	. . . . . . . 8 |- (((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) -> if(x <_ (1 / 2), ((F(*p` J)G)` (2 x. x)), (H` ((2 x. x) - 1))) = if(x <_ (1 / 2), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` if(x <_ (1 / 4), (2 x. x), (x + (1 / 4)))), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` ((x / 2) + (1 / 2)))))
326		fvif 15692	. . . . . . . 8 |- ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2)))) = if(x <_ (1 / 2), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` if(x <_ (1 / 4), (2 x. x), (x + (1 / 4)))), ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` ((x / 2) + (1 / 2))))
327	325, 326	syl6eqr 1946	. . . . . . 7 |- (((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) -> if(x <_ (1 / 2), ((F(*p` J)G)` (2 x. x)), (H` ((2 x. x) - 1))) = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2)))))
328	327	eqeq2d 1895	. . . . . 6 |- (((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) /\ x e. (0[,]1)) -> (y = if(x <_ (1 / 2), ((F(*p` J)G)` (2 x. x)), (H` ((2 x. x) - 1))) <-> y = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))))
329	328	pm5.32da 711	. . . . 5 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> ((x e. (0[,]1) /\ y = if(x <_ (1 / 2), ((F(*p` J)G)` (2 x. x)), (H` ((2 x. x) - 1)))) <-> (x e. (0[,]1) /\ y = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2)))))))
330	329	opabbidv 3401	. . . 4 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), ((F(*p` J)G)` (2 x. x)), (H` ((2 x. x) - 1))))} = {<.x, y>. | (x e. (0[,]1) /\ y = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2)))))})
331		fvex 4689	. . . . . . 7 |- (F` (2 x. v)) e. _V
332		fvex 4689	. . . . . . 7 |- ((G(*p` J)H)` ((2 x. v) - 1)) e. _V
333	331, 332	ifex 3031	. . . . . 6 |- if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))) e. _V
334	333, 62	fnopab2 4549	. . . . 5 |- {<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))} Fn (0[,]1)
335	92, 112	syl 12	. . . . . . . . . . . 12 |- (x e. (0[,](1 / 2)) -> (x + (1 / 4)) e. (0[,]1))
336		ifcl 3007	. . . . . . . . . . . 12 |- (((2 x. x) e. (0[,]1) /\ (x + (1 / 4)) e. (0[,]1)) -> if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))) e. (0[,]1))
337	24, 335, 336	syl11anc 524	. . . . . . . . . . 11 |- (x e. (0[,](1 / 2)) -> if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))) e. (0[,]1))
338	16, 337	sylbir 218	. . . . . . . . . 10 |- ((x e. RR /\ 0 <_ x /\ x <_ (1 / 2)) -> if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))) e. (0[,]1))
339	338	3expa 1067	. . . . . . . . 9 |- (((x e. RR /\ 0 <_ x) /\ x <_ (1 / 2)) -> if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))) e. (0[,]1))
340	339	3adantl3 1034	. . . . . . . 8 |- (((x e. RR /\ 0 <_ x /\ x <_ 1) /\ x <_ (1 / 2)) -> if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))) e. (0[,]1))
341	340, 22	sylanb 498	. . . . . . 7 |- ((x e. (0[,]1) /\ x <_ (1 / 2)) -> if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))) e. (0[,]1))
342	341, 255	ifclda 15695	. . . . . 6 |- (x e. (0[,]1) -> if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))) e. (0[,]1))
343		eqid 1884	. . . . . 6 |- {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))} = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))}
344		eqid 1884	. . . . . 6 |- {<.x, y>. | (x e. (0[,]1) /\ y = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2)))))} = {<.x, y>. | (x e. (0[,]1) /\ y = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2)))))}
345	342, 343, 344	fnopabco 15711	. . . . 5 |- ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))} Fn (0[,]1) -> {<.x, y>. | (x e. (0[,]1) /\ y = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2)))))} = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))} o. {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))}))
346	334, 345	ax-mp 7	. . . 4 |- {<.x, y>. | (x e. (0[,]1) /\ y = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))}` if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2)))))} = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))} o. {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))})
347	330, 346	syl6eq 1944	. . 3 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), ((F(*p` J)G)` (2 x. x)), (H` ((2 x. x) - 1))))} = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))} o. {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))}))
348		pcocn 16076	. . . . 5 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ (F` 1) = (G` 0))) -> (F(*p` J)G) e. (II Cn J))
349	1, 3, 5, 6, 348	syl13anc 1102	. . . 4 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> (F(*p` J)G) e. (II Cn J))
350		pco1 16078	. . . . . 6 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ (F` 1) = (G` 0))) -> ((F(*p` J)G)` 1) = (G` 1))
351	1, 3, 5, 6, 350	syl13anc 1102	. . . . 5 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> ((F(*p` J)G)` 1) = (G` 1))
352	351, 144	eqtrd 1925	. . . 4 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> ((F(*p` J)G)` 1) = (H` 0))
353		pcoval 16073	. . . 4 |- ((J e. Top /\ ((F(*p` J)G) e. (II Cn J) /\ H e. (II Cn J) /\ ((F(*p` J)G)` 1) = (H` 0))) -> ((F(*p` J)G)(*p` J)H) = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), ((F(*p` J)G)` (2 x. x)), (H` ((2 x. x) - 1))))})
354	1, 349, 143, 352, 353	syl13anc 1102	. . 3 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> ((F(*p` J)G)(*p` J)H) = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), ((F(*p` J)G)` (2 x. x)), (H` ((2 x. x) - 1))))})
355		pcocn 16076	. . . . . 6 |- ((J e. Top /\ (G e. (II Cn J) /\ H e. (II Cn J) /\ (G` 1) = (H` 0))) -> (G(*p` J)H) e. (II Cn J))
356	1, 5, 143, 144, 355	syl13anc 1102	. . . . 5 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> (G(*p` J)H) e. (II Cn J))
357		pco0 16077	. . . . . . 7 |- ((J e. Top /\ (G e. (II Cn J) /\ H e. (II Cn J) /\ (G` 1) = (H` 0))) -> ((G(*p` J)H)` 0) = (G` 0))
358	1, 5, 143, 144, 357	syl13anc 1102	. . . . . 6 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> ((G(*p` J)H)` 0) = (G` 0))
359	6, 358	eqtr4d 1928	. . . . 5 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> (F` 1) = ((G(*p` J)H)` 0))
360		pcoval 16073	. . . . 5 |- ((J e. Top /\ (F e. (II Cn J) /\ (G(*p` J)H) e. (II Cn J) /\ (F` 1) = ((G(*p` J)H)` 0))) -> (F(*p` J)(G(*p` J)H)) = {<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))})
361	1, 3, 356, 359, 360	syl13anc 1102	. . . 4 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> (F(*p` J)(G(*p` J)H)) = {<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))})
362	361	coeq1d 4127	. . 3 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> ((F(*p` J)(G(*p` J)H)) o. {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))}) = ({<.v, w>. | (v e. (0[,]1) /\ w = if(v <_ (1 / 2), (F` (2 x. v)), ((G(*p` J)H)` ((2 x. v) - 1))))} o. {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))}))
363	347, 354, 362	3eqtr4d 1937	. 2 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> ((F(*p` J)G)(*p` J)H) = ((F(*p` J)(G(*p` J)H)) o. {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))}))
364		reparpht 16065	. . 3 |- (((J e. Top /\ (F(*p` J)(G(*p` J)H)) e. (II Cn J)) /\ ({<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))} e. (II Cn II) /\ ({<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))}` 0) = 0 /\ ({<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))}` 1) = 1)) -> ((F(*p` J)(G(*p` J)H)) o. {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))})(~=ph` J)(F(*p` J)(G(*p` J)H)))
365		pcocn 16076	. . . . 5 |- ((J e. Top /\ (F e. (II Cn J) /\ (G(*p` J)H) e. (II Cn J) /\ (F` 1) = ((G(*p` J)H)` 0))) -> (F(*p` J)(G(*p` J)H)) e. (II Cn J))
366	1, 3, 356, 359, 365	syl13anc 1102	. . . 4 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> (F(*p` J)(G(*p` J)H)) e. (II Cn J))
367	1, 366	jca 310	. . 3 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> (J e. Top /\ (F(*p` J)(G(*p` J)H)) e. (II Cn J)))
368		elicc2 7560	. . . . . . 7 |- ((0 e. RR /\ 1 e. RR) -> ((1 / 2) e. (0[,]1) <-> ((1 / 2) e. RR /\ 0 <_ (1 / 2) /\ (1 / 2) <_ 1)))
369	11, 20, 368	mp2an 761	. . . . . 6 |- ((1 / 2) e. (0[,]1) <-> ((1 / 2) e. RR /\ 0 <_ (1 / 2) /\ (1 / 2) <_ 1))
370		halflt1 7216	. . . . . . 7 |- (1 / 2) < 1
371	14, 20, 370	ltleii 6756	. . . . . 6 |- (1 / 2) <_ 1
372	369, 14, 241, 371	mpbir3an 1052	. . . . 5 |- (1 / 2) e. (0[,]1)
373		oprex 4907	. . . . . 6 |- (2 x. x) e. _V
374		oprex 4907	. . . . . 6 |- (x + (1 / 4)) e. _V
375	373, 374	ifex 3031	. . . . 5 |- if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))) e. _V
376		oprex 4907	. . . . 5 |- ((x / 2) + (1 / 2)) e. _V
377		dfii2 15869	. . . . 5 |- II = (subSp` <.(0[,]1), (topGen` ran (,))>.)
378		eqid 1884	. . . . 5 |- (subSp` <.(0[,](1 / 2)), (topGen` ran (,))>.) = (subSp` <.(0[,](1 / 2)), (topGen` ran (,))>.)
379		eqid 1884	. . . . 5 |- (subSp` <.((1 / 2)[,]1), (topGen` ran (,))>.) = (subSp` <.((1 / 2)[,]1), (topGen` ran (,))>.)
380	151, 79	addcomi 6475	. . . . . . 7 |- ((1 / 2) + (1 / 4)) = ((1 / 4) + (1 / 2))
381	52	opreq1i 4892	. . . . . . 7 |- (((1 / 2) / 2) + (1 / 2)) = ((1 / 4) + (1 / 2))
382	380, 381	eqtr4i 1911	. . . . . 6 |- ((1 / 2) + (1 / 4)) = (((1 / 2) / 2) + (1 / 2))
383		breq1 3341	. . . . . . . 8 |- (x = (1 / 2) -> (x <_ (1 / 4) <-> (1 / 2) <_ (1 / 4)))
384		opreq2 4890	. . . . . . . 8 |- (x = (1 / 2) -> (2 x. x) = (2 x. (1 / 2)))
385		opreq1 4889	. . . . . . . 8 |- (x = (1 / 2) -> (x + (1 / 4)) = ((1 / 2) + (1 / 4)))
386	383, 384, 385	ifbieq12d 2998	. . . . . . 7 |- (x = (1 / 2) -> if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))) = if((1 / 2) <_ (1 / 4), (2 x. (1 / 2)), ((1 / 2) + (1 / 4))))
387	12, 41, 12, 41, 12, 41	sqrlem15 7937	. . . . . . . . . . 11 |- 2 < (2 + 2)
388		2p2e4 7185	. . . . . . . . . . 11 |- (2 + 2) = 4
389	387, 388	breqtri 3360	. . . . . . . . . 10 |- 2 < 4
390	12, 75, 41, 76	ltrecii 7061	. . . . . . . . . 10 |- (2 < 4 <-> (1 / 4) < (1 / 2))
391	389, 390	mpbi 206	. . . . . . . . 9 |- (1 / 4) < (1 / 2)
392	78, 14	ltnlei 6754	. . . . . . . . 9 |- ((1 / 4) < (1 / 2) <-> -. (1 / 2) <_ (1 / 4))
393	391, 392	mpbi 206	. . . . . . . 8 |- -. (1 / 2) <_ (1 / 4)
394		iffalse 2991	. . . . . . . 8 |- (-. (1 / 2) <_ (1 / 4) -> if((1 / 2) <_ (1 / 4), (2 x. (1 / 2)), ((1 / 2) + (1 / 4))) = ((1 / 2) + (1 / 4)))
395	393, 394	ax-mp 7	. . . . . . 7 |- if((1 / 2) <_ (1 / 4), (2 x. (1 / 2)), ((1 / 2) + (1 / 4))) = ((1 / 2) + (1 / 4))
396	386, 395	syl6eq 1944	. . . . . 6 |- (x = (1 / 2) -> if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))) = ((1 / 2) + (1 / 4)))
397		opreq1 4889	. . . . . . 7 |- (x = (1 / 2) -> (x / 2) = ((1 / 2) / 2))
398	397	opreq1d 4897	. . . . . 6 |- (x = (1 / 2) -> ((x / 2) + (1 / 2)) = (((1 / 2) / 2) + (1 / 2)))
399	382, 396, 398	3eqtr4a 1954	. . . . 5 |- (x = (1 / 2) -> if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))) = ((x / 2) + (1 / 2)))
400		eqid 1884	. . . . 5 |- {<.x, y>. | (x e. (0[,](1 / 2)) /\ y = if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))))} = {<.x, y>. | (x e. (0[,](1 / 2)) /\ y = if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))))}
401		eqid 1884	. . . . 5 |- {<.x, y>. | (x e. ((1 / 2)[,]1) /\ y = ((x / 2) + (1 / 2)))} = {<.x, y>. | (x e. ((1 / 2)[,]1) /\ y = ((x / 2) + (1 / 2)))}
402		iitop 15867	. . . . 5 |- II e. Top
403		elicc2 7560	. . . . . . . 8 |- ((0 e. RR /\ (1 / 2) e. RR) -> ((1 / 4) e. (0[,](1 / 2)) <-> ((1 / 4) e. RR /\ 0 <_ (1 / 4) /\ (1 / 4) <_ (1 / 2))))
404	11, 14, 403	mp2an 761	. . . . . . 7 |- ((1 / 4) e. (0[,](1 / 2)) <-> ((1 / 4) e. RR /\ 0 <_ (1 / 4) /\ (1 / 4) <_ (1 / 2)))
405	78, 14, 391	ltleii 6756	. . . . . . 7 |- (1 / 4) <_ (1 / 2)
406	404, 78, 98, 405	mpbir3an 1052	. . . . . 6 |- (1 / 4) e. (0[,](1 / 2))
407		eqid 1884	. . . . . 6 |- (subSp` <.(0[,](1 / 4)), (topGen` ran (,))>.) = (subSp` <.(0[,](1 / 4)), (topGen` ran (,))>.)
408		eqid 1884	. . . . . 6 |- (subSp` <.((1 / 4)[,](1 / 2)), (topGen` ran (,))>.) = (subSp` <.((1 / 4)[,](1 / 2)), (topGen` ran (,))>.)
409	79	2timesi 7187	. . . . . . 7 |- (2 x. (1 / 4)) = ((1 / 4) + (1 / 4))
410		opreq2 4890	. . . . . . 7 |- (x = (1 / 4) -> (2 x. x) = (2 x. (1 / 4)))
411		opreq1 4889	. . . . . . 7 |- (x = (1 / 4) -> (x + (1 / 4)) = ((1 / 4) + (1 / 4)))
412	409, 410, 411	3eqtr4a 1954	. . . . . 6 |- (x = (1 / 4) -> (2 x. x) = (x + (1 / 4)))
413		eqid 1884	. . . . . 6 |- {<.x, y>. | (x e. (0[,](1 / 4)) /\ y = (2 x. x))} = {<.x, y>. | (x e. (0[,](1 / 4)) /\ y = (2 x. x))}
414		eqid 1884	. . . . . 6 |- {<.x, y>. | (x e. ((1 / 4)[,](1 / 2)) /\ y = (x + (1 / 4)))} = {<.x, y>. | (x e. ((1 / 4)[,](1 / 2)) /\ y = (x + (1 / 4)))}
415		iccssre 7565	. . . . . . . . 9 |- ((0 e. RR /\ (1 / 4) e. RR) -> (0[,](1 / 4)) C_ RR)
416	11, 78, 415	mp2an 761	. . . . . . . 8 |- (0[,](1 / 4)) C_ RR
417	416, 293	sstri 2626	. . . . . . 7 |- (0[,](1 / 4)) C_ CC
418		eqid 1884	. . . . . . 7 |- {<.x, y>. | (x e. CC /\ y = (2 x. x))} = {<.x, y>. | (x e. CC /\ y = (2 x. x))}
419	417	sseli 2617	. . . . . . . . . 10 |- (x e. (0[,](1 / 4)) -> x e. CC)
420	181, 46, 419	sylancr 526	. . . . . . . . 9 |- (x e. (0[,](1 / 4)) -> (2 x. x) = (x x. 2))
421	46	mul02i 6595	. . . . . . . . . 10 |- (0 x. 2) = 0
422	11, 78, 185, 421, 187	iccdili 15862	. . . . . . . . 9 |- (x e. (0[,](1 / 4)) -> (x x. 2) e. (0[,](1 / 2)))
423	420, 422	eqeltrd 1971	. . . . . . . 8 |- (x e. (0[,](1 / 4)) -> (2 x. x) e. (0[,](1 / 2)))
424	11, 14	pm3.2i 307	. . . . . . . . . 10 |- (0 e. RR /\ (1 / 2) e. RR)
425	11	leidi 6790	. . . . . . . . . . 11 |- 0 <_ 0
426	425, 371	pm3.2i 307	. . . . . . . . . 10 |- (0 <_ 0 /\ (1 / 2) <_ 1)
427		iccss 15855	. . . . . . . . . 10 |- (((0 e. RR /\ 1 e. RR) /\ (0 e. RR /\ (1 / 2) e. RR) /\ (0 <_ 0 /\ (1 / 2) <_ 1)) -> (0[,](1 / 2)) C_ (0[,]1))
428	93, 424, 426, 427	mp3an 1191	. . . . . . . . 9 |- (0[,](1 / 2)) C_ (0[,]1)
429	428	sseli 2617	. . . . . . . 8 |- ((2 x. x) e. (0[,](1 / 2)) -> (2 x. x) e. (0[,]1))
430	423, 429	syl 12	. . . . . . 7 |- (x e. (0[,](1 / 4)) -> (2 x. x) e. (0[,]1))
431	418	mulc1cncf 8541	. . . . . . . 8 |- (2 e. CC -> {<.x, y>. | (x e. CC /\ y = (2 x. x))} e. (CC-cn->CC))
432	46, 431	ax-mp 7	. . . . . . 7 |- {<.x, y>. | (x e. CC /\ y = (2 x. x))} e. (CC-cn->CC)
433		stioo 15845	. . . . . . . 8 |- ((0[,](1 / 4)) C_ RR -> (subSp` <.(0[,](1 / 4)), (topGen` ran (,))>.) = (Open` ((abs o. - ) |` ((0[,](1 / 4)) X. (0[,](1 / 4))))))
434	416, 433	ax-mp 7	. . . . . . 7 |- (subSp` <.(0[,](1 / 4)), (topGen` ran (,))>.) = (Open` ((abs o. - ) |` ((0[,](1 / 4)) X. (0[,](1 / 4)))))
435		df-ii 15866	. . . . . . 7 |- II = (Open` ((abs o. - ) |` ((0[,]1) X. (0[,]1))))
436	417, 294, 418, 413, 430, 432, 434, 435	cncfres 15895	. . . . . 6 |- {<.x, y>. | (x e. (0[,](1 / 4)) /\ y = (2 x. x))} e. ((subSp` <.(0[,](1 / 4)), (topGen` ran (,))>.) Cn II)
437		iccssre 7565	. . . . . . . . 9 |- (((1 / 4) e. RR /\ (1 / 2) e. RR) -> ((1 / 4)[,](1 / 2)) C_ RR)
438	78, 14, 437	mp2an 761	. . . . . . . 8 |- ((1 / 4)[,](1 / 2)) C_ RR
439	438, 293	sstri 2626	. . . . . . 7 |- ((1 / 4)[,](1 / 2)) C_ CC
440		eqid 1884	. . . . . . 7 |- {<.x, y>. | (x e. CC /\ y = (x + (1 / 4)))} = {<.x, y>. | (x e. CC /\ y = (x + (1 / 4)))}
441		eqid 1884	. . . . . . . . 9 |- ((1 / 4) + (1 / 4)) = ((1 / 4) + (1 / 4))
442	78, 14, 78, 441, 91	iccshftri 15858	. . . . . . . 8 |- (x e. ((1 / 4)[,](1 / 2)) -> (x + (1 / 4)) e. (((1 / 4) + (1 / 4))[,](3 / 4)))
443	78, 78	readdcli 6487	. . . . . . . . . . 11 |- ((1 / 4) + (1 / 4)) e. RR
444	443, 95	pm3.2i 307	. . . . . . . . . 10 |- (((1 / 4) + (1 / 4)) e. RR /\ (3 / 4) e. RR)
445	78, 78	addge0i 6777	. . . . . . . . . . . 12 |- ((0 <_ (1 / 4) /\ 0 <_ (1 / 4)) -> 0 <_ ((1 / 4) + (1 / 4)))
446	98, 98, 445	mp2an 761	. . . . . . . . . . 11 |- 0 <_ ((1 / 4) + (1 / 4))
447	446, 108	pm3.2i 307	. . . . . . . . . 10 |- (0 <_ ((1 / 4) + (1 / 4)) /\ (3 / 4) <_ 1)
448		iccss 15855	. . . . . . . . . 10 |- (((0 e. RR /\ 1 e. RR) /\ (((1 / 4) + (1 / 4)) e. RR /\ (3 / 4) e. RR) /\ (0 <_ ((1 / 4) + (1 / 4)) /\ (3 / 4) <_ 1)) -> (((1 / 4) + (1 / 4))[,](3 / 4)) C_ (0[,]1))
449	93, 444, 447, 448	mp3an 1191	. . . . . . . . 9 |- (((1 / 4) + (1 / 4))[,](3 / 4)) C_ (0[,]1)
450	449	sseli 2617	. . . . . . . 8 |- ((x + (1 / 4)) e. (((1 / 4) + (1 / 4))[,](3 / 4)) -> (x + (1 / 4)) e. (0[,]1))
451	442, 450	syl 12	. . . . . . 7 |- (x e. ((1 / 4)[,](1 / 2)) -> (x + (1 / 4)) e. (0[,]1))
452	440	addccncf 15883	. . . . . . . 8 |- ((1 / 4) e. CC -> {<.x, y>. | (x e. CC /\ y = (x + (1 / 4)))} e. (CC-cn->CC))
453	79, 452	ax-mp 7	. . . . . . 7 |- {<.x, y>. | (x e. CC /\ y = (x + (1 / 4)))} e. (CC-cn->CC)
454		stioo 15845	. . . . . . . 8 |- (((1 / 4)[,](1 / 2)) C_ RR -> (subSp` <.((1 / 4)[,](1 / 2)), (topGen` ran (,))>.) = (Open` ((abs o. - ) |` (((1 / 4)[,](1 / 2)) X. ((1 / 4)[,](1 / 2))))))
455	438, 454	ax-mp 7	. . . . . . 7 |- (subSp` <.((1 / 4)[,](1 / 2)), (topGen` ran (,))>.) = (Open` ((abs o. - ) |` (((1 / 4)[,](1 / 2)) X. ((1 / 4)[,](1 / 2)))))
456	439, 294, 440, 414, 451, 453, 455, 435	cncfres 15895	. . . . . 6 |- {<.x, y>. | (x e. ((1 / 4)[,](1 / 2)) /\ y = (x + (1 / 4)))} e. ((subSp` <.((1 / 4)[,](1 / 2)), (topGen` ran (,))>.) Cn II)
457	11, 14, 406, 373, 374, 400, 378, 407, 408, 412, 413, 414, 402, 436, 456	piececn 15894	. . . . 5 |- {<.x, y>. | (x e. (0[,](1 / 2)) /\ y = if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))))} e. ((subSp` <.(0[,](1 / 2)), (topGen` ran (,))>.) Cn II)
458		iccssre 7565	. . . . . . . 8 |- (((1 / 2) e. RR /\ 1 e. RR) -> ((1 / 2)[,]1) C_ RR)
459	14, 20, 458	mp2an 761	. . . . . . 7 |- ((1 / 2)[,]1) C_ RR
460	459, 293	sstri 2626	. . . . . 6 |- ((1 / 2)[,]1) C_ CC
461		eqid 1884	. . . . . 6 |- {<.x, y>. | (x e. CC /\ y = ((x / 2) + (1 / 2)))} = {<.x, y>. | (x e. CC /\ y = ((x / 2) + (1 / 2)))}
462		eqid 1884	. . . . . . . . . 10 |- {<.x, v>. | (x e. CC /\ v = (x / 2))} = {<.x, v>. | (x e. CC /\ v = (x / 2))}
463		halfcl 7219	. . . . . . . . . 10 |- (x e. CC -> (x / 2) e. CC)
464	462, 463	fopab 4800	. . . . . . . . 9 |- {<.x, v>. | (x e. CC /\ v = (x / 2))}:CC-->CC
465		frn 4569	. . . . . . . . 9 |- ({<.x, v>. | (x e. CC /\ v = (x / 2))}:CC-->CC -> ran {<.x, v>. | (x e. CC /\ v = (x / 2))} C_ CC)
466	464, 465	ax-mp 7	. . . . . . . 8 |- ran {<.x, v>. | (x e. CC /\ v = (x / 2))} C_ CC
467		oprex 4907	. . . . . . . . 9 |- (x / 2) e. _V
468		oprex 4907	. . . . . . . . 9 |- (w + (1 / 2)) e. _V
469		opreq1 4889	. . . . . . . . 9 |- (w = (x / 2) -> (w + (1 / 2)) = ((x / 2) + (1 / 2)))
470		eqid 1884	. . . . . . . . 9 |- {<.w, z>. | (w e. CC /\ z = (w + (1 / 2)))} = {<.w, z>. | (w e. CC /\ z = (w + (1 / 2)))}
471	467, 468, 376, 469, 462, 470, 461	fopabco 4805	. . . . . . . 8 |- (ran {<.x, v>. | (x e. CC /\ v = (x / 2))} C_ CC -> ({<.w, z>. | (w e. CC /\ z = (w + (1 / 2)))} o. {<.x, v>. | (x e. CC /\ v = (x / 2))}) = {<.x, y>. | (x e. CC /\ y = ((x / 2) + (1 / 2)))})
472	466, 471	ax-mp 7	. . . . . . 7 |- ({<.w, z>. | (w e. CC /\ z = (w + (1 / 2)))} o. {<.x, v>. | (x e. CC /\ v = (x / 2))}) = {<.x, y>. | (x e. CC /\ y = ((x / 2) + (1 / 2)))}
473		ssid 2634	. . . . . . . . 9 |- CC C_ CC
474	473, 473, 473	3pm3.2i 1048	. . . . . . . 8 |- (CC C_ CC /\ CC C_ CC /\ CC C_ CC)
475	462	divccncf 8542	. . . . . . . . . 10 |- ((2 e. CC /\ 2 =/= 0) -> {<.x, v>. | (x e. CC /\ v = (x / 2))} e. (CC-cn->CC))
476	46, 13, 475	mp2an 761	. . . . . . . . 9 |- {<.x, v>. | (x e. CC /\ v = (x / 2))} e. (CC-cn->CC)
477	470	addccncf 15883	. . . . . . . . . 10 |- ((1 / 2) e. CC -> {<.w, z>. | (w e. CC /\ z = (w + (1 / 2)))} e. (CC-cn->CC))
478	151, 477	ax-mp 7	. . . . . . . . 9 |- {<.w, z>. | (w e. CC /\ z = (w + (1 / 2)))} e. (CC-cn->CC)
479	476, 478	pm3.2i 307	. . . . . . . 8 |- ({<.x, v>. | (x e. CC /\ v = (x / 2))} e. (CC-cn->CC) /\ {<.w, z>. | (w e. CC /\ z = (w + (1 / 2)))} e. (CC-cn->CC))
480		cncfco 15887	. . . . . . . 8 |- (((CC C_ CC /\ CC C_ CC /\ CC C_ CC) /\ ({<.x, v>. | (x e. CC /\ v = (x / 2))} e. (CC-cn->CC) /\ {<.w, z>. | (w e. CC /\ z = (w + (1 / 2)))} e. (CC-cn->CC))) -> ({<.w, z>. | (w e. CC /\ z = (w + (1 / 2)))} o. {<.x, v>. | (x e. CC /\ v = (x / 2))}) e. (CC-cn->CC))
481	474, 479, 480	mp2an 761	. . . . . . 7 |- ({<.w, z>. | (w e. CC /\ z = (w + (1 / 2)))} o. {<.x, v>. | (x e. CC /\ v = (x / 2))}) e. (CC-cn->CC)
482	472, 481	eqeltrri 1968	. . . . . 6 |- {<.x, y>. | (x e. CC /\ y = ((x / 2) + (1 / 2)))} e. (CC-cn->CC)
483		stioo 15845	. . . . . . 7 |- (((1 / 2)[,]1) C_ RR -> (subSp` <.((1 / 2)[,]1), (topGen` ran (,))>.) = (Open` ((abs o. - ) |` (((1 / 2)[,]1) X. ((1 / 2)[,]1)))))
484	459, 483	ax-mp 7	. . . . . 6 |- (subSp` <.((1 / 2)[,]1), (topGen` ran (,))>.) = (Open` ((abs o. - ) |` (((1 / 2)[,]1) X. ((1 / 2)[,]1))))
485	460, 294, 461, 401, 249, 482, 484, 435	cncfres 15895	. . . . 5 |- {<.x, y>. | (x e. ((1 / 2)[,]1) /\ y = ((x / 2) + (1 / 2)))} e. ((subSp` <.((1 / 2)[,]1), (topGen` ran (,))>.) Cn II)
486	11, 20, 372, 375, 376, 343, 377, 378, 379, 399, 400, 401, 402, 457, 485	piececn 15894	. . . 4 |- {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))} e. (II Cn II)
487		lt01 6871	. . . . . . . 8 |- 0 < 1
488	11, 20, 487	ltleii 6756	. . . . . . 7 |- 0 <_ 1
489		lbicc2 7573	. . . . . . 7 |- ((0 e. RR /\ 1 e. RR /\ 0 <_ 1) -> 0 e. (0[,]1))
490	11, 20, 488, 489	mp3an 1191	. . . . . 6 |- 0 e. (0[,]1)
491		breq1 3341	. . . . . . . 8 |- (x = 0 -> (x <_ (1 / 2) <-> 0 <_ (1 / 2)))
492		breq1 3341	. . . . . . . . 9 |- (x = 0 -> (x <_ (1 / 4) <-> 0 <_ (1 / 4)))
493		opreq2 4890	. . . . . . . . 9 |- (x = 0 -> (2 x. x) = (2 x. 0))
494		opreq1 4889	. . . . . . . . 9 |- (x = 0 -> (x + (1 / 4)) = (0 + (1 / 4)))
495	492, 493, 494	ifbieq12d 2998	. . . . . . . 8 |- (x = 0 -> if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))) = if(0 <_ (1 / 4), (2 x. 0), (0 + (1 / 4))))
496		opreq1 4889	. . . . . . . . 9 |- (x = 0 -> (x / 2) = (0 / 2))
497	496	opreq1d 4897	. . . . . . . 8 |- (x = 0 -> ((x / 2) + (1 / 2)) = ((0 / 2) + (1 / 2)))
498	491, 495, 497	ifbieq12d 2998	. . . . . . 7 |- (x = 0 -> if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))) = if(0 <_ (1 / 2), if(0 <_ (1 / 4), (2 x. 0), (0 + (1 / 4))), ((0 / 2) + (1 / 2))))
499		oprex 4907	. . . . . . . . 9 |- (2 x. 0) e. _V
500		oprex 4907	. . . . . . . . 9 |- (0 + (1 / 4)) e. _V
501	499, 500	ifex 3031	. . . . . . . 8 |- if(0 <_ (1 / 4), (2 x. 0), (0 + (1 / 4))) e. _V
502		oprex 4907	. . . . . . . 8 |- ((0 / 2) + (1 / 2)) e. _V
503	501, 502	ifex 3031	. . . . . . 7 |- if(0 <_ (1 / 2), if(0 <_ (1 / 4), (2 x. 0), (0 + (1 / 4))), ((0 / 2) + (1 / 2))) e. _V
504	498, 343, 503	fvopab4 4743	. . . . . 6 |- (0 e. (0[,]1) -> ({<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))}` 0) = if(0 <_ (1 / 2), if(0 <_ (1 / 4), (2 x. 0), (0 + (1 / 4))), ((0 / 2) + (1 / 2))))
505	490, 504	ax-mp 7	. . . . 5 |- ({<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))}` 0) = if(0 <_ (1 / 2), if(0 <_ (1 / 4), (2 x. 0), (0 + (1 / 4))), ((0 / 2) + (1 / 2)))
506		iftrue 2989	. . . . . 6 |- (0 <_ (1 / 2) -> if(0 <_ (1 / 2), if(0 <_ (1 / 4), (2 x. 0), (0 + (1 / 4))), ((0 / 2) + (1 / 2))) = if(0 <_ (1 / 4), (2 x. 0), (0 + (1 / 4))))
507	241, 506	ax-mp 7	. . . . 5 |- if(0 <_ (1 / 2), if(0 <_ (1 / 4), (2 x. 0), (0 + (1 / 4))), ((0 / 2) + (1 / 2))) = if(0 <_ (1 / 4), (2 x. 0), (0 + (1 / 4)))
508		iftrue 2989	. . . . . . 7 |- (0 <_ (1 / 4) -> if(0 <_ (1 / 4), (2 x. 0), (0 + (1 / 4))) = (2 x. 0))
509	98, 508	ax-mp 7	. . . . . 6 |- if(0 <_ (1 / 4), (2 x. 0), (0 + (1 / 4))) = (2 x. 0)
510	46	mul01i 6594	. . . . . 6 |- (2 x. 0) = 0
511	509, 510	eqtri 1908	. . . . 5 |- if(0 <_ (1 / 4), (2 x. 0), (0 + (1 / 4))) = 0
512	505, 507, 511	3eqtri 1912	. . . 4 |- ({<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))}` 0) = 0
513		ubicc2 7574	. . . . . . 7 |- ((0 e. RR /\ 1 e. RR /\ 0 <_ 1) -> 1 e. (0[,]1))
514	11, 20, 488, 513	mp3an 1191	. . . . . 6 |- 1 e. (0[,]1)
515		breq1 3341	. . . . . . . 8 |- (x = 1 -> (x <_ (1 / 2) <-> 1 <_ (1 / 2)))
516		breq1 3341	. . . . . . . . 9 |- (x = 1 -> (x <_ (1 / 4) <-> 1 <_ (1 / 4)))
517		opreq2 4890	. . . . . . . . 9 |- (x = 1 -> (2 x. x) = (2 x. 1))
518		opreq1 4889	. . . . . . . . 9 |- (x = 1 -> (x + (1 / 4)) = (1 + (1 / 4)))
519	516, 517, 518	ifbieq12d 2998	. . . . . . . 8 |- (x = 1 -> if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))) = if(1 <_ (1 / 4), (2 x. 1), (1 + (1 / 4))))
520		opreq1 4889	. . . . . . . . 9 |- (x = 1 -> (x / 2) = (1 / 2))
521	520	opreq1d 4897	. . . . . . . 8 |- (x = 1 -> ((x / 2) + (1 / 2)) = ((1 / 2) + (1 / 2)))
522	515, 519, 521	ifbieq12d 2998	. . . . . . 7 |- (x = 1 -> if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))) = if(1 <_ (1 / 2), if(1 <_ (1 / 4), (2 x. 1), (1 + (1 / 4))), ((1 / 2) + (1 / 2))))
523		oprex 4907	. . . . . . . . 9 |- (2 x. 1) e. _V
524		oprex 4907	. . . . . . . . 9 |- (1 + (1 / 4)) e. _V
525	523, 524	ifex 3031	. . . . . . . 8 |- if(1 <_ (1 / 4), (2 x. 1), (1 + (1 / 4))) e. _V
526		oprex 4907	. . . . . . . 8 |- ((1 / 2) + (1 / 2)) e. _V
527	525, 526	ifex 3031	. . . . . . 7 |- if(1 <_ (1 / 2), if(1 <_ (1 / 4), (2 x. 1), (1 + (1 / 4))), ((1 / 2) + (1 / 2))) e. _V
528	522, 343, 527	fvopab4 4743	. . . . . 6 |- (1 e. (0[,]1) -> ({<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))}` 1) = if(1 <_ (1 / 2), if(1 <_ (1 / 4), (2 x. 1), (1 + (1 / 4))), ((1 / 2) + (1 / 2))))
529	514, 528	ax-mp 7	. . . . 5 |- ({<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))}` 1) = if(1 <_ (1 / 2), if(1 <_ (1 / 4), (2 x. 1), (1 + (1 / 4))), ((1 / 2) + (1 / 2)))
530	14, 20	ltnlei 6754	. . . . . . 7 |- ((1 / 2) < 1 <-> -. 1 <_ (1 / 2))
531	370, 530	mpbi 206	. . . . . 6 |- -. 1 <_ (1 / 2)
532		iffalse 2991	. . . . . 6 |- (-. 1 <_ (1 / 2) -> if(1 <_ (1 / 2), if(1 <_ (1 / 4), (2 x. 1), (1 + (1 / 4))), ((1 / 2) + (1 / 2))) = ((1 / 2) + (1 / 2)))
533	531, 532	ax-mp 7	. . . . 5 |- if(1 <_ (1 / 2), if(1 <_ (1 / 4), (2 x. 1), (1 + (1 / 4))), ((1 / 2) + (1 / 2))) = ((1 / 2) + (1 / 2))
534	529, 533, 164	3eqtri 1912	. . . 4 |- ({<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))}` 1) = 1
535	486, 512, 534	3pm3.2i 1048	. . 3 |- ({<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))} e. (II Cn II) /\ ({<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))}` 0) = 0 /\ ({<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))}` 1) = 1)
536	364, 367, 535	sylancl 525	. 2 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> ((F(*p` J)(G(*p` J)H)) o. {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), if(x <_ (1 / 4), (2 x. x), (x + (1 / 4))), ((x / 2) + (1 / 2))))})(~=ph` J)(F(*p` J)(G(*p` J)H)))
537	363, 536	eqbrtrd 3357	1 |- ((J e. Top /\ (F e. (II Cn J) /\ G e. (II Cn J) /\ H e. (II Cn J)) /\ ((F` 1) = (G` 0) /\ (G` 1) = (H` 0))) -> ((F(*p` J)G)(*p` J)H)(~=ph` J)(F(*p` J)(G(*p` J)H)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017   C_ wss 2593  ifcif 2982  <.cop 3046   class class class wbr 3338  {copab 3395   X. cxp 3984  ran crn 3987   |` cres 3988   o. ccom 3990   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445   / cdiv 6447   <_ cle 6448   < clt 6653  2c2 7145  3c3 7146  4c4 7147  (,)cioo 7524  [,]cicc 7527  abscabs 8000  -cn->ccncf 8524  Topctop 8857  topGenctg 8860   Cn ccn 9028  Opencopn 9069  subSpcsubsp 10242  IIcii 15865  ~=phcphtpc 16044  *pcpco 16067

This theorem is referenced by:  pi1gp 16095

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  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-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-r1 5750  df-rank 5751  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-rp 7232  df-n0 7309  df-z 7345  df-q 7436  df-fl 7463  df-ioo 7528  df-icc 7531  df-uz 7587  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-cncf 8525  df-top 8861  df-topsp 8862  df-bases 8863  df-topgen 8864  df-tx 8931  df-cld 8939  df-cn 9030  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200  df-subsp 10243  df-ii 15866  df-phtpy 16045  df-phtpc 16057  df-pco 16069

Copyright terms: Public domain