Theorem pcohtpylem3 16082

Description: Lemma for pcohtpy 16083.

Hypothesis

Ref	Expression
pcohtpylem.1	|- P = {<.<.x, y>., z>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ z = if(x <_ (1 / 2), ((2 x. x)My), (((2 x. x) - 1)Ny)))}

Assertion

Ref	Expression
pcohtpylem3	|- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> P e. ((II X.t II) Cn J))

Distinct variable groups:   x,M,y,z,t   x,N,y,z,t   t,P

Proof of Theorem pcohtpylem3

Step	Hyp	Ref	Expression
1		simp1 876	. . 3 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> J e. Top)
2		iitop 15867	. . . 4 |- II e. Top
3	2, 2	txtopi 15909	. . 3 |- (II X.t II) e. Top
4	1, 3	jctil 316	. 2 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> ((II X.t II) e. Top /\ J e. Top))
5		retop 8926	. . . . . 6 |- (topGen` ran (,)) e. Top
6		0re 6603	. . . . . . 7 |- 0 e. RR
7		1re 6598	. . . . . . 7 |- 1 e. RR
8		iccssre 7565	. . . . . . 7 |- ((0 e. RR /\ 1 e. RR) -> (0[,]1) C_ RR)
9	6, 7, 8	mp2an 761	. . . . . 6 |- (0[,]1) C_ RR
10		2re 7163	. . . . . . . 8 |- 2 e. RR
11		2ne0 7174	. . . . . . . 8 |- 2 =/= 0
12	10, 11	rereccli 6979	. . . . . . 7 |- (1 / 2) e. RR
13		clint3 10184	. . . . . . 7 |- ((0 e. RR /\ (1 / 2) e. RR) -> (0[,](1 / 2)) e. (Clsd` (topGen` ran (,))))
14	6, 12, 13	mp2an 761	. . . . . 6 |- (0[,](1 / 2)) e. (Clsd` (topGen` ran (,)))
15	6, 7	pm3.2i 307	. . . . . . 7 |- (0 e. RR /\ 1 e. RR)
16	6, 12	pm3.2i 307	. . . . . . 7 |- (0 e. RR /\ (1 / 2) e. RR)
17	6	leidi 6790	. . . . . . . 8 |- 0 <_ 0
18		halflt1 7216	. . . . . . . . 9 |- (1 / 2) < 1
19	12, 7, 18	ltleii 6756	. . . . . . . 8 |- (1 / 2) <_ 1
20	17, 19	pm3.2i 307	. . . . . . 7 |- (0 <_ 0 /\ (1 / 2) <_ 1)
21		iccss 15855	. . . . . . 7 |- (((0 e. RR /\ 1 e. RR) /\ (0 e. RR /\ (1 / 2) e. RR) /\ (0 <_ 0 /\ (1 / 2) <_ 1)) -> (0[,](1 / 2)) C_ (0[,]1))
22	15, 16, 20, 21	mp3an 1191	. . . . . 6 |- (0[,](1 / 2)) C_ (0[,]1)
23		uniretop 8927	. . . . . . . 8 |- U.(topGen` ran (,)) = RR
24	23	eqcomi 1888	. . . . . . 7 |- RR = U.(topGen` ran (,))
25	24	subspcld 15838	. . . . . 6 |- ((((topGen` ran (,)) e. Top /\ (0[,]1) C_ RR) /\ ((0[,](1 / 2)) e. (Clsd` (topGen` ran (,))) /\ (0[,](1 / 2)) C_ (0[,]1))) -> (0[,](1 / 2)) e. (Clsd` (subSp` <.(0[,]1), (topGen` ran (,))>.)))
26	5, 9, 14, 22, 25	mp4an 15651	. . . . 5 |- (0[,](1 / 2)) e. (Clsd` (subSp` <.(0[,]1), (topGen` ran (,))>.))
27		dfii2 15869	. . . . . 6 |- II = (subSp` <.(0[,]1), (topGen` ran (,))>.)
28	27	fveq2i 4684	. . . . 5 |- (Clsd` II) = (Clsd` (subSp` <.(0[,]1), (topGen` ran (,))>.))
29	26, 28	eleqtrri 1970	. . . 4 |- (0[,](1 / 2)) e. (Clsd` II)
30		iiuni 15868	. . . . . 6 |- (0[,]1) = U.II
31	30	topcld 8951	. . . . 5 |- (II e. Top -> (0[,]1) e. (Clsd` II))
32	2, 31	ax-mp 7	. . . 4 |- (0[,]1) e. (Clsd` II)
33		eqid 1884	. . . . 5 |- (II X.t II) = (II X.t II)
34	33	txcld 15914	. . . 4 |- (((II e. Top /\ II e. Top) /\ ((0[,](1 / 2)) e. (Clsd` II) /\ (0[,]1) e. (Clsd` II))) -> ((0[,](1 / 2)) X. (0[,]1)) e. (Clsd` (II X.t II)))
35	2, 2, 29, 32, 34	mp4an 15651	. . 3 |- ((0[,](1 / 2)) X. (0[,]1)) e. (Clsd` (II X.t II))
36	35	a1i 8	. 2 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> ((0[,](1 / 2)) X. (0[,]1)) e. (Clsd` (II X.t II)))
37		clint3 10184	. . . . . . 7 |- (((1 / 2) e. RR /\ 1 e. RR) -> ((1 / 2)[,]1) e. (Clsd` (topGen` ran (,))))
38	12, 7, 37	mp2an 761	. . . . . 6 |- ((1 / 2)[,]1) e. (Clsd` (topGen` ran (,)))
39	12, 7	pm3.2i 307	. . . . . . 7 |- ((1 / 2) e. RR /\ 1 e. RR)
40		halfgt0 7215	. . . . . . . . 9 |- 0 < (1 / 2)
41	6, 12, 40	ltleii 6756	. . . . . . . 8 |- 0 <_ (1 / 2)
42	7	leidi 6790	. . . . . . . 8 |- 1 <_ 1
43	41, 42	pm3.2i 307	. . . . . . 7 |- (0 <_ (1 / 2) /\ 1 <_ 1)
44		iccss 15855	. . . . . . 7 |- (((0 e. RR /\ 1 e. RR) /\ ((1 / 2) e. RR /\ 1 e. RR) /\ (0 <_ (1 / 2) /\ 1 <_ 1)) -> ((1 / 2)[,]1) C_ (0[,]1))
45	15, 39, 43, 44	mp3an 1191	. . . . . 6 |- ((1 / 2)[,]1) C_ (0[,]1)
46	24	subspcld 15838	. . . . . 6 |- ((((topGen` ran (,)) e. Top /\ (0[,]1) C_ RR) /\ (((1 / 2)[,]1) e. (Clsd` (topGen` ran (,))) /\ ((1 / 2)[,]1) C_ (0[,]1))) -> ((1 / 2)[,]1) e. (Clsd` (subSp` <.(0[,]1), (topGen` ran (,))>.)))
47	5, 9, 38, 45, 46	mp4an 15651	. . . . 5 |- ((1 / 2)[,]1) e. (Clsd` (subSp` <.(0[,]1), (topGen` ran (,))>.))
48	47, 28	eleqtrri 1970	. . . 4 |- ((1 / 2)[,]1) e. (Clsd` II)
49	33	txcld 15914	. . . 4 |- (((II e. Top /\ II e. Top) /\ (((1 / 2)[,]1) e. (Clsd` II) /\ (0[,]1) e. (Clsd` II))) -> (((1 / 2)[,]1) X. (0[,]1)) e. (Clsd` (II X.t II)))
50	2, 2, 48, 32, 49	mp4an 15651	. . 3 |- (((1 / 2)[,]1) X. (0[,]1)) e. (Clsd` (II X.t II))
51	50	a1i 8	. 2 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> (((1 / 2)[,]1) X. (0[,]1)) e. (Clsd` (II X.t II)))
52		xpundir 4051	. . . 4 |- (((0[,](1 / 2)) u. ((1 / 2)[,]1)) X. (0[,]1)) = (((0[,](1 / 2)) X. (0[,]1)) u. (((1 / 2)[,]1) X. (0[,]1)))
53		elicc2 7560	. . . . . . . . 9 |- ((0 e. RR /\ 1 e. RR) -> ((1 / 2) e. (0[,]1) <-> ((1 / 2) e. RR /\ 0 <_ (1 / 2) /\ (1 / 2) <_ 1)))
54	6, 7, 53	mp2an 761	. . . . . . . 8 |- ((1 / 2) e. (0[,]1) <-> ((1 / 2) e. RR /\ 0 <_ (1 / 2) /\ (1 / 2) <_ 1))
55	54, 12, 41, 19	mpbir3an 1052	. . . . . . 7 |- (1 / 2) e. (0[,]1)
56		iccsplit 15854	. . . . . . 7 |- ((0 e. RR /\ 1 e. RR /\ (1 / 2) e. (0[,]1)) -> (0[,]1) = ((0[,](1 / 2)) u. ((1 / 2)[,]1)))
57	6, 7, 55, 56	mp3an 1191	. . . . . 6 |- (0[,]1) = ((0[,](1 / 2)) u. ((1 / 2)[,]1))
58	57	eqcomi 1888	. . . . 5 |- ((0[,](1 / 2)) u. ((1 / 2)[,]1)) = (0[,]1)
59	58	xpeq1i 4021	. . . 4 |- (((0[,](1 / 2)) u. ((1 / 2)[,]1)) X. (0[,]1)) = ((0[,]1) X. (0[,]1))
60	52, 59	eqtr3i 1910	. . 3 |- (((0[,](1 / 2)) X. (0[,]1)) u. (((1 / 2)[,]1) X. (0[,]1))) = ((0[,]1) X. (0[,]1))
61	60	a1i 8	. 2 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> (((0[,](1 / 2)) X. (0[,]1)) u. (((1 / 2)[,]1) X. (0[,]1))) = ((0[,]1) X. (0[,]1)))
62		pcohtpylem.1	. . . . . . . . . . . . . 14 |- P = {<.<.x, y>., z>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ z = if(x <_ (1 / 2), ((2 x. x)My), (((2 x. x) - 1)Ny)))}
63	62	pcohtpylem1 16080	. . . . . . . . . . . . 13 |- ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) -> (uPv) = ((2 x. u)Mv))
64	63	adantl 424	. . . . . . . . . . . 12 |- (((J e. Top /\ M e. ((II X.t II) Cn J)) /\ (u e. (0[,](1 / 2)) /\ v e. (0[,]1))) -> (uPv) = ((2 x. u)Mv))
65		foprrn 4965	. . . . . . . . . . . . . . 15 |- ((M:((0[,]1) X. (0[,]1))-->U.J /\ (2 x. u) e. (0[,]1) /\ v e. (0[,]1)) -> ((2 x. u)Mv) e. U.J)
66	65	3expb 1068	. . . . . . . . . . . . . 14 |- ((M:((0[,]1) X. (0[,]1))-->U.J /\ ((2 x. u) e. (0[,]1) /\ v e. (0[,]1))) -> ((2 x. u)Mv) e. U.J)
67	2, 2, 30, 30	txunii 15910	. . . . . . . . . . . . . . . 16 |- ((0[,]1) X. (0[,]1)) = U.(II X.t II)
68		eqid 1884	. . . . . . . . . . . . . . . 16 |- U.J = U.J
69	67, 68	cnf 9038	. . . . . . . . . . . . . . 15 |- (((II X.t II) e. Top /\ J e. Top /\ M e. ((II X.t II) Cn J)) -> M:((0[,]1) X. (0[,]1))-->U.J)
70	3, 69	mp3an1 1178	. . . . . . . . . . . . . 14 |- ((J e. Top /\ M e. ((II X.t II) Cn J)) -> M:((0[,]1) X. (0[,]1))-->U.J)
71	66, 70	sylan 497	. . . . . . . . . . . . 13 |- (((J e. Top /\ M e. ((II X.t II) Cn J)) /\ ((2 x. u) e. (0[,]1) /\ v e. (0[,]1))) -> ((2 x. u)Mv) e. U.J)
72		iihalf1 15872	. . . . . . . . . . . . 13 |- (u e. (0[,](1 / 2)) -> (2 x. u) e. (0[,]1))
73	71, 72	sylanr1 511	. . . . . . . . . . . 12 |- (((J e. Top /\ M e. ((II X.t II) Cn J)) /\ (u e. (0[,](1 / 2)) /\ v e. (0[,]1))) -> ((2 x. u)Mv) e. U.J)
74	64, 73	eqeltrd 1971	. . . . . . . . . . 11 |- (((J e. Top /\ M e. ((II X.t II) Cn J)) /\ (u e. (0[,](1 / 2)) /\ v e. (0[,]1))) -> (uPv) e. U.J)
75	74	expr 418	. . . . . . . . . 10 |- (((J e. Top /\ M e. ((II X.t II) Cn J)) /\ u e. (0[,](1 / 2))) -> (v e. (0[,]1) -> (uPv) e. U.J))
76	75	adantlrr 435	. . . . . . . . 9 |- (((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J))) /\ u e. (0[,](1 / 2))) -> (v e. (0[,]1) -> (uPv) e. U.J))
77	76	3adantl3 1034	. . . . . . . 8 |- (((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) /\ u e. (0[,](1 / 2))) -> (v e. (0[,]1) -> (uPv) e. U.J))
78	62	pcohtpylem2 16081	. . . . . . . . . . . . . . . 16 |- (((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> (uPv) = (((2 x. u) - 1)Nv))
79	78	ancoms 484	. . . . . . . . . . . . . . 15 |- ((A.t e. (0[,]1)(1Mt) = (0Nt) /\ (u e. ((1 / 2)[,]1) /\ v e. (0[,]1))) -> (uPv) = (((2 x. u) - 1)Nv))
80	79	adantll 428	. . . . . . . . . . . . . 14 |- (((N:((0[,]1) X. (0[,]1))-->U.J /\ A.t e. (0[,]1)(1Mt) = (0Nt)) /\ (u e. ((1 / 2)[,]1) /\ v e. (0[,]1))) -> (uPv) = (((2 x. u) - 1)Nv))
81		foprrn 4965	. . . . . . . . . . . . . . . . 17 |- ((N:((0[,]1) X. (0[,]1))-->U.J /\ ((2 x. u) - 1) e. (0[,]1) /\ v e. (0[,]1)) -> (((2 x. u) - 1)Nv) e. U.J)
82	81	3expb 1068	. . . . . . . . . . . . . . . 16 |- ((N:((0[,]1) X. (0[,]1))-->U.J /\ (((2 x. u) - 1) e. (0[,]1) /\ v e. (0[,]1))) -> (((2 x. u) - 1)Nv) e. U.J)
83		iihalf2 15873	. . . . . . . . . . . . . . . 16 |- (u e. ((1 / 2)[,]1) -> ((2 x. u) - 1) e. (0[,]1))
84	82, 83	sylanr1 511	. . . . . . . . . . . . . . 15 |- ((N:((0[,]1) X. (0[,]1))-->U.J /\ (u e. ((1 / 2)[,]1) /\ v e. (0[,]1))) -> (((2 x. u) - 1)Nv) e. U.J)
85	84	adantlr 429	. . . . . . . . . . . . . 14 |- (((N:((0[,]1) X. (0[,]1))-->U.J /\ A.t e. (0[,]1)(1Mt) = (0Nt)) /\ (u e. ((1 / 2)[,]1) /\ v e. (0[,]1))) -> (((2 x. u) - 1)Nv) e. U.J)
86	80, 85	eqeltrd 1971	. . . . . . . . . . . . 13 |- (((N:((0[,]1) X. (0[,]1))-->U.J /\ A.t e. (0[,]1)(1Mt) = (0Nt)) /\ (u e. ((1 / 2)[,]1) /\ v e. (0[,]1))) -> (uPv) e. U.J)
87	86	ex 402	. . . . . . . . . . . 12 |- ((N:((0[,]1) X. (0[,]1))-->U.J /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) -> (uPv) e. U.J))
88	67, 68	cnf 9038	. . . . . . . . . . . . 13 |- (((II X.t II) e. Top /\ J e. Top /\ N e. ((II X.t II) Cn J)) -> N:((0[,]1) X. (0[,]1))-->U.J)
89	3, 88	mp3an1 1178	. . . . . . . . . . . 12 |- ((J e. Top /\ N e. ((II X.t II) Cn J)) -> N:((0[,]1) X. (0[,]1))-->U.J)
90	87, 89	sylan 497	. . . . . . . . . . 11 |- (((J e. Top /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) -> (uPv) e. U.J))
91	90	adantlrl 434	. . . . . . . . . 10 |- (((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J))) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) -> (uPv) e. U.J))
92	91	3impa 1062	. . . . . . . . 9 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) -> (uPv) e. U.J))
93	92	expdimp 406	. . . . . . . 8 |- (((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) /\ u e. ((1 / 2)[,]1)) -> (v e. (0[,]1) -> (uPv) e. U.J))
94	77, 93	jaodan 471	. . . . . . 7 |- (((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) /\ (u e. (0[,](1 / 2)) \/ u e. ((1 / 2)[,]1))) -> (v e. (0[,]1) -> (uPv) e. U.J))
95	94	expimpd 404	. . . . . 6 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> (((u e. (0[,](1 / 2)) \/ u e. ((1 / 2)[,]1)) /\ v e. (0[,]1)) -> (uPv) e. U.J))
96	57	eleq2i 1961	. . . . . . . 8 |- (u e. (0[,]1) <-> u e. ((0[,](1 / 2)) u. ((1 / 2)[,]1)))
97		elun 2741	. . . . . . . 8 |- (u e. ((0[,](1 / 2)) u. ((1 / 2)[,]1)) <-> (u e. (0[,](1 / 2)) \/ u e. ((1 / 2)[,]1)))
98	96, 97	bitri 190	. . . . . . 7 |- (u e. (0[,]1) <-> (u e. (0[,](1 / 2)) \/ u e. ((1 / 2)[,]1)))
99	98	anbi1i 539	. . . . . 6 |- ((u e. (0[,]1) /\ v e. (0[,]1)) <-> ((u e. (0[,](1 / 2)) \/ u e. ((1 / 2)[,]1)) /\ v e. (0[,]1)))
100	95, 99	syl5ib 223	. . . . 5 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> ((u e. (0[,]1) /\ v e. (0[,]1)) -> (uPv) e. U.J))
101	100	r19.21aivv 2183	. . . 4 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> A.u e. (0[,]1)A.v e. (0[,]1)(uPv) e. U.J)
102		oprex 4907	. . . . . 6 |- ((2 x. x)My) e. _V
103		oprex 4907	. . . . . 6 |- (((2 x. x) - 1)Ny) e. _V
104	102, 103	ifex 3031	. . . . 5 |- if(x <_ (1 / 2), ((2 x. x)My), (((2 x. x) - 1)Ny)) e. _V
105	104, 62	fnoprab2 5064	. . . 4 |- P Fn ((0[,]1) X. (0[,]1))
106	101, 105	jctil 316	. . 3 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> (P Fn ((0[,]1) X. (0[,]1)) /\ A.u e. (0[,]1)A.v e. (0[,]1)(uPv) e. U.J))
107		ffnoprv 4943	. . 3 |- (P:((0[,]1) X. (0[,]1))-->U.J <-> (P Fn ((0[,]1) X. (0[,]1)) /\ A.u e. (0[,]1)A.v e. (0[,]1)(uPv) e. U.J))
108	106, 107	sylibr 217	. 2 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> P:((0[,]1) X. (0[,]1))-->U.J)
109		ffn 4562	. . . . . . . 8 |- (M:((0[,]1) X. (0[,]1))-->U.J -> M Fn ((0[,]1) X. (0[,]1)))
110	70, 109	syl 12	. . . . . . 7 |- ((J e. Top /\ M e. ((II X.t II) Cn J)) -> M Fn ((0[,]1) X. (0[,]1)))
111	110	adantrr 431	. . . . . 6 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J))) -> M Fn ((0[,]1) X. (0[,]1)))
112	111	3adant3 896	. . . . 5 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> M Fn ((0[,]1) X. (0[,]1)))
113	72	adantr 425	. . . . . 6 |- ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) -> (2 x. u) e. (0[,]1))
114		simpr 350	. . . . . 6 |- ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) -> v e. (0[,]1))
115		eqid 1884	. . . . . 6 |- {<.<.u, v>., z>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ z = <.(2 x. u), v>.)} = {<.<.u, v>., z>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ z = <.(2 x. u), v>.)}
116		eqid 1884	. . . . . 6 |- {<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))} = {<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))}
117	113, 114, 115, 116	oprab2co 10160	. . . . 5 |- (M Fn ((0[,]1) X. (0[,]1)) -> {<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))} = (M o. {<.<.u, v>., z>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ z = <.(2 x. u), v>.)}))
118	112, 117	syl 12	. . . 4 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> {<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))} = (M o. {<.<.u, v>., z>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ z = <.(2 x. u), v>.)}))
119		ssid 2634	. . . . . . . . . . 11 |- (0[,]1) C_ (0[,]1)
120		xpss12 4089	. . . . . . . . . . 11 |- (((0[,](1 / 2)) C_ (0[,]1) /\ (0[,]1) C_ (0[,]1)) -> ((0[,](1 / 2)) X. (0[,]1)) C_ ((0[,]1) X. (0[,]1)))
121	22, 119, 120	mp2an 761	. . . . . . . . . 10 |- ((0[,](1 / 2)) X. (0[,]1)) C_ ((0[,]1) X. (0[,]1))
122	33, 30, 30	txuni 8935	. . . . . . . . . . 11 |- ((II e. Top /\ II e. Top) -> U.(II X.t II) = ((0[,]1) X. (0[,]1)))
123	2, 2, 122	mp2an 761	. . . . . . . . . 10 |- U.(II X.t II) = ((0[,]1) X. (0[,]1))
124	121, 123	sseqtr4i 2650	. . . . . . . . 9 |- ((0[,](1 / 2)) X. (0[,]1)) C_ U.(II X.t II)
125		stoig3 10253	. . . . . . . . 9 |- (((II X.t II) e. Top /\ ((0[,](1 / 2)) X. (0[,]1)) C_ U.(II X.t II)) -> (subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) e. Top)
126	3, 124, 125	mp2an 761	. . . . . . . 8 |- (subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) e. Top
127	126	a1i 8	. . . . . . 7 |- (J e. Top -> (subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) e. Top)
128	3	a1i 8	. . . . . . 7 |- (J e. Top -> (II X.t II) e. Top)
129		id 73	. . . . . . 7 |- (J e. Top -> J e. Top)
130	127, 128, 129	3jca 1050	. . . . . 6 |- (J e. Top -> ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) e. Top /\ (II X.t II) e. Top /\ J e. Top))
131	130	3ad2ant1 897	. . . . 5 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) e. Top /\ (II X.t II) e. Top /\ J e. Top))
132		simp2l 902	. . . . . 6 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> M e. ((II X.t II) Cn J))
133	22, 30	sseqtri 2649	. . . . . . . . 9 |- (0[,](1 / 2)) C_ U.II
134		stoig3 10253	. . . . . . . . 9 |- ((II e. Top /\ (0[,](1 / 2)) C_ U.II) -> (subSp` <.(0[,](1 / 2)), II>.) e. Top)
135	2, 133, 134	mp2an 761	. . . . . . . 8 |- (subSp` <.(0[,](1 / 2)), II>.) e. Top
136	30	eqimssi 2668	. . . . . . . . 9 |- (0[,]1) C_ U.II
137		stoig3 10253	. . . . . . . . 9 |- ((II e. Top /\ (0[,]1) C_ U.II) -> (subSp` <.(0[,]1), II>.) e. Top)
138	2, 136, 137	mp2an 761	. . . . . . . 8 |- (subSp` <.(0[,]1), II>.) e. Top
139	135, 138	pm3.2i 307	. . . . . . 7 |- ((subSp` <.(0[,](1 / 2)), II>.) e. Top /\ (subSp` <.(0[,]1), II>.) e. Top)
140	2, 2	pm3.2i 307	. . . . . . 7 |- (II e. Top /\ II e. Top)
141		iccssre 7565	. . . . . . . . . . 11 |- ((0 e. RR /\ (1 / 2) e. RR) -> (0[,](1 / 2)) C_ RR)
142	6, 12, 141	mp2an 761	. . . . . . . . . 10 |- (0[,](1 / 2)) C_ RR
143		axresscn 6420	. . . . . . . . . 10 |- RR C_ CC
144	142, 143	sstri 2626	. . . . . . . . 9 |- (0[,](1 / 2)) C_ CC
145	9, 143	sstri 2626	. . . . . . . . 9 |- (0[,]1) C_ CC
146		eqid 1884	. . . . . . . . 9 |- {<.x, y>. | (x e. CC /\ y = (2 x. x))} = {<.x, y>. | (x e. CC /\ y = (2 x. x))}
147		eqid 1884	. . . . . . . . 9 |- {<.x, y>. | (x e. (0[,](1 / 2)) /\ y = (2 x. x))} = {<.x, y>. | (x e. (0[,](1 / 2)) /\ y = (2 x. x))}
148		iihalf1 15872	. . . . . . . . 9 |- (x e. (0[,](1 / 2)) -> (2 x. x) e. (0[,]1))
149		2cn 7164	. . . . . . . . . 10 |- 2 e. CC
150	146	mulc1cncf 8541	. . . . . . . . . 10 |- (2 e. CC -> {<.x, y>. | (x e. CC /\ y = (2 x. x))} e. (CC-cn->CC))
151	149, 150	ax-mp 7	. . . . . . . . 9 |- {<.x, y>. | (x e. CC /\ y = (2 x. x))} e. (CC-cn->CC)
152		eqid 1884	. . . . . . . . . . . . 13 |- (abs o. - ) = (abs o. - )
153	152	cnmet 9182	. . . . . . . . . . . 12 |- (abs o. - ) e. Met
154		metres 9100	. . . . . . . . . . . 12 |- ((abs o. - ) e. Met -> ((abs o. - ) |` ((0[,]1) X. (0[,]1))) e. Met)
155	153, 154	ax-mp 7	. . . . . . . . . . 11 |- ((abs o. - ) |` ((0[,]1) X. (0[,]1))) e. Met
156		eqid 1884	. . . . . . . . . . . 12 |- (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` ((0[,](1 / 2)) X. (0[,](1 / 2)))) = (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` ((0[,](1 / 2)) X. (0[,](1 / 2))))
157	152	cnmetba 9181	. . . . . . . . . . . . . 14 |- CC = dom dom (abs o. - )
158	157	metssba2 9087	. . . . . . . . . . . . 13 |- (((abs o. - ) e. Met /\ (0[,]1) C_ CC) -> (0[,]1) = dom dom ((abs o. - ) |` ((0[,]1) X. (0[,]1))))
159	153, 145, 158	mp2an 761	. . . . . . . . . . . 12 |- (0[,]1) = dom dom ((abs o. - ) |` ((0[,]1) X. (0[,]1)))
160		df-ii 15866	. . . . . . . . . . . 12 |- II = (Open` ((abs o. - ) |` ((0[,]1) X. (0[,]1))))
161		eqid 1884	. . . . . . . . . . . 12 |- (Open` (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` ((0[,](1 / 2)) X. (0[,](1 / 2))))) = (Open` (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` ((0[,](1 / 2)) X. (0[,](1 / 2)))))
162	156, 159, 160, 161	subtopmet 10256	. . . . . . . . . . 11 |- ((((abs o. - ) |` ((0[,]1) X. (0[,]1))) e. Met /\ (0[,](1 / 2)) C_ (0[,]1)) -> (subSp` <.(0[,](1 / 2)), II>.) = (Open` (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` ((0[,](1 / 2)) X. (0[,](1 / 2))))))
163	155, 22, 162	mp2an 761	. . . . . . . . . 10 |- (subSp` <.(0[,](1 / 2)), II>.) = (Open` (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` ((0[,](1 / 2)) X. (0[,](1 / 2)))))
164		xpss12 4089	. . . . . . . . . . . . . 14 |- (((0[,](1 / 2)) C_ (0[,]1) /\ (0[,](1 / 2)) C_ (0[,]1)) -> ((0[,](1 / 2)) X. (0[,](1 / 2))) C_ ((0[,]1) X. (0[,]1)))
165	164	anidms 480	. . . . . . . . . . . . 13 |- ((0[,](1 / 2)) C_ (0[,]1) -> ((0[,](1 / 2)) X. (0[,](1 / 2))) C_ ((0[,]1) X. (0[,]1)))
166	22, 165	ax-mp 7	. . . . . . . . . . . 12 |- ((0[,](1 / 2)) X. (0[,](1 / 2))) C_ ((0[,]1) X. (0[,]1))
167		resabs1 4244	. . . . . . . . . . . 12 |- (((0[,](1 / 2)) X. (0[,](1 / 2))) C_ ((0[,]1) X. (0[,]1)) -> (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` ((0[,](1 / 2)) X. (0[,](1 / 2)))) = ((abs o. - ) |` ((0[,](1 / 2)) X. (0[,](1 / 2)))))
168	166, 167	ax-mp 7	. . . . . . . . . . 11 |- (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` ((0[,](1 / 2)) X. (0[,](1 / 2)))) = ((abs o. - ) |` ((0[,](1 / 2)) X. (0[,](1 / 2))))
169	168	fveq2i 4684	. . . . . . . . . 10 |- (Open` (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` ((0[,](1 / 2)) X. (0[,](1 / 2))))) = (Open` ((abs o. - ) |` ((0[,](1 / 2)) X. (0[,](1 / 2)))))
170	163, 169	eqtri 1908	. . . . . . . . 9 |- (subSp` <.(0[,](1 / 2)), II>.) = (Open` ((abs o. - ) |` ((0[,](1 / 2)) X. (0[,](1 / 2)))))
171	144, 145, 146, 147, 148, 151, 170, 160	cncfres 15895	. . . . . . . 8 |- {<.x, y>. | (x e. (0[,](1 / 2)) /\ y = (2 x. x))} e. ((subSp` <.(0[,](1 / 2)), II>.) Cn II)
172	30	idcn 9042	. . . . . . . . . 10 |- (II e. Top -> ( _I |` (0[,]1)) e. (II Cn II))
173	2, 172	ax-mp 7	. . . . . . . . 9 |- ( _I |` (0[,]1)) e. (II Cn II)
174	30	subspid 10249	. . . . . . . . . . 11 |- (II e. Top -> (subSp` <.(0[,]1), II>.) = II)
175	2, 174	ax-mp 7	. . . . . . . . . 10 |- (subSp` <.(0[,]1), II>.) = II
176	175	opreq1i 4892	. . . . . . . . 9 |- ((subSp` <.(0[,]1), II>.) Cn II) = (II Cn II)
177	173, 176	eleqtrri 1970	. . . . . . . 8 |- ( _I |` (0[,]1)) e. ((subSp` <.(0[,]1), II>.) Cn II)
178	171, 177	pm3.2i 307	. . . . . . 7 |- ({<.x, y>. | (x e. (0[,](1 / 2)) /\ y = (2 x. x))} e. ((subSp` <.(0[,](1 / 2)), II>.) Cn II) /\ ( _I |` (0[,]1)) e. ((subSp` <.(0[,]1), II>.) Cn II))
179	30, 30	txsubsp 15912	. . . . . . . . 9 |- (((II e. Top /\ II e. Top) /\ ((0[,](1 / 2)) C_ (0[,]1) /\ (0[,]1) C_ (0[,]1))) -> (subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) = ((subSp` <.(0[,](1 / 2)), II>.) X.t (subSp` <.(0[,]1), II>.)))
180	2, 2, 22, 119, 179	mp4an 15651	. . . . . . . 8 |- (subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) = ((subSp` <.(0[,](1 / 2)), II>.) X.t (subSp` <.(0[,]1), II>.))
181		stoig2 10252	. . . . . . . . . 10 |- ((II e. Top /\ (0[,](1 / 2)) C_ U.II) -> U.(subSp` <.(0[,](1 / 2)), II>.) = (0[,](1 / 2)))
182	2, 133, 181	mp2an 761	. . . . . . . . 9 |- U.(subSp` <.(0[,](1 / 2)), II>.) = (0[,](1 / 2))
183	182	eqcomi 1888	. . . . . . . 8 |- (0[,](1 / 2)) = U.(subSp` <.(0[,](1 / 2)), II>.)
184		stoig2 10252	. . . . . . . . . 10 |- ((II e. Top /\ (0[,]1) C_ U.II) -> U.(subSp` <.(0[,]1), II>.) = (0[,]1))
185	2, 136, 184	mp2an 761	. . . . . . . . 9 |- U.(subSp` <.(0[,]1), II>.) = (0[,]1)
186	185	eqcomi 1888	. . . . . . . 8 |- (0[,]1) = U.(subSp` <.(0[,]1), II>.)
187		opreq2 4890	. . . . . . . . . . . . . . 15 |- (x = u -> (2 x. x) = (2 x. u))
188		oprex 4907	. . . . . . . . . . . . . . 15 |- (2 x. u) e. _V
189	187, 147, 188	fvopab4 4743	. . . . . . . . . . . . . 14 |- (u e. (0[,](1 / 2)) -> ({<.x, y>. | (x e. (0[,](1 / 2)) /\ y = (2 x. x))}` u) = (2 x. u))
190	189	adantr 425	. . . . . . . . . . . . 13 |- ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) -> ({<.x, y>. | (x e. (0[,](1 / 2)) /\ y = (2 x. x))}` u) = (2 x. u))
191		fvresi 4819	. . . . . . . . . . . . . 14 |- (v e. (0[,]1) -> (( _I |` (0[,]1))` v) = v)
192	191	adantl 424	. . . . . . . . . . . . 13 |- ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) -> (( _I |` (0[,]1))` v) = v)
193	190, 192	opeq12d 3166	. . . . . . . . . . . 12 |- ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) -> <.({<.x, y>. | (x e. (0[,](1 / 2)) /\ y = (2 x. x))}` u), (( _I |` (0[,]1))` v)>. = <.(2 x. u), v>.)
194	193	eqcomd 1889	. . . . . . . . . . 11 |- ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) -> <.(2 x. u), v>. = <.({<.x, y>. | (x e. (0[,](1 / 2)) /\ y = (2 x. x))}` u), (( _I |` (0[,]1))` v)>.)
195	194	eqeq2d 1895	. . . . . . . . . 10 |- ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) -> (z = <.(2 x. u), v>. <-> z = <.({<.x, y>. | (x e. (0[,](1 / 2)) /\ y = (2 x. x))}` u), (( _I |` (0[,]1))` v)>.))
196	195	pm5.32i 707	. . . . . . . . 9 |- (((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ z = <.(2 x. u), v>.) <-> ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ z = <.({<.x, y>. | (x e. (0[,](1 / 2)) /\ y = (2 x. x))}` u), (( _I |` (0[,]1))` v)>.))
197	196	oprabbii 4923	. . . . . . . 8 |- {<.<.u, v>., z>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ z = <.(2 x. u), v>.)} = {<.<.u, v>., z>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ z = <.({<.x, y>. | (x e. (0[,](1 / 2)) /\ y = (2 x. x))}` u), (( _I |` (0[,]1))` v)>.)}
198	180, 33, 183, 186, 197	2txcn 10229	. . . . . . 7 |- ((((subSp` <.(0[,](1 / 2)), II>.) e. Top /\ (subSp` <.(0[,]1), II>.) e. Top) /\ (II e. Top /\ II e. Top) /\ ({<.x, y>. | (x e. (0[,](1 / 2)) /\ y = (2 x. x))} e. ((subSp` <.(0[,](1 / 2)), II>.) Cn II) /\ ( _I |` (0[,]1)) e. ((subSp` <.(0[,]1), II>.) Cn II))) -> {<.<.u, v>., z>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ z = <.(2 x. u), v>.)} e. ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) Cn (II X.t II)))
199	139, 140, 178, 198	mp3an 1191	. . . . . 6 |- {<.<.u, v>., z>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ z = <.(2 x. u), v>.)} e. ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) Cn (II X.t II))
200	132, 199	jctil 316	. . . . 5 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> ({<.<.u, v>., z>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ z = <.(2 x. u), v>.)} e. ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) Cn (II X.t II)) /\ M e. ((II X.t II) Cn J)))
201		cnco 9045	. . . . 5 |- ((((subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) e. Top /\ (II X.t II) e. Top /\ J e. Top) /\ ({<.<.u, v>., z>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ z = <.(2 x. u), v>.)} e. ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) Cn (II X.t II)) /\ M e. ((II X.t II) Cn J))) -> (M o. {<.<.u, v>., z>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ z = <.(2 x. u), v>.)}) e. ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) Cn J))
202	131, 200, 201	syl11anc 524	. . . 4 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> (M o. {<.<.u, v>., z>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ z = <.(2 x. u), v>.)}) e. ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) Cn J))
203	118, 202	eqeltrd 1971	. . 3 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> {<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))} e. ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) Cn J))
204		fnssres 4526	. . . . . 6 |- ((P Fn ((0[,]1) X. (0[,]1)) /\ ((0[,](1 / 2)) X. (0[,]1)) C_ ((0[,]1) X. (0[,]1))) -> (P |` ((0[,](1 / 2)) X. (0[,]1))) Fn ((0[,](1 / 2)) X. (0[,]1)))
205	105, 121, 204	mp2an 761	. . . . 5 |- (P |` ((0[,](1 / 2)) X. (0[,]1))) Fn ((0[,](1 / 2)) X. (0[,]1))
206		oprex 4907	. . . . . 6 |- ((2 x. u)Mv) e. _V
207	206, 116	fnoprab2 5064	. . . . 5 |- {<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))} Fn ((0[,](1 / 2)) X. (0[,]1))
208		eqfnfv 4766	. . . . 5 |- (((P |` ((0[,](1 / 2)) X. (0[,]1))) Fn ((0[,](1 / 2)) X. (0[,]1)) /\ {<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))} Fn ((0[,](1 / 2)) X. (0[,]1))) -> ((P |` ((0[,](1 / 2)) X. (0[,]1))) = {<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))} <-> (((0[,](1 / 2)) X. (0[,]1)) = ((0[,](1 / 2)) X. (0[,]1)) /\ A.t e. ((0[,](1 / 2)) X. (0[,]1))((P |` ((0[,](1 / 2)) X. (0[,]1)))` t) = ({<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))}` t))))
209	205, 207, 208	mp2an 761	. . . 4 |- ((P |` ((0[,](1 / 2)) X. (0[,]1))) = {<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))} <-> (((0[,](1 / 2)) X. (0[,]1)) = ((0[,](1 / 2)) X. (0[,]1)) /\ A.t e. ((0[,](1 / 2)) X. (0[,]1))((P |` ((0[,](1 / 2)) X. (0[,]1)))` t) = ({<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))}` t)))
210		eqid 1884	. . . 4 |- ((0[,](1 / 2)) X. (0[,]1)) = ((0[,](1 / 2)) X. (0[,]1))
211		elxp6 5041	. . . . . . 7 |- (t e. ((0[,](1 / 2)) X. (0[,]1)) <-> (t = <.(1st` t), (2nd` t)>. /\ ((1st` t) e. (0[,](1 / 2)) /\ (2nd` t) e. (0[,]1))))
212		fveq2 4681	. . . . . . . . 9 |- (t = <.(1st` t), (2nd` t)>. -> (P` t) = (P` <.(1st` t), (2nd` t)>.))
213		df-opr 4886	. . . . . . . . 9 |- ((1st` t)P(2nd` t)) = (P` <.(1st` t), (2nd` t)>.)
214	212, 213	syl6eqr 1946	. . . . . . . 8 |- (t = <.(1st` t), (2nd` t)>. -> (P` t) = ((1st` t)P(2nd` t)))
215	62	pcohtpylem1 16080	. . . . . . . 8 |- (((1st` t) e. (0[,](1 / 2)) /\ (2nd` t) e. (0[,]1)) -> ((1st` t)P(2nd` t)) = ((2 x. (1st` t))M(2nd` t)))
216	214, 215	sylan9eq 1948	. . . . . . 7 |- ((t = <.(1st` t), (2nd` t)>. /\ ((1st` t) e. (0[,](1 / 2)) /\ (2nd` t) e. (0[,]1))) -> (P` t) = ((2 x. (1st` t))M(2nd` t)))
217	211, 216	sylbi 216	. . . . . 6 |- (t e. ((0[,](1 / 2)) X. (0[,]1)) -> (P` t) = ((2 x. (1st` t))M(2nd` t)))
218		fvres 4691	. . . . . 6 |- (t e. ((0[,](1 / 2)) X. (0[,]1)) -> ((P |` ((0[,](1 / 2)) X. (0[,]1)))` t) = (P` t))
219		fveq2 4681	. . . . . . . . 9 |- (t = <.(1st` t), (2nd` t)>. -> ({<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))}` t) = ({<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))}` <.(1st` t), (2nd` t)>.))
220		df-opr 4886	. . . . . . . . 9 |- ((1st` t){<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))} (2nd` t)) = ({<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))}` <.(1st` t), (2nd` t)>.)
221	219, 220	syl6eqr 1946	. . . . . . . 8 |- (t = <.(1st` t), (2nd` t)>. -> ({<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))}` t) = ((1st` t){<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))} (2nd` t)))
222		oprex 4907	. . . . . . . . 9 |- ((2 x. (1st` t))M(2nd` t)) e. _V
223		opreq2 4890	. . . . . . . . . 10 |- (u = (1st` t) -> (2 x. u) = (2 x. (1st` t)))
224	223	opreq1d 4897	. . . . . . . . 9 |- (u = (1st` t) -> ((2 x. u)Mv) = ((2 x. (1st` t))Mv))
225		opreq2 4890	. . . . . . . . 9 |- (v = (2nd` t) -> ((2 x. (1st` t))Mv) = ((2 x. (1st` t))M(2nd` t)))
226	222, 224, 225, 116	oprabval2 4957	. . . . . . . 8 |- (((1st` t) e. (0[,](1 / 2)) /\ (2nd` t) e. (0[,]1)) -> ((1st` t){<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))} (2nd` t)) = ((2 x. (1st` t))M(2nd` t)))
227	221, 226	sylan9eq 1948	. . . . . . 7 |- ((t = <.(1st` t), (2nd` t)>. /\ ((1st` t) e. (0[,](1 / 2)) /\ (2nd` t) e. (0[,]1))) -> ({<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))}` t) = ((2 x. (1st` t))M(2nd` t)))
228	211, 227	sylbi 216	. . . . . 6 |- (t e. ((0[,](1 / 2)) X. (0[,]1)) -> ({<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))}` t) = ((2 x. (1st` t))M(2nd` t)))
229	217, 218, 228	3eqtr4d 1937	. . . . 5 |- (t e. ((0[,](1 / 2)) X. (0[,]1)) -> ((P |` ((0[,](1 / 2)) X. (0[,]1)))` t) = ({<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))}` t))
230	229	rgen 2159	. . . 4 |- A.t e. ((0[,](1 / 2)) X. (0[,]1))((P |` ((0[,](1 / 2)) X. (0[,]1)))` t) = ({<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))}` t)
231	209, 210, 230	mpbir2an 800	. . 3 |- (P |` ((0[,](1 / 2)) X. (0[,]1))) = {<.<.u, v>., w>. | ((u e. (0[,](1 / 2)) /\ v e. (0[,]1)) /\ w = ((2 x. u)Mv))}
232	203, 231	syl5eqel 1975	. 2 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> (P |` ((0[,](1 / 2)) X. (0[,]1))) e. ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) Cn J))
233		xpss12 4089	. . . . . . . . 9 |- ((((1 / 2)[,]1) C_ (0[,]1) /\ (0[,]1) C_ (0[,]1)) -> (((1 / 2)[,]1) X. (0[,]1)) C_ ((0[,]1) X. (0[,]1)))
234	45, 119, 233	mp2an 761	. . . . . . . 8 |- (((1 / 2)[,]1) X. (0[,]1)) C_ ((0[,]1) X. (0[,]1))
235		fnssres 4526	. . . . . . . 8 |- ((P Fn ((0[,]1) X. (0[,]1)) /\ (((1 / 2)[,]1) X. (0[,]1)) C_ ((0[,]1) X. (0[,]1))) -> (P |` (((1 / 2)[,]1) X. (0[,]1))) Fn (((1 / 2)[,]1) X. (0[,]1)))
236	105, 234, 235	mp2an 761	. . . . . . 7 |- (P |` (((1 / 2)[,]1) X. (0[,]1))) Fn (((1 / 2)[,]1) X. (0[,]1))
237		oprex 4907	. . . . . . . 8 |- (((2 x. u) - 1)Nv) e. _V
238		eqid 1884	. . . . . . . 8 |- {<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))} = {<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))}
239	237, 238	fnoprab2 5064	. . . . . . 7 |- {<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))} Fn (((1 / 2)[,]1) X. (0[,]1))
240		eqfnfv 4766	. . . . . . 7 |- (((P |` (((1 / 2)[,]1) X. (0[,]1))) Fn (((1 / 2)[,]1) X. (0[,]1)) /\ {<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))} Fn (((1 / 2)[,]1) X. (0[,]1))) -> ((P |` (((1 / 2)[,]1) X. (0[,]1))) = {<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))} <-> ((((1 / 2)[,]1) X. (0[,]1)) = (((1 / 2)[,]1) X. (0[,]1)) /\ A.s e. (((1 / 2)[,]1) X. (0[,]1))((P |` (((1 / 2)[,]1) X. (0[,]1)))` s) = ({<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))}` s))))
241	236, 239, 240	mp2an 761	. . . . . 6 |- ((P |` (((1 / 2)[,]1) X. (0[,]1))) = {<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))} <-> ((((1 / 2)[,]1) X. (0[,]1)) = (((1 / 2)[,]1) X. (0[,]1)) /\ A.s e. (((1 / 2)[,]1) X. (0[,]1))((P |` (((1 / 2)[,]1) X. (0[,]1)))` s) = ({<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))}` s)))
242		eqidd 1885	. . . . . 6 |- (A.t e. (0[,]1)(1Mt) = (0Nt) -> (((1 / 2)[,]1) X. (0[,]1)) = (((1 / 2)[,]1) X. (0[,]1)))
243		fveq2 4681	. . . . . . . . . . . 12 |- (s = <.(1st` s), (2nd` s)>. -> (P` s) = (P` <.(1st` s), (2nd` s)>.))
244		df-opr 4886	. . . . . . . . . . . 12 |- ((1st` s)P(2nd` s)) = (P` <.(1st` s), (2nd` s)>.)
245	243, 244	syl6eqr 1946	. . . . . . . . . . 11 |- (s = <.(1st` s), (2nd` s)>. -> (P` s) = ((1st` s)P(2nd` s)))
246	245	ad2antrl 442	. . . . . . . . . 10 |- ((A.t e. (0[,]1)(1Mt) = (0Nt) /\ (s = <.(1st` s), (2nd` s)>. /\ ((1st` s) e. ((1 / 2)[,]1) /\ (2nd` s) e. (0[,]1)))) -> (P` s) = ((1st` s)P(2nd` s)))
247	62	pcohtpylem2 16081	. . . . . . . . . . . 12 |- ((((1st` s) e. ((1 / 2)[,]1) /\ (2nd` s) e. (0[,]1)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> ((1st` s)P(2nd` s)) = (((2 x. (1st` s)) - 1)N(2nd` s)))
248	247	ancoms 484	. . . . . . . . . . 11 |- ((A.t e. (0[,]1)(1Mt) = (0Nt) /\ ((1st` s) e. ((1 / 2)[,]1) /\ (2nd` s) e. (0[,]1))) -> ((1st` s)P(2nd` s)) = (((2 x. (1st` s)) - 1)N(2nd` s)))
249	248	adantrl 430	. . . . . . . . . 10 |- ((A.t e. (0[,]1)(1Mt) = (0Nt) /\ (s = <.(1st` s), (2nd` s)>. /\ ((1st` s) e. ((1 / 2)[,]1) /\ (2nd` s) e. (0[,]1)))) -> ((1st` s)P(2nd` s)) = (((2 x. (1st` s)) - 1)N(2nd` s)))
250	246, 249	eqtrd 1925	. . . . . . . . 9 |- ((A.t e. (0[,]1)(1Mt) = (0Nt) /\ (s = <.(1st` s), (2nd` s)>. /\ ((1st` s) e. ((1 / 2)[,]1) /\ (2nd` s) e. (0[,]1)))) -> (P` s) = (((2 x. (1st` s)) - 1)N(2nd` s)))
251		elxp6 5041	. . . . . . . . 9 |- (s e. (((1 / 2)[,]1) X. (0[,]1)) <-> (s = <.(1st` s), (2nd` s)>. /\ ((1st` s) e. ((1 / 2)[,]1) /\ (2nd` s) e. (0[,]1))))
252	250, 251	sylan2b 501	. . . . . . . 8 |- ((A.t e. (0[,]1)(1Mt) = (0Nt) /\ s e. (((1 / 2)[,]1) X. (0[,]1))) -> (P` s) = (((2 x. (1st` s)) - 1)N(2nd` s)))
253		fvres 4691	. . . . . . . . 9 |- (s e. (((1 / 2)[,]1) X. (0[,]1)) -> ((P |` (((1 / 2)[,]1) X. (0[,]1)))` s) = (P` s))
254	253	adantl 424	. . . . . . . 8 |- ((A.t e. (0[,]1)(1Mt) = (0Nt) /\ s e. (((1 / 2)[,]1) X. (0[,]1))) -> ((P |` (((1 / 2)[,]1) X. (0[,]1)))` s) = (P` s))
255		fveq2 4681	. . . . . . . . . . . 12 |- (s = <.(1st` s), (2nd` s)>. -> ({<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))}` s) = ({<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))}` <.(1st` s), (2nd` s)>.))
256		df-opr 4886	. . . . . . . . . . . 12 |- ((1st` s){<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))} (2nd` s)) = ({<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))}` <.(1st` s), (2nd` s)>.)
257	255, 256	syl6eqr 1946	. . . . . . . . . . 11 |- (s = <.(1st` s), (2nd` s)>. -> ({<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))}` s) = ((1st` s){<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))} (2nd` s)))
258		oprex 4907	. . . . . . . . . . . 12 |- (((2 x. (1st` s)) - 1)N(2nd` s)) e. _V
259		opreq2 4890	. . . . . . . . . . . . . 14 |- (u = (1st` s) -> (2 x. u) = (2 x. (1st` s)))
260	259	opreq1d 4897	. . . . . . . . . . . . 13 |- (u = (1st` s) -> ((2 x. u) - 1) = ((2 x. (1st` s)) - 1))
261	260	opreq1d 4897	. . . . . . . . . . . 12 |- (u = (1st` s) -> (((2 x. u) - 1)Nv) = (((2 x. (1st` s)) - 1)Nv))
262		opreq2 4890	. . . . . . . . . . . 12 |- (v = (2nd` s) -> (((2 x. (1st` s)) - 1)Nv) = (((2 x. (1st` s)) - 1)N(2nd` s)))
263	258, 261, 262, 238	oprabval2 4957	. . . . . . . . . . 11 |- (((1st` s) e. ((1 / 2)[,]1) /\ (2nd` s) e. (0[,]1)) -> ((1st` s){<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))} (2nd` s)) = (((2 x. (1st` s)) - 1)N(2nd` s)))
264	257, 263	sylan9eq 1948	. . . . . . . . . 10 |- ((s = <.(1st` s), (2nd` s)>. /\ ((1st` s) e. ((1 / 2)[,]1) /\ (2nd` s) e. (0[,]1))) -> ({<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))}` s) = (((2 x. (1st` s)) - 1)N(2nd` s)))
265	251, 264	sylbi 216	. . . . . . . . 9 |- (s e. (((1 / 2)[,]1) X. (0[,]1)) -> ({<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))}` s) = (((2 x. (1st` s)) - 1)N(2nd` s)))
266	265	adantl 424	. . . . . . . 8 |- ((A.t e. (0[,]1)(1Mt) = (0Nt) /\ s e. (((1 / 2)[,]1) X. (0[,]1))) -> ({<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))}` s) = (((2 x. (1st` s)) - 1)N(2nd` s)))
267	252, 254, 266	3eqtr4d 1937	. . . . . . 7 |- ((A.t e. (0[,]1)(1Mt) = (0Nt) /\ s e. (((1 / 2)[,]1) X. (0[,]1))) -> ((P |` (((1 / 2)[,]1) X. (0[,]1)))` s) = ({<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))}` s))
268	267	r19.21aiva 2176	. . . . . 6 |- (A.t e. (0[,]1)(1Mt) = (0Nt) -> A.s e. (((1 / 2)[,]1) X. (0[,]1))((P |` (((1 / 2)[,]1) X. (0[,]1)))` s) = ({<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))}` s))
269	241, 242, 268	sylanbrc 527	. . . . 5 |- (A.t e. (0[,]1)(1Mt) = (0Nt) -> (P |` (((1 / 2)[,]1) X. (0[,]1))) = {<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))})
270	269	3ad2ant3 899	. . . 4 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> (P |` (((1 / 2)[,]1) X. (0[,]1))) = {<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))})
271		ffn 4562	. . . . . . . 8 |- (N:((0[,]1) X. (0[,]1))-->U.J -> N Fn ((0[,]1) X. (0[,]1)))
272	89, 271	syl 12	. . . . . . 7 |- ((J e. Top /\ N e. ((II X.t II) Cn J)) -> N Fn ((0[,]1) X. (0[,]1)))
273	272	adantrl 430	. . . . . 6 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J))) -> N Fn ((0[,]1) X. (0[,]1)))
274	273	3adant3 896	. . . . 5 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> N Fn ((0[,]1) X. (0[,]1)))
275	83	adantr 425	. . . . . 6 |- ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) -> ((2 x. u) - 1) e. (0[,]1))
276		simpr 350	. . . . . 6 |- ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) -> v e. (0[,]1))
277		eqid 1884	. . . . . 6 |- {<.<.u, v>., z>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ z = <.((2 x. u) - 1), v>.)} = {<.<.u, v>., z>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ z = <.((2 x. u) - 1), v>.)}
278	275, 276, 277, 238	oprab2co 10160	. . . . 5 |- (N Fn ((0[,]1) X. (0[,]1)) -> {<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))} = (N o. {<.<.u, v>., z>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ z = <.((2 x. u) - 1), v>.)}))
279	274, 278	syl 12	. . . 4 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> {<.<.u, v>., w>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ w = (((2 x. u) - 1)Nv))} = (N o. {<.<.u, v>., z>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ z = <.((2 x. u) - 1), v>.)}))
280	270, 279	eqtrd 1925	. . 3 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> (P |` (((1 / 2)[,]1) X. (0[,]1))) = (N o. {<.<.u, v>., z>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ z = <.((2 x. u) - 1), v>.)}))
281	234, 123	sseqtr4i 2650	. . . . . . . 8 |- (((1 / 2)[,]1) X. (0[,]1)) C_ U.(II X.t II)
282		stoig3 10253	. . . . . . . 8 |- (((II X.t II) e. Top /\ (((1 / 2)[,]1) X. (0[,]1)) C_ U.(II X.t II)) -> (subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) e. Top)
283	3, 281, 282	mp2an 761	. . . . . . 7 |- (subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) e. Top
284	283	a1i 8	. . . . . 6 |- (J e. Top -> (subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) e. Top)
285	284, 128, 129	3jca 1050	. . . . 5 |- (J e. Top -> ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) e. Top /\ (II X.t II) e. Top /\ J e. Top))
286	285	3ad2ant1 897	. . . 4 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) e. Top /\ (II X.t II) e. Top /\ J e. Top))
287		simp2r 903	. . . . 5 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> N e. ((II X.t II) Cn J))
288	45, 30	sseqtri 2649	. . . . . . . 8 |- ((1 / 2)[,]1) C_ U.II
289		stoig3 10253	. . . . . . . 8 |- ((II e. Top /\ ((1 / 2)[,]1) C_ U.II) -> (subSp` <.((1 / 2)[,]1), II>.) e. Top)
290	2, 288, 289	mp2an 761	. . . . . . 7 |- (subSp` <.((1 / 2)[,]1), II>.) e. Top
291	290, 138	pm3.2i 307	. . . . . 6 |- ((subSp` <.((1 / 2)[,]1), II>.) e. Top /\ (subSp` <.(0[,]1), II>.) e. Top)
292		iccssre 7565	. . . . . . . . . 10 |- (((1 / 2) e. RR /\ 1 e. RR) -> ((1 / 2)[,]1) C_ RR)
293	12, 7, 292	mp2an 761	. . . . . . . . 9 |- ((1 / 2)[,]1) C_ RR
294	293, 143	sstri 2626	. . . . . . . 8 |- ((1 / 2)[,]1) C_ CC
295		eqid 1884	. . . . . . . . . . 11 |- {<.x, z>. | (x e. CC /\ z = (2 x. x))} = {<.x, z>. | (x e. CC /\ z = (2 x. x))}
296		mulcl 6456	. . . . . . . . . . . 12 |- ((2 e. CC /\ x e. CC) -> (2 x. x) e. CC)
297	149, 296	mpan 759	. . . . . . . . . . 11 |- (x e. CC -> (2 x. x) e. CC)
298	295, 297	fopab 4800	. . . . . . . . . 10 |- {<.x, z>. | (x e. CC /\ z = (2 x. x))}:CC-->CC
299		frn 4569	. . . . . . . . . 10 |- ({<.x, z>. | (x e. CC /\ z = (2 x. x))}:CC-->CC -> ran {<.x, z>. | (x e. CC /\ z = (2 x. x))} C_ CC)
300	298, 299	ax-mp 7	. . . . . . . . 9 |- ran {<.x, z>. | (x e. CC /\ z = (2 x. x))} C_ CC
301		oprex 4907	. . . . . . . . . 10 |- (2 x. x) e. _V
302		oprex 4907	. . . . . . . . . 10 |- (v - 1) e. _V
303		oprex 4907	. . . . . . . . . 10 |- ((2 x. x) - 1) e. _V
304		opreq1 4889	. . . . . . . . . 10 |- (v = (2 x. x) -> (v - 1) = ((2 x. x) - 1))
305		eqid 1884	. . . . . . . . . 10 |- {<.v, w>. | (v e. CC /\ w = (v - 1))} = {<.v, w>. | (v e. CC /\ w = (v - 1))}
306		eqid 1884	. . . . . . . . . 10 |- {<.x, y>. | (x e. CC /\ y = ((2 x. x) - 1))} = {<.x, y>. | (x e. CC /\ y = ((2 x. x) - 1))}
307	301, 302, 303, 304, 295, 305, 306	fopabco 4805	. . . . . . . . 9 |- (ran {<.x, z>. | (x e. CC /\ z = (2 x. x))} C_ CC -> ({<.v, w>. | (v e. CC /\ w = (v - 1))} o. {<.x, z>. | (x e. CC /\ z = (2 x. x))}) = {<.x, y>. | (x e. CC /\ y = ((2 x. x) - 1))})
308	300, 307	ax-mp 7	. . . . . . . 8 |- ({<.v, w>. | (v e. CC /\ w = (v - 1))} o. {<.x, z>. | (x e. CC /\ z = (2 x. x))}) = {<.x, y>. | (x e. CC /\ y = ((2 x. x) - 1))}
309		eqid 1884	. . . . . . . 8 |- {<.x, y>. | (x e. ((1 / 2)[,]1) /\ y = ((2 x. x) - 1))} = {<.x, y>. | (x e. ((1 / 2)[,]1) /\ y = ((2 x. x) - 1))}
310		iihalf2 15873	. . . . . . . 8 |- (x e. ((1 / 2)[,]1) -> ((2 x. x) - 1) e. (0[,]1))
311		ssid 2634	. . . . . . . . . 10 |- CC C_ CC
312	311, 311, 311	3pm3.2i 1048	. . . . . . . . 9 |- (CC C_ CC /\ CC C_ CC /\ CC C_ CC)
313	295	mulc1cncf 8541	. . . . . . . . . . 11 |- (2 e. CC -> {<.x, z>. | (x e. CC /\ z = (2 x. x))} e. (CC-cn->CC))
314	149, 313	ax-mp 7	. . . . . . . . . 10 |- {<.x, z>. | (x e. CC /\ z = (2 x. x))} e. (CC-cn->CC)
315		ax1cn 6422	. . . . . . . . . . 11 |- 1 e. CC
316	305	sub1cncf 15885	. . . . . . . . . . 11 |- (1 e. CC -> {<.v, w>. | (v e. CC /\ w = (v - 1))} e. (CC-cn->CC))
317	315, 316	ax-mp 7	. . . . . . . . . 10 |- {<.v, w>. | (v e. CC /\ w = (v - 1))} e. (CC-cn->CC)
318	314, 317	pm3.2i 307	. . . . . . . . 9 |- ({<.x, z>. | (x e. CC /\ z = (2 x. x))} e. (CC-cn->CC) /\ {<.v, w>. | (v e. CC /\ w = (v - 1))} e. (CC-cn->CC))
319		cncfco 15887	. . . . . . . . 9 |- (((CC C_ CC /\ CC C_ CC /\ CC C_ CC) /\ ({<.x, z>. | (x e. CC /\ z = (2 x. x))} e. (CC-cn->CC) /\ {<.v, w>. | (v e. CC /\ w = (v - 1))} e. (CC-cn->CC))) -> ({<.v, w>. | (v e. CC /\ w = (v - 1))} o. {<.x, z>. | (x e. CC /\ z = (2 x. x))}) e. (CC-cn->CC))
320	312, 318, 319	mp2an 761	. . . . . . . 8 |- ({<.v, w>. | (v e. CC /\ w = (v - 1))} o. {<.x, z>. | (x e. CC /\ z = (2 x. x))}) e. (CC-cn->CC)
321		eqid 1884	. . . . . . . . . . 11 |- (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` (((1 / 2)[,]1) X. ((1 / 2)[,]1))) = (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` (((1 / 2)[,]1) X. ((1 / 2)[,]1)))
322		eqid 1884	. . . . . . . . . . 11 |- (Open` (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` (((1 / 2)[,]1) X. ((1 / 2)[,]1)))) = (Open` (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` (((1 / 2)[,]1) X. ((1 / 2)[,]1))))
323	321, 159, 160, 322	subtopmet 10256	. . . . . . . . . 10 |- ((((abs o. - ) |` ((0[,]1) X. (0[,]1))) e. Met /\ ((1 / 2)[,]1) C_ (0[,]1)) -> (subSp` <.((1 / 2)[,]1), II>.) = (Open` (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` (((1 / 2)[,]1) X. ((1 / 2)[,]1)))))
324	155, 45, 323	mp2an 761	. . . . . . . . 9 |- (subSp` <.((1 / 2)[,]1), II>.) = (Open` (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` (((1 / 2)[,]1) X. ((1 / 2)[,]1))))
325		xpss12 4089	. . . . . . . . . . . 12 |- ((((1 / 2)[,]1) C_ (0[,]1) /\ ((1 / 2)[,]1) C_ (0[,]1)) -> (((1 / 2)[,]1) X. ((1 / 2)[,]1)) C_ ((0[,]1) X. (0[,]1)))
326	45, 45, 325	mp2an 761	. . . . . . . . . . 11 |- (((1 / 2)[,]1) X. ((1 / 2)[,]1)) C_ ((0[,]1) X. (0[,]1))
327		resabs1 4244	. . . . . . . . . . 11 |- ((((1 / 2)[,]1) X. ((1 / 2)[,]1)) C_ ((0[,]1) X. (0[,]1)) -> (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` (((1 / 2)[,]1) X. ((1 / 2)[,]1))) = ((abs o. - ) |` (((1 / 2)[,]1) X. ((1 / 2)[,]1))))
328	326, 327	ax-mp 7	. . . . . . . . . 10 |- (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` (((1 / 2)[,]1) X. ((1 / 2)[,]1))) = ((abs o. - ) |` (((1 / 2)[,]1) X. ((1 / 2)[,]1)))
329	328	fveq2i 4684	. . . . . . . . 9 |- (Open` (((abs o. - ) |` ((0[,]1) X. (0[,]1))) |` (((1 / 2)[,]1) X. ((1 / 2)[,]1)))) = (Open` ((abs o. - ) |` (((1 / 2)[,]1) X. ((1 / 2)[,]1))))
330	324, 329	eqtri 1908	. . . . . . . 8 |- (subSp` <.((1 / 2)[,]1), II>.) = (Open` ((abs o. - ) |` (((1 / 2)[,]1) X. ((1 / 2)[,]1))))
331	294, 145, 308, 309, 310, 320, 330, 160	cncfres 15895	. . . . . . 7 |- {<.x, y>. | (x e. ((1 / 2)[,]1) /\ y = ((2 x. x) - 1))} e. ((subSp` <.((1 / 2)[,]1), II>.) Cn II)
332	331, 177	pm3.2i 307	. . . . . 6 |- ({<.x, y>. | (x e. ((1 / 2)[,]1) /\ y = ((2 x. x) - 1))} e. ((subSp` <.((1 / 2)[,]1), II>.) Cn II) /\ ( _I |` (0[,]1)) e. ((subSp` <.(0[,]1), II>.) Cn II))
333	30, 30	txsubsp 15912	. . . . . . . 8 |- (((II e. Top /\ II e. Top) /\ (((1 / 2)[,]1) C_ (0[,]1) /\ (0[,]1) C_ (0[,]1))) -> (subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) = ((subSp` <.((1 / 2)[,]1), II>.) X.t (subSp` <.(0[,]1), II>.)))
334	2, 2, 45, 119, 333	mp4an 15651	. . . . . . 7 |- (subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) = ((subSp` <.((1 / 2)[,]1), II>.) X.t (subSp` <.(0[,]1), II>.))
335		stoig2 10252	. . . . . . . . 9 |- ((II e. Top /\ ((1 / 2)[,]1) C_ U.II) -> U.(subSp` <.((1 / 2)[,]1), II>.) = ((1 / 2)[,]1))
336	2, 288, 335	mp2an 761	. . . . . . . 8 |- U.(subSp` <.((1 / 2)[,]1), II>.) = ((1 / 2)[,]1)
337	336	eqcomi 1888	. . . . . . 7 |- ((1 / 2)[,]1) = U.(subSp` <.((1 / 2)[,]1), II>.)
338	187	opreq1d 4897	. . . . . . . . . . . . . 14 |- (x = u -> ((2 x. x) - 1) = ((2 x. u) - 1))
339		oprex 4907	. . . . . . . . . . . . . 14 |- ((2 x. u) - 1) e. _V
340	338, 309, 339	fvopab4 4743	. . . . . . . . . . . . 13 |- (u e. ((1 / 2)[,]1) -> ({<.x, y>. | (x e. ((1 / 2)[,]1) /\ y = ((2 x. x) - 1))}` u) = ((2 x. u) - 1))
341	340	adantr 425	. . . . . . . . . . . 12 |- ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) -> ({<.x, y>. | (x e. ((1 / 2)[,]1) /\ y = ((2 x. x) - 1))}` u) = ((2 x. u) - 1))
342	191	adantl 424	. . . . . . . . . . . 12 |- ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) -> (( _I |` (0[,]1))` v) = v)
343	341, 342	opeq12d 3166	. . . . . . . . . . 11 |- ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) -> <.({<.x, y>. | (x e. ((1 / 2)[,]1) /\ y = ((2 x. x) - 1))}` u), (( _I |` (0[,]1))` v)>. = <.((2 x. u) - 1), v>.)
344	343	eqcomd 1889	. . . . . . . . . 10 |- ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) -> <.((2 x. u) - 1), v>. = <.({<.x, y>. | (x e. ((1 / 2)[,]1) /\ y = ((2 x. x) - 1))}` u), (( _I |` (0[,]1))` v)>.)
345	344	eqeq2d 1895	. . . . . . . . 9 |- ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) -> (z = <.((2 x. u) - 1), v>. <-> z = <.({<.x, y>. | (x e. ((1 / 2)[,]1) /\ y = ((2 x. x) - 1))}` u), (( _I |` (0[,]1))` v)>.))
346	345	pm5.32i 707	. . . . . . . 8 |- (((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ z = <.((2 x. u) - 1), v>.) <-> ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ z = <.({<.x, y>. | (x e. ((1 / 2)[,]1) /\ y = ((2 x. x) - 1))}` u), (( _I |` (0[,]1))` v)>.))
347	346	oprabbii 4923	. . . . . . 7 |- {<.<.u, v>., z>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ z = <.((2 x. u) - 1), v>.)} = {<.<.u, v>., z>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ z = <.({<.x, y>. | (x e. ((1 / 2)[,]1) /\ y = ((2 x. x) - 1))}` u), (( _I |` (0[,]1))` v)>.)}
348	334, 33, 337, 186, 347	2txcn 10229	. . . . . 6 |- ((((subSp` <.((1 / 2)[,]1), II>.) e. Top /\ (subSp` <.(0[,]1), II>.) e. Top) /\ (II e. Top /\ II e. Top) /\ ({<.x, y>. | (x e. ((1 / 2)[,]1) /\ y = ((2 x. x) - 1))} e. ((subSp` <.((1 / 2)[,]1), II>.) Cn II) /\ ( _I |` (0[,]1)) e. ((subSp` <.(0[,]1), II>.) Cn II))) -> {<.<.u, v>., z>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ z = <.((2 x. u) - 1), v>.)} e. ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) Cn (II X.t II)))
349	291, 140, 332, 348	mp3an 1191	. . . . 5 |- {<.<.u, v>., z>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ z = <.((2 x. u) - 1), v>.)} e. ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) Cn (II X.t II))
350	287, 349	jctil 316	. . . 4 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> ({<.<.u, v>., z>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ z = <.((2 x. u) - 1), v>.)} e. ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) Cn (II X.t II)) /\ N e. ((II X.t II) Cn J)))
351		cnco 9045	. . . 4 |- ((((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) e. Top /\ (II X.t II) e. Top /\ J e. Top) /\ ({<.<.u, v>., z>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ z = <.((2 x. u) - 1), v>.)} e. ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) Cn (II X.t II)) /\ N e. ((II X.t II) Cn J))) -> (N o. {<.<.u, v>., z>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ z = <.((2 x. u) - 1), v>.)}) e. ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) Cn J))
352	286, 350, 351	syl11anc 524	. . 3 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> (N o. {<.<.u, v>., z>. | ((u e. ((1 / 2)[,]1) /\ v e. (0[,]1)) /\ z = <.((2 x. u) - 1), v>.)}) e. ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) Cn J))
353	280, 352	eqeltrd 1971	. 2 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> (P |` (((1 / 2)[,]1) X. (0[,]1))) e. ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) Cn J))
354	67, 68	paste 15893	. 2 |- ((((II X.t II) e. Top /\ J e. Top) /\ (((0[,](1 / 2)) X. (0[,]1)) e. (Clsd` (II X.t II)) /\ (((1 / 2)[,]1) X. (0[,]1)) e. (Clsd` (II X.t II)) /\ (((0[,](1 / 2)) X. (0[,]1)) u. (((1 / 2)[,]1) X. (0[,]1))) = ((0[,]1) X. (0[,]1))) /\ (P:((0[,]1) X. (0[,]1))-->U.J /\ (P |` ((0[,](1 / 2)) X. (0[,]1))) e. ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) Cn J) /\ (P |` (((1 / 2)[,]1) X. (0[,]1))) e. ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) Cn J))) -> P e. ((II X.t II) Cn J))
355	4, 36, 51, 61, 108, 232, 353, 354	syl133anc 1120	1 |- ((J e. Top /\ (M e. ((II X.t II) Cn J) /\ N e. ((II X.t II) Cn J)) /\ A.t e. (0[,]1)(1Mt) = (0Nt)) -> P e. ((II X.t II) Cn J))

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   u. cun 2591   C_ wss 2593  ifcif 2982  <.cop 3046  U.cuni 3177   class class class wbr 3338  {copab 3395   _I cid 3582   X. cxp 3984  dom cdm 3986  ran crn 3987   |` cres 3988   o. ccom 3990   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  1stc1st 5018  2ndc2nd 5019  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   x. cmul 6391   - cmin 6445   / cdiv 6447   <_ cle 6448  2c2 7145  (,)cioo 7524  [,]cicc 7527  abscabs 8000  -cn->ccncf 8524  Topctop 8857  topGenctg 8860   X.t ctx 8930  Clsdccld 8936   Cn ccn 9028  Metcme 9066  Opencopn 9069  subSpcsubsp 10242  IIcii 15865

This theorem is referenced by:  pcohtpy 16083

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

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-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-rp 7232  df-n0 7309  df-z 7345  df-q 7436  df-ioo 7528  df-icc 7531  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  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-subsp 10243  df-ii 15866

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