Theorem phtpycolem5 12097

Description: Lemma for phtpyco 12098.

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. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) -> M e. ((IItXII) Cn J))

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

Proof of Theorem phtpycolem5

Step	Hyp	Ref	Expression
1		iitop 11932	. . . . 5 |- II e. Top
2		eqid 1518	. . . . . 6 |- (IItXII) = (IItXII)
3		iiuni 11933	. . . . . . 7 |- U.II = (0[,]1)
4	3	eqcomi 1522	. . . . . 6 |- (0[,]1) = U.II
5	2, 4, 4	txuni 11975	. . . . 5 |- ((II e. Top /\ II e. Top) -> U.(IItXII) = ((0[,]1) X. (0[,]1)))
6	1, 1, 5	mp2an 701	. . . 4 |- U.(IItXII) = ((0[,]1) X. (0[,]1))
7	6	eqcomi 1522	. . 3 |- ((0[,]1) X. (0[,]1)) = U.(IItXII)
8		eqid 1518	. . 3 |- U.J = U.J
9	7, 8	paste 11954	. 2 |- ((((IItXII) e. Top /\ J e. Top) /\ (((0[,]1) X. (0[,](1 / 2))) e. (Clsd` (IItXII)) /\ ((0[,]1) X. ((1 / 2)[,]1)) e. (Clsd` (IItXII)) /\ (((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))), (IItXII)>.) Cn J) /\ (M |` ((0[,]1) X. ((1 / 2)[,]1))) e. ((subSp` <.((0[,]1) X. ((1 / 2)[,]1)), (IItXII)>.) Cn J))) -> M e. ((IItXII) Cn J))
10		simpll 412	. . 3 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) -> J e. Top)
11	2	txtop 11974	. . . 4 |- ((II e. Top /\ II e. Top) -> (IItXII) e. Top)
12	1, 1, 11	mp2an 701	. . 3 |- (IItXII) e. Top
13	10, 12	jctil 290	. 2 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) -> ((IItXII) e. Top /\ J e. Top))
14	1, 1	pm3.2i 283	. . . . 5 |- (II e. Top /\ II e. Top)
15	4	topcld 7885	. . . . . . 7 |- (II e. Top -> (0[,]1) e. (Clsd` II))
16	1, 15	ax-mp 7	. . . . . 6 |- (0[,]1) e. (Clsd` II)
17		retop 7866	. . . . . . . . 9 |- (topGen` ran (,)) e. Top
18		0re 5594	. . . . . . . . . 10 |- 0 e. RR
19		1re 5589	. . . . . . . . . 10 |- 1 e. RR
20		iccssre 6523	. . . . . . . . . 10 |- ((0 e. RR /\ 1 e. RR) -> (0[,]1) (_ RR)
21	18, 19, 20	mp2an 701	. . . . . . . . 9 |- (0[,]1) (_ RR
22	17, 21	pm3.2i 283	. . . . . . . 8 |- ((topGen` ran (,)) e. Top /\ (0[,]1) (_ RR)
23		2re 6125	. . . . . . . . . . 11 |- 2 e. RR
24		2ne0 6136	. . . . . . . . . . 11 |- 2 =/= 0
25	23, 24	rereccli 5941	. . . . . . . . . 10 |- (1 / 2) e. RR
26		clint3 11002	. . . . . . . . . 10 |- ((0 e. RR /\ (1 / 2) e. RR) -> (0[,](1 / 2)) e. (Clsd` (topGen` ran (,))))
27	18, 25, 26	mp2an 701	. . . . . . . . 9 |- (0[,](1 / 2)) e. (Clsd` (topGen` ran (,)))
28	18, 19	pm3.2i 283	. . . . . . . . . 10 |- (0 e. RR /\ 1 e. RR)
29	18, 25	pm3.2i 283	. . . . . . . . . 10 |- (0 e. RR /\ (1 / 2) e. RR)
30	18	leidi 5764	. . . . . . . . . . 11 |- 0 <_ 0
31		halflt1 6176	. . . . . . . . . . . 12 |- (1 / 2) < 1
32	25, 19, 31	ltleii 5735	. . . . . . . . . . 11 |- (1 / 2) <_ 1
33	30, 32	pm3.2i 283	. . . . . . . . . 10 |- (0 <_ 0 /\ (1 / 2) <_ 1)
34		iccss 11920	. . . . . . . . . 10 |- (((0 e. RR /\ 1 e. RR) /\ (0 e. RR /\ (1 / 2) e. RR) /\ (0 <_ 0 /\ (1 / 2) <_ 1)) -> (0[,](1 / 2)) (_ (0[,]1))
35	28, 29, 33, 34	mp3an 922	. . . . . . . . 9 |- (0[,](1 / 2)) (_ (0[,]1)
36	27, 35	pm3.2i 283	. . . . . . . 8 |- ((0[,](1 / 2)) e. (Clsd` (topGen` ran (,))) /\ (0[,](1 / 2)) (_ (0[,]1))
37		uniretop 7867	. . . . . . . . . 10 |- U.(topGen` ran (,)) = RR
38	37	eqcomi 1522	. . . . . . . . 9 |- RR = U.(topGen` ran (,))
39	38	subspcld 11901	. . . . . . . 8 |- ((((topGen` ran (,)) e. Top /\ (0[,]1) (_ RR) /\ ((0[,](1 / 2)) e. (Clsd` (topGen` ran (,))) /\ (0[,](1 / 2)) (_ (0[,]1))) -> (0[,](1 / 2)) e. (Clsd` (subSp` <.(0[,]1), (topGen` ran (,))>.)))
40	22, 36, 39	mp2an 701	. . . . . . 7 |- (0[,](1 / 2)) e. (Clsd` (subSp` <.(0[,]1), (topGen` ran (,))>.))
41		dfii2 11934	. . . . . . . 8 |- II = (subSp` <.(0[,]1), (topGen` ran (,))>.)
42	41	fveq2i 3838	. . . . . . 7 |- (Clsd` II) = (Clsd` (subSp` <.(0[,]1), (topGen` ran (,))>.))
43	40, 42	eleqtrri 1590	. . . . . 6 |- (0[,](1 / 2)) e. (Clsd` II)
44	16, 43	pm3.2i 283	. . . . 5 |- ((0[,]1) e. (Clsd` II) /\ (0[,](1 / 2)) e. (Clsd` II))
45	2	txcld 11985	. . . . 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` (IItXII)))
46	14, 44, 45	mp2an 701	. . . 4 |- ((0[,]1) X. (0[,](1 / 2))) e. (Clsd` (IItXII))
47		clint3 11002	. . . . . . . . . 10 |- (((1 / 2) e. RR /\ 1 e. RR) -> ((1 / 2)[,]1) e. (Clsd` (topGen` ran (,))))
48	25, 19, 47	mp2an 701	. . . . . . . . 9 |- ((1 / 2)[,]1) e. (Clsd` (topGen` ran (,)))
49	25, 19	pm3.2i 283	. . . . . . . . . 10 |- ((1 / 2) e. RR /\ 1 e. RR)
50		halfgt0 6175	. . . . . . . . . . . 12 |- 0 < (1 / 2)
51	18, 25, 50	ltleii 5735	. . . . . . . . . . 11 |- 0 <_ (1 / 2)
52	19	leidi 5764	. . . . . . . . . . 11 |- 1 <_ 1
53	51, 52	pm3.2i 283	. . . . . . . . . 10 |- (0 <_ (1 / 2) /\ 1 <_ 1)
54		iccss 11920	. . . . . . . . . 10 |- (((0 e. RR /\ 1 e. RR) /\ ((1 / 2) e. RR /\ 1 e. RR) /\ (0 <_ (1 / 2) /\ 1 <_ 1)) -> ((1 / 2)[,]1) (_ (0[,]1))
55	28, 49, 53, 54	mp3an 922	. . . . . . . . 9 |- ((1 / 2)[,]1) (_ (0[,]1)
56	48, 55	pm3.2i 283	. . . . . . . 8 |- (((1 / 2)[,]1) e. (Clsd` (topGen` ran (,))) /\ ((1 / 2)[,]1) (_ (0[,]1))
57	38	subspcld 11901	. . . . . . . 8 |- ((((topGen` ran (,)) e. Top /\ (0[,]1) (_ RR) /\ (((1 / 2)[,]1) e. (Clsd` (topGen` ran (,))) /\ ((1 / 2)[,]1) (_ (0[,]1))) -> ((1 / 2)[,]1) e. (Clsd` (subSp` <.(0[,]1), (topGen` ran (,))>.)))
58	22, 56, 57	mp2an 701	. . . . . . 7 |- ((1 / 2)[,]1) e. (Clsd` (subSp` <.(0[,]1), (topGen` ran (,))>.))
59	58, 42	eleqtrri 1590	. . . . . 6 |- ((1 / 2)[,]1) e. (Clsd` II)
60	16, 59	pm3.2i 283	. . . . 5 |- ((0[,]1) e. (Clsd` II) /\ ((1 / 2)[,]1) e. (Clsd` II))
61	2	txcld 11985	. . . . 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` (IItXII)))
62	14, 60, 61	mp2an 701	. . . 4 |- ((0[,]1) X. ((1 / 2)[,]1)) e. (Clsd` (IItXII))
63		xpundi 3310	. . . . 5 |- ((0[,]1) X. ((0[,](1 / 2)) u. ((1 / 2)[,]1))) = (((0[,]1) X. (0[,](1 / 2))) u. ((0[,]1) X. ((1 / 2)[,]1)))
64	25, 51, 32	3pm3.2i 824	. . . . . . . . 9 |- ((1 / 2) e. RR /\ 0 <_ (1 / 2) /\ (1 / 2) <_ 1)
65		elicc2 6518	. . . . . . . . . 10 |- ((0 e. RR /\ 1 e. RR) -> ((1 / 2) e. (0[,]1) <-> ((1 / 2) e. RR /\ 0 <_ (1 / 2) /\ (1 / 2) <_ 1)))
66	18, 19, 65	mp2an 701	. . . . . . . . 9 |- ((1 / 2) e. (0[,]1) <-> ((1 / 2) e. RR /\ 0 <_ (1 / 2) /\ (1 / 2) <_ 1))
67	64, 66	mpbir 188	. . . . . . . 8 |- (1 / 2) e. (0[,]1)
68		iccsplit 11919	. . . . . . . 8 |- ((0 e. RR /\ 1 e. RR /\ (1 / 2) e. (0[,]1)) -> (0[,]1) = ((0[,](1 / 2)) u. ((1 / 2)[,]1)))
69	18, 19, 67, 68	mp3an 922	. . . . . . 7 |- (0[,]1) = ((0[,](1 / 2)) u. ((1 / 2)[,]1))
70	69	eqcomi 1522	. . . . . 6 |- ((0[,](1 / 2)) u. ((1 / 2)[,]1)) = (0[,]1)
71	70	xpeq2i 3286	. . . . 5 |- ((0[,]1) X. ((0[,](1 / 2)) u. ((1 / 2)[,]1))) = ((0[,]1) X. (0[,]1))
72	63, 71	eqtr3i 1540	. . . 4 |- (((0[,]1) X. (0[,](1 / 2))) u. ((0[,]1) X. ((1 / 2)[,]1))) = ((0[,]1) X. (0[,]1))
73	46, 62, 72	3pm3.2i 824	. . 3 |- (((0[,]1) X. (0[,](1 / 2))) e. (Clsd` (IItXII)) /\ ((0[,]1) X. ((1 / 2)[,]1)) e. (Clsd` (IItXII)) /\ (((0[,]1) X. (0[,](1 / 2))) u. ((0[,]1) X. ((1 / 2)[,]1))) = ((0[,]1) X. (0[,]1)))
74	73	a1i 8	. 2 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) -> (((0[,]1) X. (0[,](1 / 2))) e. (Clsd` (IItXII)) /\ ((0[,]1) X. ((1 / 2)[,]1)) e. (Clsd` (IItXII)) /\ (((0[,]1) X. (0[,](1 / 2))) u. ((0[,]1) X. ((1 / 2)[,]1))) = ((0[,]1) X. (0[,]1))))
75		fveq2 3835	. . . . . . . . . . . 12 |- (v = <.(1st` v), (2nd` v)>. -> (M` v) = (M` <.(1st` v), (2nd` v)>.))
76		df-opr 4023	. . . . . . . . . . . 12 |- ((1st` v)M(2nd` v)) = (M` <.(1st` v), (2nd` v)>.)
77	75, 76	syl6eqr 1568	. . . . . . . . . . 11 |- (v = <.(1st` v), (2nd` v)>. -> (M` v) = ((1st` v)M(2nd` v)))
78	77	eleq1d 1583	. . . . . . . . . 10 |- (v = <.(1st` v), (2nd` v)>. -> ((M` v) e. U.J <-> ((1st` v)M(2nd` v)) e. U.J))
79		phtpyco.1	. . . . . . . . . . . . . . . 16 |- 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))))}
80	79	phtpycolem1 12093	. . . . . . . . . . . . . . 15 |- (((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,](1 / 2))) -> ((1st` v)M(2nd` v)) = ((1st` v)K(2 x. (2nd` v))))
81	80	adantll 392	. . . . . . . . . . . . . 14 |- (((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) 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))))
82		ffvelrn 3928	. . . . . . . . . . . . . . . . . . . . 21 |- ((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)
83		df-opr 4023	. . . . . . . . . . . . . . . . . . . . 21 |- ((1st` v)K(2 x. (2nd` v))) = (K` <.(1st` v), (2 x. (2nd` v))>.)
84	82, 83	syl5eqel 1595	. . . . . . . . . . . . . . . . . . . 20 |- ((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)
85		oprex 4041	. . . . . . . . . . . . . . . . . . . . 21 |- (2 x. (2nd` v)) e. V
86	85	opelxp 3297	. . . . . . . . . . . . . . . . . . . 20 |- (<.(1st` v), (2 x. (2nd` v))>. e. ((0[,]1) X. (0[,]1)) <-> ((1st` v) e. (0[,]1) /\ (2 x. (2nd` v)) e. (0[,]1)))
87	84, 86	sylan2br 455	. . . . . . . . . . . . . . . . . . 19 |- ((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)
88		iihalf1 11936	. . . . . . . . . . . . . . . . . . 19 |- ((2nd` v) e. (0[,](1 / 2)) -> (2 x. (2nd` v)) e. (0[,]1))
89	87, 88	sylanr2 465	. . . . . . . . . . . . . . . . . 18 |- ((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)
90	7, 8	cnf 7972	. . . . . . . . . . . . . . . . . . 19 |- (((IItXII) e. Top /\ J e. Top /\ K e. ((IItXII) Cn J)) -> K:((0[,]1) X. (0[,]1))-->U.J)
91	12, 90	mp3an1 909	. . . . . . . . . . . . . . . . . 18 |- ((J e. Top /\ K e. ((IItXII) Cn J)) -> K:((0[,]1) X. (0[,]1))-->U.J)
92	89, 91	sylan 450	. . . . . . . . . . . . . . . . 17 |- (((J e. Top /\ K e. ((IItXII) Cn J)) /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,](1 / 2)))) -> ((1st` v)K(2 x. (2nd` v))) e. U.J)
93	92	adantlrr 399	. . . . . . . . . . . . . . . 16 |- (((J e. Top /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,](1 / 2)))) -> ((1st` v)K(2 x. (2nd` v))) e. U.J)
94	93	adantllr 397	. . . . . . . . . . . . . . 15 |- ((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,](1 / 2)))) -> ((1st` v)K(2 x. (2nd` v))) e. U.J)
95	94	anassrs 443	. . . . . . . . . . . . . 14 |- (((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) /\ (1st` v) e. (0[,]1)) /\ (2nd` v) e. (0[,](1 / 2))) -> ((1st` v)K(2 x. (2nd` v))) e. U.J)
96	81, 95	eqeltrd 1591	. . . . . . . . . . . . 13 |- (((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) /\ (1st` v) e. (0[,]1)) /\ (2nd` v) e. (0[,](1 / 2))) -> ((1st` v)M(2nd` v)) e. U.J)
97	79	phtpycolem2 12094	. . . . . . . . . . . . . . . . 17 |- ((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)))
98	97	3expa 839	. . . . . . . . . . . . . . . 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)))
99	98	adantlll 396	. . . . . . . . . . . . . . 15 |- ((((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)))
100	99	adantllr 397	. . . . . . . . . . . . . 14 |- (((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) 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)))
101		ffvelrn 3928	. . . . . . . . . . . . . . . . . . . . 21 |- ((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)
102		df-opr 4023	. . . . . . . . . . . . . . . . . . . . 21 |- ((1st` v)L((2 x. (2nd` v)) - 1)) = (L` <.(1st` v), ((2 x. (2nd` v)) - 1)>.)
103	101, 102	syl5eqel 1595	. . . . . . . . . . . . . . . . . . . 20 |- ((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)
104		oprex 4041	. . . . . . . . . . . . . . . . . . . . 21 |- ((2 x. (2nd` v)) - 1) e. V
105	104	opelxp 3297	. . . . . . . . . . . . . . . . . . . 20 |- (<.(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)))
106	103, 105	sylan2br 455	. . . . . . . . . . . . . . . . . . 19 |- ((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)
107		iihalf2 11937	. . . . . . . . . . . . . . . . . . 19 |- ((2nd` v) e. ((1 / 2)[,]1) -> ((2 x. (2nd` v)) - 1) e. (0[,]1))
108	106, 107	sylanr2 465	. . . . . . . . . . . . . . . . . 18 |- ((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)
109	7, 8	cnf 7972	. . . . . . . . . . . . . . . . . . 19 |- (((IItXII) e. Top /\ J e. Top /\ L e. ((IItXII) Cn J)) -> L:((0[,]1) X. (0[,]1))-->U.J)
110	12, 109	mp3an1 909	. . . . . . . . . . . . . . . . . 18 |- ((J e. Top /\ L e. ((IItXII) Cn J)) -> L:((0[,]1) X. (0[,]1))-->U.J)
111	108, 110	sylan 450	. . . . . . . . . . . . . . . . 17 |- (((J e. Top /\ L e. ((IItXII) Cn J)) /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. ((1 / 2)[,]1))) -> ((1st` v)L((2 x. (2nd` v)) - 1)) e. U.J)
112	111	adantlrl 398	. . . . . . . . . . . . . . . 16 |- (((J e. Top /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. ((1 / 2)[,]1))) -> ((1st` v)L((2 x. (2nd` v)) - 1)) e. U.J)
113	112	adantllr 397	. . . . . . . . . . . . . . 15 |- ((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. ((1 / 2)[,]1))) -> ((1st` v)L((2 x. (2nd` v)) - 1)) e. U.J)
114	113	anassrs 443	. . . . . . . . . . . . . 14 |- (((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) /\ (1st` v) e. (0[,]1)) /\ (2nd` v) e. ((1 / 2)[,]1)) -> ((1st` v)L((2 x. (2nd` v)) - 1)) e. U.J)
115	100, 114	eqeltrd 1591	. . . . . . . . . . . . 13 |- (((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) /\ (1st` v) e. (0[,]1)) /\ (2nd` v) e. ((1 / 2)[,]1)) -> ((1st` v)M(2nd` v)) e. U.J)
116	96, 115	jaodan 426	. . . . . . . . . . . 12 |- (((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) 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)
117	69	eleq2i 1581	. . . . . . . . . . . . 13 |- ((2nd` v) e. (0[,]1) <-> (2nd` v) e. ((0[,](1 / 2)) u. ((1 / 2)[,]1)))
118		elun 2225	. . . . . . . . . . . . 13 |- ((2nd` v) e. ((0[,](1 / 2)) u. ((1 / 2)[,]1)) <-> ((2nd` v) e. (0[,](1 / 2)) \/ (2nd` v) e. ((1 / 2)[,]1)))
119	117, 118	bitri 171	. . . . . . . . . . . 12 |- ((2nd` v) e. (0[,]1) <-> ((2nd` v) e. (0[,](1 / 2)) \/ (2nd` v) e. ((1 / 2)[,]1)))
120	116, 119	sylan2b 454	. . . . . . . . . . 11 |- (((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) /\ (1st` v) e. (0[,]1)) /\ (2nd` v) e. (0[,]1)) -> ((1st` v)M(2nd` v)) e. U.J)
121	120	anasss 442	. . . . . . . . . 10 |- ((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,]1))) -> ((1st` v)M(2nd` v)) e. U.J)
122	78, 121	syl5cbir 209	. . . . . . . . 9 |- ((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,]1))) -> (v = <.(1st` v), (2nd` v)>. -> (M` v) e. U.J))
123	122	impr 11359	. . . . . . . 8 |- ((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) /\ (((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,]1)) /\ v = <.(1st` v), (2nd` v)>.)) -> (M` v) e. U.J)
124	123	ancom2s 490	. . . . . . 7 |- ((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) /\ (v = <.(1st` v), (2nd` v)>. /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,]1)))) -> (M` v) e. U.J)
125		elxp6 4161	. . . . . . 7 |- (v e. ((0[,]1) X. (0[,]1)) <-> (v = <.(1st` v), (2nd` v)>. /\ ((1st` v) e. (0[,]1) /\ (2nd` v) e. (0[,]1))))
126	124, 125	sylan2b 454	. . . . . 6 |- ((((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) /\ v e. ((0[,]1) X. (0[,]1))) -> (M` v) e. U.J)
127	126	r19.21aiva 1760	. . . . 5 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) -> A.v e. ((0[,]1) X. (0[,]1))(M` v) e. U.J)
128		oprex 4041	. . . . . . 7 |- (xK(2 x. y)) e. V
129		oprex 4041	. . . . . . 7 |- (xL((2 x. y) - 1)) e. V
130	128, 129	ifex 2457	. . . . . 6 |- if(y <_ (1 / 2), (xK(2 x. y)), (xL((2 x. y) - 1))) e. V
131	130, 79	fnoprab2 4184	. . . . 5 |- M Fn ((0[,]1) X. (0[,]1))
132	127, 131	jctil 290	. . . 4 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) -> (M Fn ((0[,]1) X. (0[,]1)) /\ A.v e. ((0[,]1) X. (0[,]1))(M` v) e. U.J))
133		ffnfv 3942	. . . 4 |- (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))
134	132, 133	sylibr 198	. . 3 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) -> M:((0[,]1) X. (0[,]1))-->U.J)
135	79	phtpycolem3 12095	. . . 4 |- ((J e. Top /\ K e. ((IItXII) Cn J)) -> (M |` ((0[,]1) X. (0[,](1 / 2)))) e. ((subSp` <.((0[,]1) X. (0[,](1 / 2))), (IItXII)>.) Cn J))
136	135	ad2ant2r 409	. . 3 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) -> (M |` ((0[,]1) X. (0[,](1 / 2)))) e. ((subSp` <.((0[,]1) X. (0[,](1 / 2))), (IItXII)>.) Cn J))
137	79	phtpycolem4 12096	. . . . . 6 |- ((J e. Top /\ L e. ((IItXII) Cn J) /\ A.w e. (0[,]1)(wK1) = (wL0)) -> (M |` ((0[,]1) X. ((1 / 2)[,]1))) e. ((subSp` <.((0[,]1) X. ((1 / 2)[,]1)), (IItXII)>.) Cn J))
138	137	3expa 839	. . . . 5 |- (((J e. Top /\ L e. ((IItXII) Cn J)) /\ A.w e. (0[,]1)(wK1) = (wL0)) -> (M |` ((0[,]1) X. ((1 / 2)[,]1))) e. ((subSp` <.((0[,]1) X. ((1 / 2)[,]1)), (IItXII)>.) Cn J))
139	138	an1rs 492	. . . 4 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ L e. ((IItXII) Cn J)) -> (M |` ((0[,]1) X. ((1 / 2)[,]1))) e. ((subSp` <.((0[,]1) X. ((1 / 2)[,]1)), (IItXII)>.) Cn J))
140	139	adantrl 394	. . 3 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) -> (M |` ((0[,]1) X. ((1 / 2)[,]1))) e. ((subSp` <.((0[,]1) X. ((1 / 2)[,]1)), (IItXII)>.) Cn J))
141	134, 136, 140	3jca 825	. 2 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) -> (M:((0[,]1) X. (0[,]1))-->U.J /\ (M |` ((0[,]1) X. (0[,](1 / 2)))) e. ((subSp` <.((0[,]1) X. (0[,](1 / 2))), (IItXII)>.) Cn J) /\ (M |` ((0[,]1) X. ((1 / 2)[,]1))) e. ((subSp` <.((0[,]1) X. ((1 / 2)[,]1)), (IItXII)>.) Cn J)))
142	9, 13, 74, 141	syl3anc 864	1 |- (((J e. Top /\ A.w e. (0[,]1)(wK1) = (wL0)) /\ (K e. ((IItXII) Cn J) /\ L e. ((IItXII) Cn J))) -> M e. ((IItXII) Cn J))

Syntax hints:   -> wi 3   <-> wb 144   \/ wo 220   /\ wa 221   /\ w3a 781   = wceq 992   e. wcel 994  A.wral 1691   u. cun 2097   (_ wss 2099  ifcif 2415  <.cop 2469  U.cuni 2569   class class class wbr 2692   X. cxp 3249  ran crn 3252   |` cres 3253   Fn wfn 3258  -->wf 3259  ` cfv 3263  (class class class)co 4021  {copab2 4022  1stc1st 4138  2ndc2nd 4139  RRcr 5387  0cc0 5388  1c1 5389   x. cmul 5393   - cmin 5446   / cdiv 5448   <_ cle 5449  2c2 6107  (,)cioo 6483  [,]cicc 6486  Topctop 7800  topGenctg 7803  Clsdccld 7870   Cn ccn 7962  subSpcsubsp 11054  IIcii 11930  tXctx 11970

This theorem is referenced by:  phtpyco 12098

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-reg 4736  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-nel 1631  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-iin 2636  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-map 4465  df-en 4509  df-dom 4510  df-sdom 4511  df-sup 4717  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-1p 5241  df-plp 5242  df-mp 5243  df-ltp 5244  df-plpr 5318  df-mpr 5319  df-enr 5320  df-nr 5321  df-plr 5322  df-mr 5323  df-ltr 5324  df-0r 5325  df-1r 5326  df-m1r 5327  df-c 5394  df-0 5395  df-1 5396  df-i 5397  df-r 5398  df-plus 5399  df-mul 5400  df-lt 5401  df-sub 5510  df-neg 5512  df-pnf 5641  df-mnf 5642  df-xr 5643  df-ltxr 5644  df-le 5645  df-div 5855  df-n 6070  df-2 6116  df-rp 6191  df-n0 6268  df-z 6304  df-q 6395  df-ioo 6487  df-icc 6490  df-seq1 6673  df-exp 6764  df-sqr 6871  df-re 6952  df-im 6953  df-cj 6954  df-abs 6955  df-cncf 7468  df-top 7804  df-topsp 7805  df-bases 7806  df-topgen 7807  df-cld 7873  df-cn 7964  df-cnp 7965  df-met 8003  df-bl 8005  df-opn 8006  df-subsp 11055  df-ii 11931  df-tx 11971

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