Theorem phtpycolem5 16055

Description: Lemma for phtpyco 16056.

Hypothesis

Ref	Expression
phtpyco.1	|- M = {<.<.x, y>., z>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ z = if(y <_ (1 / 2), (xK(2 x. y)), (xL((2 x. y) - 1))))}

Assertion

Ref	Expression
phtpycolem5	|- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) -> M e. ((II X.t II) Cn J))

Distinct variable groups:   w,J,x,y,z   w,L,x,y,z   w,M   w,K,x,y,z

Proof of Theorem phtpycolem5

Step	Hyp	Ref	Expression
1		simpll 448	. . 3 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) -> J e. Top)
2		iitop 15867	. . . 4 |- II e. Top
3		eqid 1884	. . . . 5 |- (II X.t II) = (II X.t II)
4	3	txtop 8934	. . . 4 |- ((II e. Top /\ II e. Top) -> (II X.t II) e. Top)
5	2, 2, 4	mp2an 761	. . 3 |- (II X.t II) e. Top
6	1, 5	jctil 316	. 2 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) -> ((II X.t II) e. Top /\ J e. Top))
7		iiuni 15868	. . . . . . 7 |- (0[,]1) = U.II
8	7	topcld 8951	. . . . . 6 |- (II e. Top -> (0[,]1) e. (Clsd` II))
9	2, 8	ax-mp 7	. . . . 5 |- (0[,]1) e. (Clsd` II)
10		retop 8926	. . . . . . 7 |- (topGen` ran (,)) e. Top
11		0re 6603	. . . . . . . 8 |- 0 e. RR
12		1re 6598	. . . . . . . 8 |- 1 e. RR
13		iccssre 7565	. . . . . . . 8 |- ((0 e. RR /\ 1 e. RR) -> (0[,]1) C_ RR)
14	11, 12, 13	mp2an 761	. . . . . . 7 |- (0[,]1) C_ RR
15		2re 7163	. . . . . . . . 9 |- 2 e. RR
16		2ne0 7174	. . . . . . . . 9 |- 2 =/= 0
17	15, 16	rereccli 6979	. . . . . . . 8 |- (1 / 2) e. RR
18		clint3 10184	. . . . . . . 8 |- ((0 e. RR /\ (1 / 2) e. RR) -> (0[,](1 / 2)) e. (Clsd` (topGen` ran (,))))
19	11, 17, 18	mp2an 761	. . . . . . 7 |- (0[,](1 / 2)) e. (Clsd` (topGen` ran (,)))
20	11, 12	pm3.2i 307	. . . . . . . 8 |- (0 e. RR /\ 1 e. RR)
21	11, 17	pm3.2i 307	. . . . . . . 8 |- (0 e. RR /\ (1 / 2) e. RR)
22	11	leidi 6790	. . . . . . . . 9 |- 0 <_ 0
23		halflt1 7216	. . . . . . . . . 10 |- (1 / 2) < 1
24	17, 12, 23	ltleii 6756	. . . . . . . . 9 |- (1 / 2) <_ 1
25	22, 24	pm3.2i 307	. . . . . . . 8 |- (0 <_ 0 /\ (1 / 2) <_ 1)
26		iccss 15855	. . . . . . . 8 |- (((0 e. RR /\ 1 e. RR) /\ (0 e. RR /\ (1 / 2) e. RR) /\ (0 <_ 0 /\ (1 / 2) <_ 1)) -> (0[,](1 / 2)) C_ (0[,]1))
27	20, 21, 25, 26	mp3an 1191	. . . . . . 7 |- (0[,](1 / 2)) C_ (0[,]1)
28		uniretop 8927	. . . . . . . . 9 |- U.(topGen` ran (,)) = RR
29	28	eqcomi 1888	. . . . . . . 8 |- RR = U.(topGen` ran (,))
30	29	subspcld 15838	. . . . . . 7 |- ((((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 (,))>.)))
31	10, 14, 19, 27, 30	mp4an 15651	. . . . . 6 |- (0[,](1 / 2)) e. (Clsd` (subSp` <.(0[,]1), (topGen` ran (,))>.))
32		dfii2 15869	. . . . . . 7 |- II = (subSp` <.(0[,]1), (topGen` ran (,))>.)
33	32	fveq2i 4684	. . . . . 6 |- (Clsd` II) = (Clsd` (subSp` <.(0[,]1), (topGen` ran (,))>.))
34	31, 33	eleqtrri 1970	. . . . 5 |- (0[,](1 / 2)) e. (Clsd` II)
35	3	txcld 15914	. . . . 5 |- (((II e. Top /\ II e. Top) /\ ((0[,]1) e. (Clsd` II) /\ (0[,](1 / 2)) e. (Clsd` II))) -> ((0[,]1) X. (0[,](1 / 2))) e. (Clsd` (II X.t II)))
36	2, 2, 9, 34, 35	mp4an 15651	. . . 4 |- ((0[,]1) X. (0[,](1 / 2))) e. (Clsd` (II X.t II))
37		clint3 10184	. . . . . . . 8 |- (((1 / 2) e. RR /\ 1 e. RR) -> ((1 / 2)[,]1) e. (Clsd` (topGen` ran (,))))
38	17, 12, 37	mp2an 761	. . . . . . 7 |- ((1 / 2)[,]1) e. (Clsd` (topGen` ran (,)))
39	17, 12	pm3.2i 307	. . . . . . . 8 |- ((1 / 2) e. RR /\ 1 e. RR)
40		halfgt0 7215	. . . . . . . . . 10 |- 0 < (1 / 2)
41	11, 17, 40	ltleii 6756	. . . . . . . . 9 |- 0 <_ (1 / 2)
42	12	leidi 6790	. . . . . . . . 9 |- 1 <_ 1
43	41, 42	pm3.2i 307	. . . . . . . 8 |- (0 <_ (1 / 2) /\ 1 <_ 1)
44		iccss 15855	. . . . . . . 8 |- (((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	20, 39, 43, 44	mp3an 1191	. . . . . . 7 |- ((1 / 2)[,]1) C_ (0[,]1)
46	29	subspcld 15838	. . . . . . 7 |- ((((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	10, 14, 38, 45, 46	mp4an 15651	. . . . . 6 |- ((1 / 2)[,]1) e. (Clsd` (subSp` <.(0[,]1), (topGen` ran (,))>.))
48	47, 33	eleqtrri 1970	. . . . 5 |- ((1 / 2)[,]1) e. (Clsd` II)
49	3	txcld 15914	. . . . 5 |- (((II e. Top /\ II e. Top) /\ ((0[,]1) e. (Clsd` II) /\ ((1 / 2)[,]1) e. (Clsd` II))) -> ((0[,]1) X. ((1 / 2)[,]1)) e. (Clsd` (II X.t II)))
50	2, 2, 9, 48, 49	mp4an 15651	. . . 4 |- ((0[,]1) X. ((1 / 2)[,]1)) e. (Clsd` (II X.t II))
51		xpundi 4050	. . . . 5 |- ((0[,]1) X. ((0[,](1 / 2)) u. ((1 / 2)[,]1))) = (((0[,]1) X. (0[,](1 / 2))) u. ((0[,]1) X. ((1 / 2)[,]1)))
52		elicc2 7560	. . . . . . . . . 10 |- ((0 e. RR /\ 1 e. RR) -> ((1 / 2) e. (0[,]1) <-> ((1 / 2) e. RR /\ 0 <_ (1 / 2) /\ (1 / 2) <_ 1)))
53	11, 12, 52	mp2an 761	. . . . . . . . 9 |- ((1 / 2) e. (0[,]1) <-> ((1 / 2) e. RR /\ 0 <_ (1 / 2) /\ (1 / 2) <_ 1))
54	53, 17, 41, 24	mpbir3an 1052	. . . . . . . 8 |- (1 / 2) e. (0[,]1)
55		iccsplit 15854	. . . . . . . 8 |- ((0 e. RR /\ 1 e. RR /\ (1 / 2) e. (0[,]1)) -> (0[,]1) = ((0[,](1 / 2)) u. ((1 / 2)[,]1)))
56	11, 12, 54, 55	mp3an 1191	. . . . . . 7 |- (0[,]1) = ((0[,](1 / 2)) u. ((1 / 2)[,]1))
57	56	eqcomi 1888	. . . . . 6 |- ((0[,](1 / 2)) u. ((1 / 2)[,]1)) = (0[,]1)
58	57	xpeq2i 4022	. . . . 5 |- ((0[,]1) X. ((0[,](1 / 2)) u. ((1 / 2)[,]1))) = ((0[,]1) X. (0[,]1))
59	51, 58	eqtr3i 1910	. . . 4 |- (((0[,]1) X. (0[,](1 / 2))) u. ((0[,]1) X. ((1 / 2)[,]1))) = ((0[,]1) X. (0[,]1))
60	36, 50, 59	3pm3.2i 1048	. . 3 |- (((0[,]1) X. (0[,](1 / 2))) e. (Clsd` (II X.t II)) /\ ((0[,]1) X. ((1 / 2)[,]1)) e. (Clsd` (II X.t II)) /\ (((0[,]1) X. (0[,](1 / 2))) u. ((0[,]1) X. ((1 / 2)[,]1))) = ((0[,]1) X. (0[,]1)))
61	60	a1i 8	. 2 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) -> (((0[,]1) X. (0[,](1 / 2))) e. (Clsd` (II X.t II)) /\ ((0[,]1) X. ((1 / 2)[,]1)) e. (Clsd` (II X.t II)) /\ (((0[,]1) X. (0[,](1 / 2))) u. ((0[,]1) X. ((1 / 2)[,]1))) = ((0[,]1) X. (0[,]1))))
62		fveq2 4681	. . . . . . . . . . 11 |- (v = <.(1st` v), (2nd` v)>. -> (M` v) = (M` <.(1st` v), (2nd` v)>.))
63		df-opr 4886	. . . . . . . . . . 11 |- ((1st` v)M(2nd` v)) = (M` <.(1st` v), (2nd` v)>.)
64	62, 63	syl6eqr 1946	. . . . . . . . . 10 |- (v = <.(1st` v), (2nd` v)>. -> (M` v) = ((1st` v)M(2nd` v)))
65	64	eleq1d 1963	. . . . . . . . 9 |- (v = <.(1st` v), (2nd` v)>. -> ((M` v) e. U.J <-> ((1st` v)M(2nd` v)) e. U.J))
66		phtpyco.1	. . . . . . . . . . . . . . 15 |- M = {<.<.x, y>., z>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ z = if(y <_ (1 / 2), (xK(2 x. y)), (xL((2 x. y) - 1))))}
67	66	phtpycolem1 16051	. . . . . . . . . . . . . 14 |- (((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,](1 / 2))) -> ((1st` v)M(2nd` v)) = ((1st` v)K(2 x. (2nd` v))))
68	67	adantll 428	. . . . . . . . . . . . 13 |- (((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) /\ (1st` v) e. (0[,]1)) /\ (2nd` v) e. (0[,](1 / 2))) -> ((1st` v)M(2nd` v)) = ((1st` v)K(2 x. (2nd` v))))
69		ffvelrn 4787	. . . . . . . . . . . . . . . . . . . 20 |- ((K:((0[,]1) X. (0[,]1))-->U.J /\ <.(1st` v), (2 x. (2nd` v))>. e. ((0[,]1) X. (0[,]1))) -> (K` <.(1st` v), (2 x. (2nd` v))>.) e. U.J)
70		df-opr 4886	. . . . . . . . . . . . . . . . . . . 20 |- ((1st` v)K(2 x. (2nd` v))) = (K` <.(1st` v), (2 x. (2nd` v))>.)
71	69, 70	syl5eqel 1975	. . . . . . . . . . . . . . . . . . 19 |- ((K:((0[,]1) X. (0[,]1))-->U.J /\ <.(1st` v), (2 x. (2nd` v))>. e. ((0[,]1) X. (0[,]1))) -> ((1st` v)K(2 x. (2nd` v))) e. U.J)
72		oprex 4907	. . . . . . . . . . . . . . . . . . . 20 |- (2 x. (2nd` v)) e. _V
73	72	opelxp 4036	. . . . . . . . . . . . . . . . . . 19 |- (<.(1st` v), (2 x. (2nd` v))>. e. ((0[,]1) X. (0[,]1)) <-> ((1st` v) e. (0[,]1) /\ (2 x. (2nd` v)) e. (0[,]1)))
74	71, 73	sylan2br 502	. . . . . . . . . . . . . . . . . 18 |- ((K:((0[,]1) X. (0[,]1))-->U.J /\ ((1st` v) e. (0[,]1) /\ (2 x. (2nd` v)) e. (0[,]1))) -> ((1st` v)K(2 x. (2nd` v))) e. U.J)
75		iihalf1 15872	. . . . . . . . . . . . . . . . . 18 |- ((2nd` v) e. (0[,](1 / 2)) -> (2 x. (2nd` v)) e. (0[,]1))
76	74, 75	sylanr2 512	. . . . . . . . . . . . . . . . 17 |- ((K:((0[,]1) X. (0[,]1))-->U.J /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,](1 / 2)))) -> ((1st` v)K(2 x. (2nd` v))) e. U.J)
77	3, 7, 7	txuni 8935	. . . . . . . . . . . . . . . . . . . . 21 |- ((II e. Top /\ II e. Top) -> U.(II X.t II) = ((0[,]1) X. (0[,]1)))
78	2, 2, 77	mp2an 761	. . . . . . . . . . . . . . . . . . . 20 |- U.(II X.t II) = ((0[,]1) X. (0[,]1))
79	78	eqcomi 1888	. . . . . . . . . . . . . . . . . . 19 |- ((0[,]1) X. (0[,]1)) = U.(II X.t II)
80		eqid 1884	. . . . . . . . . . . . . . . . . . 19 |- U.J = U.J
81	79, 80	cnf 9038	. . . . . . . . . . . . . . . . . 18 |- (((II X.t II) e. Top /\ J e. Top /\ K e. ((II X.t II) Cn J)) -> K:((0[,]1) X. (0[,]1))-->U.J)
82	5, 81	mp3an1 1178	. . . . . . . . . . . . . . . . 17 |- ((J e. Top /\ K e. ((II X.t II) Cn J)) -> K:((0[,]1) X. (0[,]1))-->U.J)
83	76, 82	sylan 497	. . . . . . . . . . . . . . . 16 |- (((J e. Top /\ K e. ((II X.t II) Cn J)) /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,](1 / 2)))) -> ((1st` v)K(2 x. (2nd` v))) e. U.J)
84	83	adantlrr 435	. . . . . . . . . . . . . . 15 |- (((J e. Top /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,](1 / 2)))) -> ((1st` v)K(2 x. (2nd` v))) e. U.J)
85	84	adantllr 433	. . . . . . . . . . . . . 14 |- ((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,](1 / 2)))) -> ((1st` v)K(2 x. (2nd` v))) e. U.J)
86	85	anassrs 489	. . . . . . . . . . . . 13 |- (((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) /\ (1st` v) e. (0[,]1)) /\ (2nd` v) e. (0[,](1 / 2))) -> ((1st` v)K(2 x. (2nd` v))) e. U.J)
87	68, 86	eqeltrd 1971	. . . . . . . . . . . 12 |- (((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) /\ (1st` v) e. (0[,]1)) /\ (2nd` v) e. (0[,](1 / 2))) -> ((1st` v)M(2nd` v)) e. U.J)
88	66	phtpycolem2 16052	. . . . . . . . . . . . . . . 16 |- ((A.w e. (0[,]1)(wK1) = (wL0) /\ (1st` v) e. (0[,]1) /\ (2nd` v) e. ((1 / 2)[,]1)) -> ((1st` v)M(2nd` v)) = ((1st` v)L((2 x. (2nd` v)) - 1)))
89	88	3expa 1067	. . . . . . . . . . . . . . 15 |- (((A.w e. (0[,]1)(wK1) = (wL0) /\ (1st` v) e. (0[,]1)) /\ (2nd` v) e. ((1 / 2)[,]1)) -> ((1st` v)M(2nd` v)) = ((1st` v)L((2 x. (2nd` v)) - 1)))
90	89	adantlll 432	. . . . . . . . . . . . . 14 |- ((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (1st` v) e. (0[,]1)) /\ (2nd` v) e. ((1 / 2)[,]1)) -> ((1st` v)M(2nd` v)) = ((1st` v)L((2 x. (2nd` v)) - 1)))
91	90	adantllr 433	. . . . . . . . . . . . 13 |- (((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) /\ (1st` v) e. (0[,]1)) /\ (2nd` v) e. ((1 / 2)[,]1)) -> ((1st` v)M(2nd` v)) = ((1st` v)L((2 x. (2nd` v)) - 1)))
92		ffvelrn 4787	. . . . . . . . . . . . . . . . . . . 20 |- ((L:((0[,]1) X. (0[,]1))-->U.J /\ <.(1st` v), ((2 x. (2nd` v)) - 1)>. e. ((0[,]1) X. (0[,]1))) -> (L` <.(1st` v), ((2 x. (2nd` v)) - 1)>.) e. U.J)
93		df-opr 4886	. . . . . . . . . . . . . . . . . . . 20 |- ((1st` v)L((2 x. (2nd` v)) - 1)) = (L` <.(1st` v), ((2 x. (2nd` v)) - 1)>.)
94	92, 93	syl5eqel 1975	. . . . . . . . . . . . . . . . . . 19 |- ((L:((0[,]1) X. (0[,]1))-->U.J /\ <.(1st` v), ((2 x. (2nd` v)) - 1)>. e. ((0[,]1) X. (0[,]1))) -> ((1st` v)L((2 x. (2nd` v)) - 1)) e. U.J)
95		oprex 4907	. . . . . . . . . . . . . . . . . . . 20 |- ((2 x. (2nd` v)) - 1) e. _V
96	95	opelxp 4036	. . . . . . . . . . . . . . . . . . 19 |- (<.(1st` v), ((2 x. (2nd` v)) - 1)>. e. ((0[,]1) X. (0[,]1)) <-> ((1st` v) e. (0[,]1) /\ ((2 x. (2nd` v)) - 1) e. (0[,]1)))
97	94, 96	sylan2br 502	. . . . . . . . . . . . . . . . . 18 |- ((L:((0[,]1) X. (0[,]1))-->U.J /\ ((1st` v) e. (0[,]1) /\ ((2 x. (2nd` v)) - 1) e. (0[,]1))) -> ((1st` v)L((2 x. (2nd` v)) - 1)) e. U.J)
98		iihalf2 15873	. . . . . . . . . . . . . . . . . 18 |- ((2nd` v) e. ((1 / 2)[,]1) -> ((2 x. (2nd` v)) - 1) e. (0[,]1))
99	97, 98	sylanr2 512	. . . . . . . . . . . . . . . . 17 |- ((L:((0[,]1) X. (0[,]1))-->U.J /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. ((1 / 2)[,]1))) -> ((1st` v)L((2 x. (2nd` v)) - 1)) e. U.J)
100	79, 80	cnf 9038	. . . . . . . . . . . . . . . . . 18 |- (((II X.t II) e. Top /\ J e. Top /\ L e. ((II X.t II) Cn J)) -> L:((0[,]1) X. (0[,]1))-->U.J)
101	5, 100	mp3an1 1178	. . . . . . . . . . . . . . . . 17 |- ((J e. Top /\ L e. ((II X.t II) Cn J)) -> L:((0[,]1) X. (0[,]1))-->U.J)
102	99, 101	sylan 497	. . . . . . . . . . . . . . . 16 |- (((J e. Top /\ L e. ((II X.t II) Cn J)) /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. ((1 / 2)[,]1))) -> ((1st` v)L((2 x. (2nd` v)) - 1)) e. U.J)
103	102	adantlrl 434	. . . . . . . . . . . . . . 15 |- (((J e. Top /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. ((1 / 2)[,]1))) -> ((1st` v)L((2 x. (2nd` v)) - 1)) e. U.J)
104	103	adantllr 433	. . . . . . . . . . . . . 14 |- ((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. ((1 / 2)[,]1))) -> ((1st` v)L((2 x. (2nd` v)) - 1)) e. U.J)
105	104	anassrs 489	. . . . . . . . . . . . 13 |- (((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) /\ (1st` v) e. (0[,]1)) /\ (2nd` v) e. ((1 / 2)[,]1)) -> ((1st` v)L((2 x. (2nd` v)) - 1)) e. U.J)
106	91, 105	eqeltrd 1971	. . . . . . . . . . . 12 |- (((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) /\ (1st` v) e. (0[,]1)) /\ (2nd` v) e. ((1 / 2)[,]1)) -> ((1st` v)M(2nd` v)) e. U.J)
107	87, 106	jaodan 471	. . . . . . . . . . 11 |- (((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) /\ (1st` v) e. (0[,]1)) /\ ((2nd` v) e. (0[,](1 / 2)) \/ (2nd` v) e. ((1 / 2)[,]1))) -> ((1st` v)M(2nd` v)) e. U.J)
108	56	eleq2i 1961	. . . . . . . . . . . 12 |- ((2nd` v) e. (0[,]1) <-> (2nd` v) e. ((0[,](1 / 2)) u. ((1 / 2)[,]1)))
109		elun 2741	. . . . . . . . . . . 12 |- ((2nd` v) e. ((0[,](1 / 2)) u. ((1 / 2)[,]1)) <-> ((2nd` v) e. (0[,](1 / 2)) \/ (2nd` v) e. ((1 / 2)[,]1)))
110	108, 109	bitri 190	. . . . . . . . . . 11 |- ((2nd` v) e. (0[,]1) <-> ((2nd` v) e. (0[,](1 / 2)) \/ (2nd` v) e. ((1 / 2)[,]1)))
111	107, 110	sylan2b 501	. . . . . . . . . 10 |- (((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) /\ (1st` v) e. (0[,]1)) /\ (2nd` v) e. (0[,]1)) -> ((1st` v)M(2nd` v)) e. U.J)
112	111	anasss 488	. . . . . . . . 9 |- ((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,]1))) -> ((1st` v)M(2nd` v)) e. U.J)
113	65, 112	syl5cbir 228	. . . . . . . 8 |- ((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,]1))) -> (v = <.(1st` v), (2nd` v)>. -> (M` v) e. U.J))
114	113	impr 422	. . . . . . 7 |- ((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) /\ (((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,]1)) /\ v = <.(1st` v), (2nd` v)>.)) -> (M` v) e. U.J)
115	114	ancom2s 545	. . . . . 6 |- ((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) /\ (v = <.(1st` v), (2nd` v)>. /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,]1)))) -> (M` v) e. U.J)
116		elxp6 5041	. . . . . 6 |- (v e. ((0[,]1) X. (0[,]1)) <-> (v = <.(1st` v), (2nd` v)>. /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,]1))))
117	115, 116	sylan2b 501	. . . . 5 |- ((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) /\ v e. ((0[,]1) X. (0[,]1))) -> (M` v) e. U.J)
118	117	r19.21aiva 2176	. . . 4 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) -> A.v e. ((0[,]1) X. (0[,]1))(M` v) e. U.J)
119		oprex 4907	. . . . . 6 |- (xK(2 x. y)) e. _V
120		oprex 4907	. . . . . 6 |- (xL((2 x. y) - 1)) e. _V
121	119, 120	ifex 3031	. . . . 5 |- if(y <_ (1 / 2), (xK(2 x. y)), (xL((2 x. y) - 1))) e. _V
122	121, 66	fnoprab2 5064	. . . 4 |- M Fn ((0[,]1) X. (0[,]1))
123	118, 122	jctil 316	. . 3 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) -> (M Fn ((0[,]1) X. (0[,]1)) /\ A.v e. ((0[,]1) X. (0[,]1))(M` v) e. U.J))
124		ffnfv 4801	. . 3 |- (M:((0[,]1) X. (0[,]1))-->U.J <-> (M Fn ((0[,]1) X. (0[,]1)) /\ A.v e. ((0[,]1) X. (0[,]1))(M` v) e. U.J))
125	123, 124	sylibr 217	. 2 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) -> M:((0[,]1) X. (0[,]1))-->U.J)
126	66	phtpycolem3 16053	. . 3 |- ((J e. Top /\ K e. ((II X.t II) Cn J)) -> (M |` ((0[,]1) X. (0[,](1 / 2)))) e. ((subSp` <.((0[,]1) X. (0[,](1 / 2))), (II X.t II)>.) Cn J))
127	126	ad2ant2r 445	. 2 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) -> (M |` ((0[,]1) X. (0[,](1 / 2)))) e. ((subSp` <.((0[,]1) X. (0[,](1 / 2))), (II X.t II)>.) Cn J))
128	66	phtpycolem4 16054	. . . . 5 |- ((J e. Top /\ L e. ((II X.t II) Cn J) /\ A.w e. (0[,]1)(wK1) = (wL0)) -> (M |` ((0[,]1) X. ((1 / 2)[,]1))) e. ((subSp` <.((0[,]1) X. ((1 / 2)[,]1)), (II X.t II)>.) Cn J))
129	128	3expa 1067	. . . 4 |- (((J e. Top /\ L e. ((II X.t II) Cn J)) /\ A.w e. (0[,]1)(wK1) = (wL0)) -> (M |` ((0[,]1) X. ((1 / 2)[,]1))) e. ((subSp` <.((0[,]1) X. ((1 / 2)[,]1)), (II X.t II)>.) Cn J))
130	129	an1rs 547	. . 3 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ L e. ((II X.t II) Cn J)) -> (M |` ((0[,]1) X. ((1 / 2)[,]1))) e. ((subSp` <.((0[,]1) X. ((1 / 2)[,]1)), (II X.t II)>.) Cn J))
131	130	adantrl 430	. 2 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) -> (M |` ((0[,]1) X. ((1 / 2)[,]1))) e. ((subSp` <.((0[,]1) X. ((1 / 2)[,]1)), (II X.t II)>.) Cn J))
132	79, 80	paste 15893	. 2 |- ((((II X.t II) e. Top /\ J e. Top) /\ (((0[,]1) X. (0[,](1 / 2))) e. (Clsd` (II X.t II)) /\ ((0[,]1) X. ((1 / 2)[,]1)) e. (Clsd` (II X.t II)) /\ (((0[,]1) X. (0[,](1 / 2))) u. ((0[,]1) X. ((1 / 2)[,]1))) = ((0[,]1) X. (0[,]1))) /\ (M:((0[,]1) X. (0[,]1))-->U.J /\ (M |` ((0[,]1) X. (0[,](1 / 2)))) e. ((subSp` <.((0[,]1) X. (0[,](1 / 2))), (II X.t II)>.) Cn J) /\ (M |` ((0[,]1) X. ((1 / 2)[,]1))) e. ((subSp` <.((0[,]1) X. ((1 / 2)[,]1)), (II X.t II)>.) Cn J))) -> M e. ((II X.t II) Cn J))
133	6, 61, 125, 127, 131, 132	syl113anc 1112	1 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((II X.t II) Cn J) /\ L e. ((II X.t II) Cn J))) -> M 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   X. cxp 3984  ran crn 3987   |` cres 3988   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  1stc1st 5018  2ndc2nd 5019  RRcr 6385  0cc0 6386  1c1 6387   x. cmul 6391   - cmin 6445   / cdiv 6447   <_ cle 6448  2c2 7145  (,)cioo 7524  [,]cicc 7527  Topctop 8857  topGenctg 8860   X.t ctx 8930  Clsdccld 8936   Cn ccn 9028  subSpcsubsp 10242  IIcii 15865

This theorem is referenced by:  phtpyco 16056

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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