Theorem pcorevlem 16086

Description: Lemma for pcorev 16087. Prove continuity of the homotopy function.

Hypothesis

Ref	Expression
pcorevlem.1	|- H = {<.<.s, t>., u>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ u = if(s <_ (1 / 2), (F` (1 - ((1 - t) x. (2 x. s)))), (F` (t + ((1 - t) x. ((2 x. s) - 1))))))}

Assertion

Ref	Expression
pcorevlem	|- ((J e. Top /\ F e. (II Cn J)) -> H e. ((II X.t II) Cn J))

Distinct variable groups:   J,s,t,u   F,s,t,u

Proof of Theorem pcorevlem

Step	Hyp	Ref	Expression
1		iitop 15867	. . . 4 |- II e. Top
2		iiuni 15868	. . . . 5 |- (0[,]1) = U.II
3		eqid 1884	. . . . 5 |- U.J = U.J
4	2, 3	cnf 9038	. . . 4 |- ((II e. Top /\ J e. Top /\ F e. (II Cn J)) -> F:(0[,]1)-->U.J)
5	1, 4	mp3an1 1178	. . 3 |- ((J e. Top /\ F e. (II Cn J)) -> F:(0[,]1)-->U.J)
6		ffn 4562	. . 3 |- (F:(0[,]1)-->U.J -> F Fn (0[,]1))
7		iimulcl 15874	. . . . . . . . 9 |- (((1 - t) e. (0[,]1) /\ (2 x. s) e. (0[,]1)) -> ((1 - t) x. (2 x. s)) e. (0[,]1))
8		iirev 15871	. . . . . . . . 9 |- (((1 - t) x. (2 x. s)) e. (0[,]1) -> (1 - ((1 - t) x. (2 x. s))) e. (0[,]1))
9	7, 8	syl 12	. . . . . . . 8 |- (((1 - t) e. (0[,]1) /\ (2 x. s) e. (0[,]1)) -> (1 - ((1 - t) x. (2 x. s))) e. (0[,]1))
10	9	ancoms 484	. . . . . . 7 |- (((2 x. s) e. (0[,]1) /\ (1 - t) e. (0[,]1)) -> (1 - ((1 - t) x. (2 x. s))) e. (0[,]1))
11		0re 6603	. . . . . . . . . . . . 13 |- 0 e. RR
12		1re 6598	. . . . . . . . . . . . 13 |- 1 e. RR
13		iccssre 7565	. . . . . . . . . . . . 13 |- ((0 e. RR /\ 1 e. RR) -> (0[,]1) C_ RR)
14	11, 12, 13	mp2an 761	. . . . . . . . . . . 12 |- (0[,]1) C_ RR
15	14	sseli 2617	. . . . . . . . . . 11 |- (s e. (0[,]1) -> s e. RR)
16	15	adantr 425	. . . . . . . . . 10 |- ((s e. (0[,]1) /\ s <_ (1 / 2)) -> s e. RR)
17		elicc2 7560	. . . . . . . . . . . . 13 |- ((0 e. RR /\ 1 e. RR) -> (s e. (0[,]1) <-> (s e. RR /\ 0 <_ s /\ s <_ 1)))
18	11, 12, 17	mp2an 761	. . . . . . . . . . . 12 |- (s e. (0[,]1) <-> (s e. RR /\ 0 <_ s /\ s <_ 1))
19	18	simp2bi 892	. . . . . . . . . . 11 |- (s e. (0[,]1) -> 0 <_ s)
20	19	adantr 425	. . . . . . . . . 10 |- ((s e. (0[,]1) /\ s <_ (1 / 2)) -> 0 <_ s)
21		simpr 350	. . . . . . . . . 10 |- ((s e. (0[,]1) /\ s <_ (1 / 2)) -> s <_ (1 / 2))
22	16, 20, 21	3jca 1050	. . . . . . . . 9 |- ((s e. (0[,]1) /\ s <_ (1 / 2)) -> (s e. RR /\ 0 <_ s /\ s <_ (1 / 2)))
23		2re 7163	. . . . . . . . . . 11 |- 2 e. RR
24		2ne0 7174	. . . . . . . . . . 11 |- 2 =/= 0
25	23, 24	rereccli 6979	. . . . . . . . . 10 |- (1 / 2) e. RR
26		elicc2 7560	. . . . . . . . . 10 |- ((0 e. RR /\ (1 / 2) e. RR) -> (s e. (0[,](1 / 2)) <-> (s e. RR /\ 0 <_ s /\ s <_ (1 / 2))))
27	11, 25, 26	mp2an 761	. . . . . . . . 9 |- (s e. (0[,](1 / 2)) <-> (s e. RR /\ 0 <_ s /\ s <_ (1 / 2)))
28	22, 27	sylibr 217	. . . . . . . 8 |- ((s e. (0[,]1) /\ s <_ (1 / 2)) -> s e. (0[,](1 / 2)))
29		iihalf1 15872	. . . . . . . 8 |- (s e. (0[,](1 / 2)) -> (2 x. s) e. (0[,]1))
30	28, 29	syl 12	. . . . . . 7 |- ((s e. (0[,]1) /\ s <_ (1 / 2)) -> (2 x. s) e. (0[,]1))
31		iirev 15871	. . . . . . 7 |- (t e. (0[,]1) -> (1 - t) e. (0[,]1))
32	10, 30, 31	syl2an 503	. . . . . 6 |- (((s e. (0[,]1) /\ s <_ (1 / 2)) /\ t e. (0[,]1)) -> (1 - ((1 - t) x. (2 x. s))) e. (0[,]1))
33	32	an1rs 547	. . . . 5 |- (((s e. (0[,]1) /\ t e. (0[,]1)) /\ s <_ (1 / 2)) -> (1 - ((1 - t) x. (2 x. s))) e. (0[,]1))
34		simpr 350	. . . . . . . . . . 11 |- ((((2 x. s) - 1) e. CC /\ t e. CC) -> t e. CC)
35		mulcl 6456	. . . . . . . . . . . . 13 |- (((1 - t) e. CC /\ ((2 x. s) - 1) e. CC) -> ((1 - t) x. ((2 x. s) - 1)) e. CC)
36		ax1cn 6422	. . . . . . . . . . . . . 14 |- 1 e. CC
37		subcl 6524	. . . . . . . . . . . . . 14 |- ((1 e. CC /\ t e. CC) -> (1 - t) e. CC)
38	36, 37	mpan 759	. . . . . . . . . . . . 13 |- (t e. CC -> (1 - t) e. CC)
39	35, 38	sylan 497	. . . . . . . . . . . 12 |- ((t e. CC /\ ((2 x. s) - 1) e. CC) -> ((1 - t) x. ((2 x. s) - 1)) e. CC)
40	39	ancoms 484	. . . . . . . . . . 11 |- ((((2 x. s) - 1) e. CC /\ t e. CC) -> ((1 - t) x. ((2 x. s) - 1)) e. CC)
41		addcom 6458	. . . . . . . . . . 11 |- ((t e. CC /\ ((1 - t) x. ((2 x. s) - 1)) e. CC) -> (t + ((1 - t) x. ((2 x. s) - 1))) = (((1 - t) x. ((2 x. s) - 1)) + t))
42	34, 40, 41	syl11anc 524	. . . . . . . . . 10 |- ((((2 x. s) - 1) e. CC /\ t e. CC) -> (t + ((1 - t) x. ((2 x. s) - 1))) = (((1 - t) x. ((2 x. s) - 1)) + t))
43		ax1id 6435	. . . . . . . . . . . 12 |- (t e. CC -> (t x. 1) = t)
44	43	adantl 424	. . . . . . . . . . 11 |- ((((2 x. s) - 1) e. CC /\ t e. CC) -> (t x. 1) = t)
45	44	opreq2d 4898	. . . . . . . . . 10 |- ((((2 x. s) - 1) e. CC /\ t e. CC) -> (((1 - t) x. ((2 x. s) - 1)) + (t x. 1)) = (((1 - t) x. ((2 x. s) - 1)) + t))
46	42, 45	eqtr4d 1928	. . . . . . . . 9 |- ((((2 x. s) - 1) e. CC /\ t e. CC) -> (t + ((1 - t) x. ((2 x. s) - 1))) = (((1 - t) x. ((2 x. s) - 1)) + (t x. 1)))
47		axresscn 6420	. . . . . . . . . . 11 |- RR C_ CC
48	14, 47	sstri 2626	. . . . . . . . . 10 |- (0[,]1) C_ CC
49	48	sseli 2617	. . . . . . . . 9 |- (((2 x. s) - 1) e. (0[,]1) -> ((2 x. s) - 1) e. CC)
50	48	sseli 2617	. . . . . . . . 9 |- (t e. (0[,]1) -> t e. CC)
51	46, 49, 50	syl2an 503	. . . . . . . 8 |- ((((2 x. s) - 1) e. (0[,]1) /\ t e. (0[,]1)) -> (t + ((1 - t) x. ((2 x. s) - 1))) = (((1 - t) x. ((2 x. s) - 1)) + (t x. 1)))
52		lt01 6871	. . . . . . . . . . 11 |- 0 < 1
53	11, 12, 52	ltleii 6756	. . . . . . . . . 10 |- 0 <_ 1
54		ubicc2 7574	. . . . . . . . . 10 |- ((0 e. RR /\ 1 e. RR /\ 0 <_ 1) -> 1 e. (0[,]1))
55	11, 12, 53, 54	mp3an 1191	. . . . . . . . 9 |- 1 e. (0[,]1)
56		lincmb01icc 15879	. . . . . . . . . 10 |- ((0 e. RR /\ 1 e. RR) -> ((((2 x. s) - 1) e. (0[,]1) /\ 1 e. (0[,]1) /\ t e. (0[,]1)) -> (((1 - t) x. ((2 x. s) - 1)) + (t x. 1)) e. (0[,]1)))
57	11, 12, 56	mp2an 761	. . . . . . . . 9 |- ((((2 x. s) - 1) e. (0[,]1) /\ 1 e. (0[,]1) /\ t e. (0[,]1)) -> (((1 - t) x. ((2 x. s) - 1)) + (t x. 1)) e. (0[,]1))
58	55, 57	mp3an2 1179	. . . . . . . 8 |- ((((2 x. s) - 1) e. (0[,]1) /\ t e. (0[,]1)) -> (((1 - t) x. ((2 x. s) - 1)) + (t x. 1)) e. (0[,]1))
59	51, 58	eqeltrd 1971	. . . . . . 7 |- ((((2 x. s) - 1) e. (0[,]1) /\ t e. (0[,]1)) -> (t + ((1 - t) x. ((2 x. s) - 1))) e. (0[,]1))
60	15	adantr 425	. . . . . . . . . 10 |- ((s e. (0[,]1) /\ -. s <_ (1 / 2)) -> s e. RR)
61		ltnle 6680	. . . . . . . . . . . . . 14 |- (((1 / 2) e. RR /\ s e. RR) -> ((1 / 2) < s <-> -. s <_ (1 / 2)))
62	25, 61	mpan 759	. . . . . . . . . . . . 13 |- (s e. RR -> ((1 / 2) < s <-> -. s <_ (1 / 2)))
63		ltle 6690	. . . . . . . . . . . . . 14 |- (((1 / 2) e. RR /\ s e. RR) -> ((1 / 2) < s -> (1 / 2) <_ s))
64	25, 63	mpan 759	. . . . . . . . . . . . 13 |- (s e. RR -> ((1 / 2) < s -> (1 / 2) <_ s))
65	62, 64	sylbird 222	. . . . . . . . . . . 12 |- (s e. RR -> (-. s <_ (1 / 2) -> (1 / 2) <_ s))
66	65	imp 377	. . . . . . . . . . 11 |- ((s e. RR /\ -. s <_ (1 / 2)) -> (1 / 2) <_ s)
67	66, 15	sylan 497	. . . . . . . . . 10 |- ((s e. (0[,]1) /\ -. s <_ (1 / 2)) -> (1 / 2) <_ s)
68	18	simp3bi 893	. . . . . . . . . . 11 |- (s e. (0[,]1) -> s <_ 1)
69	68	adantr 425	. . . . . . . . . 10 |- ((s e. (0[,]1) /\ -. s <_ (1 / 2)) -> s <_ 1)
70	60, 67, 69	3jca 1050	. . . . . . . . 9 |- ((s e. (0[,]1) /\ -. s <_ (1 / 2)) -> (s e. RR /\ (1 / 2) <_ s /\ s <_ 1))
71		elicc2 7560	. . . . . . . . . 10 |- (((1 / 2) e. RR /\ 1 e. RR) -> (s e. ((1 / 2)[,]1) <-> (s e. RR /\ (1 / 2) <_ s /\ s <_ 1)))
72	25, 12, 71	mp2an 761	. . . . . . . . 9 |- (s e. ((1 / 2)[,]1) <-> (s e. RR /\ (1 / 2) <_ s /\ s <_ 1))
73	70, 72	sylibr 217	. . . . . . . 8 |- ((s e. (0[,]1) /\ -. s <_ (1 / 2)) -> s e. ((1 / 2)[,]1))
74		iihalf2 15873	. . . . . . . 8 |- (s e. ((1 / 2)[,]1) -> ((2 x. s) - 1) e. (0[,]1))
75	73, 74	syl 12	. . . . . . 7 |- ((s e. (0[,]1) /\ -. s <_ (1 / 2)) -> ((2 x. s) - 1) e. (0[,]1))
76	59, 75	sylan 497	. . . . . 6 |- (((s e. (0[,]1) /\ -. s <_ (1 / 2)) /\ t e. (0[,]1)) -> (t + ((1 - t) x. ((2 x. s) - 1))) e. (0[,]1))
77	76	an1rs 547	. . . . 5 |- (((s e. (0[,]1) /\ t e. (0[,]1)) /\ -. s <_ (1 / 2)) -> (t + ((1 - t) x. ((2 x. s) - 1))) e. (0[,]1))
78	33, 77	ifclda 15695	. . . 4 |- ((s e. (0[,]1) /\ t e. (0[,]1)) -> if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))) e. (0[,]1))
79		eqid 1884	. . . 4 |- {<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))} = {<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))}
80		pcorevlem.1	. . . . 5 |- H = {<.<.s, t>., u>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ u = if(s <_ (1 / 2), (F` (1 - ((1 - t) x. (2 x. s)))), (F` (t + ((1 - t) x. ((2 x. s) - 1))))))}
81		fvif 15692	. . . . . . . . 9 |- (F` if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1))))) = if(s <_ (1 / 2), (F` (1 - ((1 - t) x. (2 x. s)))), (F` (t + ((1 - t) x. ((2 x. s) - 1)))))
82	81	eqcomi 1888	. . . . . . . 8 |- if(s <_ (1 / 2), (F` (1 - ((1 - t) x. (2 x. s)))), (F` (t + ((1 - t) x. ((2 x. s) - 1))))) = (F` if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))
83	82	eqeq2i 1894	. . . . . . 7 |- (u = if(s <_ (1 / 2), (F` (1 - ((1 - t) x. (2 x. s)))), (F` (t + ((1 - t) x. ((2 x. s) - 1))))) <-> u = (F` if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1))))))
84	83	anbi2i 538	. . . . . 6 |- (((s e. (0[,]1) /\ t e. (0[,]1)) /\ u = if(s <_ (1 / 2), (F` (1 - ((1 - t) x. (2 x. s)))), (F` (t + ((1 - t) x. ((2 x. s) - 1)))))) <-> ((s e. (0[,]1) /\ t e. (0[,]1)) /\ u = (F` if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))))
85	84	oprabbii 4923	. . . . 5 |- {<.<.s, t>., u>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ u = if(s <_ (1 / 2), (F` (1 - ((1 - t) x. (2 x. s)))), (F` (t + ((1 - t) x. ((2 x. s) - 1))))))} = {<.<.s, t>., u>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ u = (F` if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1))))))}
86	80, 85	eqtri 1908	. . . 4 |- H = {<.<.s, t>., u>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ u = (F` if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1))))))}
87	78, 79, 86	oprabco 10159	. . 3 |- (F Fn (0[,]1) -> H = (F o. {<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))}))
88	5, 6, 87	3syl 24	. 2 |- ((J e. Top /\ F e. (II Cn J)) -> H = (F o. {<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))}))
89	1, 1	txtopi 15909	. . . 4 |- (II X.t II) e. Top
90	89	a1i 8	. . 3 |- ((J e. Top /\ F e. (II Cn J)) -> (II X.t II) e. Top)
91	1	a1i 8	. . 3 |- ((J e. Top /\ F e. (II Cn J)) -> II e. Top)
92		simpl 346	. . 3 |- ((J e. Top /\ F e. (II Cn J)) -> J e. Top)
93		simpr 350	. . . 4 |- ((J e. Top /\ F e. (II Cn J)) -> F e. (II Cn J))
94	89, 1	pm3.2i 307	. . . . 5 |- ((II X.t II) e. Top /\ II e. Top)
95		retop 8926	. . . . . . . . 9 |- (topGen` ran (,)) e. Top
96		clint3 10184	. . . . . . . . . 10 |- ((0 e. RR /\ (1 / 2) e. RR) -> (0[,](1 / 2)) e. (Clsd` (topGen` ran (,))))
97	11, 25, 96	mp2an 761	. . . . . . . . 9 |- (0[,](1 / 2)) e. (Clsd` (topGen` ran (,)))
98	11, 12	pm3.2i 307	. . . . . . . . . 10 |- (0 e. RR /\ 1 e. RR)
99	11, 25	pm3.2i 307	. . . . . . . . . 10 |- (0 e. RR /\ (1 / 2) e. RR)
100	11	leidi 6790	. . . . . . . . . . 11 |- 0 <_ 0
101		halflt1 7216	. . . . . . . . . . . 12 |- (1 / 2) < 1
102	25, 12, 101	ltleii 6756	. . . . . . . . . . 11 |- (1 / 2) <_ 1
103	100, 102	pm3.2i 307	. . . . . . . . . 10 |- (0 <_ 0 /\ (1 / 2) <_ 1)
104		iccss 15855	. . . . . . . . . 10 |- (((0 e. RR /\ 1 e. RR) /\ (0 e. RR /\ (1 / 2) e. RR) /\ (0 <_ 0 /\ (1 / 2) <_ 1)) -> (0[,](1 / 2)) C_ (0[,]1))
105	98, 99, 103, 104	mp3an 1191	. . . . . . . . 9 |- (0[,](1 / 2)) C_ (0[,]1)
106		uniretop 8927	. . . . . . . . . . 11 |- U.(topGen` ran (,)) = RR
107	106	eqcomi 1888	. . . . . . . . . 10 |- RR = U.(topGen` ran (,))
108	107	subspcld 15838	. . . . . . . . 9 |- ((((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 (,))>.)))
109	95, 14, 97, 105, 108	mp4an 15651	. . . . . . . 8 |- (0[,](1 / 2)) e. (Clsd` (subSp` <.(0[,]1), (topGen` ran (,))>.))
110		dfii2 15869	. . . . . . . . 9 |- II = (subSp` <.(0[,]1), (topGen` ran (,))>.)
111	110	fveq2i 4684	. . . . . . . 8 |- (Clsd` II) = (Clsd` (subSp` <.(0[,]1), (topGen` ran (,))>.))
112	109, 111	eleqtrri 1970	. . . . . . 7 |- (0[,](1 / 2)) e. (Clsd` II)
113	2	topcld 8951	. . . . . . . 8 |- (II e. Top -> (0[,]1) e. (Clsd` II))
114	1, 113	ax-mp 7	. . . . . . 7 |- (0[,]1) e. (Clsd` II)
115		eqid 1884	. . . . . . . 8 |- (II X.t II) = (II X.t II)
116	115	txcld 15914	. . . . . . 7 |- (((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)))
117	1, 1, 112, 114, 116	mp4an 15651	. . . . . 6 |- ((0[,](1 / 2)) X. (0[,]1)) e. (Clsd` (II X.t II))
118		clint3 10184	. . . . . . . . . 10 |- (((1 / 2) e. RR /\ 1 e. RR) -> ((1 / 2)[,]1) e. (Clsd` (topGen` ran (,))))
119	25, 12, 118	mp2an 761	. . . . . . . . 9 |- ((1 / 2)[,]1) e. (Clsd` (topGen` ran (,)))
120	25, 12	pm3.2i 307	. . . . . . . . . 10 |- ((1 / 2) e. RR /\ 1 e. RR)
121		halfgt0 7215	. . . . . . . . . . . 12 |- 0 < (1 / 2)
122	11, 25, 121	ltleii 6756	. . . . . . . . . . 11 |- 0 <_ (1 / 2)
123	12	leidi 6790	. . . . . . . . . . 11 |- 1 <_ 1
124	122, 123	pm3.2i 307	. . . . . . . . . 10 |- (0 <_ (1 / 2) /\ 1 <_ 1)
125		iccss 15855	. . . . . . . . . 10 |- (((0 e. RR /\ 1 e. RR) /\ ((1 / 2) e. RR /\ 1 e. RR) /\ (0 <_ (1 / 2) /\ 1 <_ 1)) -> ((1 / 2)[,]1) C_ (0[,]1))
126	98, 120, 124, 125	mp3an 1191	. . . . . . . . 9 |- ((1 / 2)[,]1) C_ (0[,]1)
127	107	subspcld 15838	. . . . . . . . 9 |- ((((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 (,))>.)))
128	95, 14, 119, 126, 127	mp4an 15651	. . . . . . . 8 |- ((1 / 2)[,]1) e. (Clsd` (subSp` <.(0[,]1), (topGen` ran (,))>.))
129	128, 111	eleqtrri 1970	. . . . . . 7 |- ((1 / 2)[,]1) e. (Clsd` II)
130	115	txcld 15914	. . . . . . 7 |- (((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)))
131	1, 1, 129, 114, 130	mp4an 15651	. . . . . 6 |- (((1 / 2)[,]1) X. (0[,]1)) e. (Clsd` (II X.t II))
132		elicc2 7560	. . . . . . . . . . 11 |- ((0 e. RR /\ 1 e. RR) -> ((1 / 2) e. (0[,]1) <-> ((1 / 2) e. RR /\ 0 <_ (1 / 2) /\ (1 / 2) <_ 1)))
133	11, 12, 132	mp2an 761	. . . . . . . . . 10 |- ((1 / 2) e. (0[,]1) <-> ((1 / 2) e. RR /\ 0 <_ (1 / 2) /\ (1 / 2) <_ 1))
134	133, 25, 122, 102	mpbir3an 1052	. . . . . . . . 9 |- (1 / 2) e. (0[,]1)
135		iccsplit 15854	. . . . . . . . 9 |- ((0 e. RR /\ 1 e. RR /\ (1 / 2) e. (0[,]1)) -> (0[,]1) = ((0[,](1 / 2)) u. ((1 / 2)[,]1)))
136	11, 12, 134, 135	mp3an 1191	. . . . . . . 8 |- (0[,]1) = ((0[,](1 / 2)) u. ((1 / 2)[,]1))
137	136	xpeq1i 4021	. . . . . . 7 |- ((0[,]1) X. (0[,]1)) = (((0[,](1 / 2)) u. ((1 / 2)[,]1)) X. (0[,]1))
138		xpundir 4051	. . . . . . 7 |- (((0[,](1 / 2)) u. ((1 / 2)[,]1)) X. (0[,]1)) = (((0[,](1 / 2)) X. (0[,]1)) u. (((1 / 2)[,]1) X. (0[,]1)))
139	137, 138	eqtr2i 1909	. . . . . 6 |- (((0[,](1 / 2)) X. (0[,]1)) u. (((1 / 2)[,]1) X. (0[,]1))) = ((0[,]1) X. (0[,]1))
140	117, 131, 139	3pm3.2i 1048	. . . . 5 |- (((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)))
141	79, 78	foprab 5062	. . . . . 6 |- {<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))}:((0[,]1) X. (0[,]1))-->(0[,]1)
142		oprex 4907	. . . . . . . 8 |- (1 - ((1 - t) x. (2 x. s))) e. _V
143		oprex 4907	. . . . . . . 8 |- (t + ((1 - t) x. ((2 x. s) - 1))) e. _V
144		eqid 1884	. . . . . . . 8 |- {<.<.s, t>., v>. | ((s e. (0[,](1 / 2)) /\ t e. (0[,]1)) /\ v = (1 - ((1 - t) x. (2 x. s))))} = {<.<.s, t>., v>. | ((s e. (0[,](1 / 2)) /\ t e. (0[,]1)) /\ v = (1 - ((1 - t) x. (2 x. s))))}
145	11, 12, 53, 142, 143, 134, 79, 144	oprpiece1res1 15880	. . . . . . 7 |- ({<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))} |` ((0[,](1 / 2)) X. (0[,]1))) = {<.<.s, t>., v>. | ((s e. (0[,](1 / 2)) /\ t e. (0[,]1)) /\ v = (1 - ((1 - t) x. (2 x. s))))}
146		eqid 1884	. . . . . . . . . . . . 13 |- (abs o. - ) = (abs o. - )
147	146	cnmet 9182	. . . . . . . . . . . 12 |- (abs o. - ) e. Met
148	146	cnmetba 9181	. . . . . . . . . . . . 13 |- CC = dom dom (abs o. - )
149		eqid 1884	. . . . . . . . . . . . 13 |- (Open` (abs o. - )) = (Open` (abs o. - ))
150	148, 149	uniopn2 9138	. . . . . . . . . . . 12 |- ((abs o. - ) e. Met -> U.(Open` (abs o. - )) = CC)
151	147, 150	ax-mp 7	. . . . . . . . . . 11 |- U.(Open` (abs o. - )) = CC
152	151	eqcomi 1888	. . . . . . . . . 10 |- CC = U.(Open` (abs o. - ))
153		iccssre 7565	. . . . . . . . . . . 12 |- ((0 e. RR /\ (1 / 2) e. RR) -> (0[,](1 / 2)) C_ RR)
154	11, 25, 153	mp2an 761	. . . . . . . . . . 11 |- (0[,](1 / 2)) C_ RR
155	154, 47	sstri 2626	. . . . . . . . . 10 |- (0[,](1 / 2)) C_ CC
156	7, 31, 29	syl2an 503	. . . . . . . . . . . 12 |- ((t e. (0[,]1) /\ s e. (0[,](1 / 2))) -> ((1 - t) x. (2 x. s)) e. (0[,]1))
157	156	ancoms 484	. . . . . . . . . . 11 |- ((s e. (0[,](1 / 2)) /\ t e. (0[,]1)) -> ((1 - t) x. (2 x. s)) e. (0[,]1))
158	157, 8	syl 12	. . . . . . . . . 10 |- ((s e. (0[,](1 / 2)) /\ t e. (0[,]1)) -> (1 - ((1 - t) x. (2 x. s))) e. (0[,]1))
159		eqid 1884	. . . . . . . . . 10 |- {<.<.s, t>., v>. | ((s e. CC /\ t e. CC) /\ v = (1 - ((1 - t) x. (2 x. s))))} = {<.<.s, t>., v>. | ((s e. CC /\ t e. CC) /\ v = (1 - ((1 - t) x. (2 x. s))))}
160	149	opntop 9147	. . . . . . . . . . 11 |- ((abs o. - ) e. Met -> (Open` (abs o. - )) e. Top)
161	147, 160	ax-mp 7	. . . . . . . . . 10 |- (Open` (abs o. - )) e. Top
162	36	elisseti 2301	. . . . . . . . . . 11 |- 1 e. _V
163		oprex 4907	. . . . . . . . . . 11 |- ((1 - t) x. (2 x. s)) e. _V
164		eqid 1884	. . . . . . . . . . 11 |- {<.<.s, t>., u>. | ((s e. CC /\ t e. CC) /\ u = 1)} = {<.<.s, t>., u>. | ((s e. CC /\ t e. CC) /\ u = 1)}
165		eqid 1884	. . . . . . . . . . 11 |- {<.<.s, t>., w>. | ((s e. CC /\ t e. CC) /\ w = ((1 - t) x. (2 x. s)))} = {<.<.s, t>., w>. | ((s e. CC /\ t e. CC) /\ w = ((1 - t) x. (2 x. s)))}
166		eqid 1884	. . . . . . . . . . . 12 |- {<.s, v>. | (s e. CC /\ v = 1)} = {<.s, v>. | (s e. CC /\ v = 1)}
167	166	constcncf 15882	. . . . . . . . . . . . 13 |- (1 e. CC -> {<.s, v>. | (s e. CC /\ v = 1)} e. (CC-cn->CC))
168	36, 167	ax-mp 7	. . . . . . . . . . . 12 |- {<.s, v>. | (s e. CC /\ v = 1)} e. (CC-cn->CC)
169	149, 162, 166, 168, 164	cnoprab1c 15923	. . . . . . . . . . 11 |- {<.<.s, t>., u>. | ((s e. CC /\ t e. CC) /\ u = 1)} e. (((Open` (abs o. - )) X.t (Open` (abs o. - ))) Cn (Open` (abs o. - )))
170		oprex 4907	. . . . . . . . . . . 12 |- (1 - t) e. _V
171		oprex 4907	. . . . . . . . . . . 12 |- (2 x. s) e. _V
172		eqid 1884	. . . . . . . . . . . 12 |- {<.<.s, t>., u>. | ((s e. CC /\ t e. CC) /\ u = (1 - t))} = {<.<.s, t>., u>. | ((s e. CC /\ t e. CC) /\ u = (1 - t))}
173		eqid 1884	. . . . . . . . . . . 12 |- {<.<.s, t>., v>. | ((s e. CC /\ t e. CC) /\ v = (2 x. s))} = {<.<.s, t>., v>. | ((s e. CC /\ t e. CC) /\ v = (2 x. s))}
174		eqid 1884	. . . . . . . . . . . . 13 |- {<.t, v>. | (t e. CC /\ v = (1 - t))} = {<.t, v>. | (t e. CC /\ v = (1 - t))}
175	174	sub2cncf 15886	. . . . . . . . . . . . . 14 |- (1 e. CC -> {<.t, v>. | (t e. CC /\ v = (1 - t))} e. (CC-cn->CC))
176	36, 175	ax-mp 7	. . . . . . . . . . . . 13 |- {<.t, v>. | (t e. CC /\ v = (1 - t))} e. (CC-cn->CC)
177	149, 170, 174, 176, 172	cnoprab2c 15924	. . . . . . . . . . . 12 |- {<.<.s, t>., u>. | ((s e. CC /\ t e. CC) /\ u = (1 - t))} e. (((Open` (abs o. - )) X.t (Open` (abs o. - ))) Cn (Open` (abs o. - )))
178		eqid 1884	. . . . . . . . . . . . 13 |- {<.s, u>. | (s e. CC /\ u = (2 x. s))} = {<.s, u>. | (s e. CC /\ u = (2 x. s))}
179		2cn 7164	. . . . . . . . . . . . . 14 |- 2 e. CC
180	178	mulc1cncf 8541	. . . . . . . . . . . . . 14 |- (2 e. CC -> {<.s, u>. | (s e. CC /\ u = (2 x. s))} e. (CC-cn->CC))
181	179, 180	ax-mp 7	. . . . . . . . . . . . 13 |- {<.s, u>. | (s e. CC /\ u = (2 x. s))} e. (CC-cn->CC)
182	149, 171, 178, 181, 173	cnoprab1c 15923	. . . . . . . . . . . 12 |- {<.<.s, t>., v>. | ((s e. CC /\ t e. CC) /\ v = (2 x. s))} e. (((Open` (abs o. - )) X.t (Open` (abs o. - ))) Cn (Open` (abs o. - )))
183	146, 149	mulcntx 15929	. . . . . . . . . . . 12 |- x. e. (((Open` (abs o. - )) X.t (Open` (abs o. - ))) Cn (Open` (abs o. - )))
184	149, 170, 171, 172, 173, 165, 177, 182, 183	cnoproprabcoc 15920	. . . . . . . . . . 11 |- {<.<.s, t>., w>. | ((s e. CC /\ t e. CC) /\ w = ((1 - t) x. (2 x. s)))} e. (((Open` (abs o. - )) X.t (Open` (abs o. - ))) Cn (Open` (abs o. - )))
185	146, 149	subcntx 15928	. . . . . . . . . . 11 |- - e. (((Open` (abs o. - )) X.t (Open` (abs o. - ))) Cn (Open` (abs o. - )))
186	149, 162, 163, 164, 165, 159, 169, 184, 185	cnoproprabcoc 15920	. . . . . . . . . 10 |- {<.<.s, t>., v>. | ((s e. CC /\ t e. CC) /\ v = (1 - ((1 - t) x. (2 x. s))))} e. (((Open` (abs o. - )) X.t (Open` (abs o. - ))) Cn (Open` (abs o. - )))
187	152, 152, 152, 155, 48, 48, 158, 159, 144, 161, 161, 161, 186	cnresoprab 15915	. . . . . . . . 9 |- {<.<.s, t>., v>. | ((s e. (0[,](1 / 2)) /\ t e. (0[,]1)) /\ v = (1 - ((1 - t) x. (2 x. s))))} e. ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.) Cn (subSp` <.(0[,]1), (Open` (abs o. - ))>.))
188		dfii3 15870	. . . . . . . . . 10 |- II = (subSp` <.(0[,]1), (Open` (abs o. - ))>.)
189	188	opreq2i 4893	. . . . . . . . 9 |- ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.) Cn II) = ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.) Cn (subSp` <.(0[,]1), (Open` (abs o. - ))>.))
190	187, 189	eleqtrri 1970	. . . . . . . 8 |- {<.<.s, t>., v>. | ((s e. (0[,](1 / 2)) /\ t e. (0[,]1)) /\ v = (1 - ((1 - t) x. (2 x. s))))} e. ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.) Cn II)
191	188, 188	opreq12i 4894	. . . . . . . . . . . . 13 |- (II X.t II) = ((subSp` <.(0[,]1), (Open` (abs o. - ))>.) X.t (subSp` <.(0[,]1), (Open` (abs o. - ))>.))
192	152, 152	txsubsp 15912	. . . . . . . . . . . . . 14 |- ((((Open` (abs o. - )) e. Top /\ (Open` (abs o. - )) e. Top) /\ ((0[,]1) C_ CC /\ (0[,]1) C_ CC)) -> (subSp` <.((0[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.) = ((subSp` <.(0[,]1), (Open` (abs o. - ))>.) X.t (subSp` <.(0[,]1), (Open` (abs o. - ))>.)))
193	161, 161, 48, 48, 192	mp4an 15651	. . . . . . . . . . . . 13 |- (subSp` <.((0[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.) = ((subSp` <.(0[,]1), (Open` (abs o. - ))>.) X.t (subSp` <.(0[,]1), (Open` (abs o. - ))>.))
194	191, 193	eqtr4i 1911	. . . . . . . . . . . 12 |- (II X.t II) = (subSp` <.((0[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.)
195	194	opeq2i 3162	. . . . . . . . . . 11 |- <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>. = <.((0[,](1 / 2)) X. (0[,]1)), (subSp` <.((0[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.)>.
196	195	fveq2i 4684	. . . . . . . . . 10 |- (subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) = (subSp` <.((0[,](1 / 2)) X. (0[,]1)), (subSp` <.((0[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.)>.)
197	161, 161	txtopi 15909	. . . . . . . . . . 11 |- ((Open` (abs o. - )) X.t (Open` (abs o. - ))) e. Top
198		xpss12 4089	. . . . . . . . . . . 12 |- (((0[,]1) C_ CC /\ (0[,]1) C_ CC) -> ((0[,]1) X. (0[,]1)) C_ (CC X. CC))
199	48, 48, 198	mp2an 761	. . . . . . . . . . 11 |- ((0[,]1) X. (0[,]1)) C_ (CC X. CC)
200		ssid 2634	. . . . . . . . . . . 12 |- (0[,]1) C_ (0[,]1)
201		xpss12 4089	. . . . . . . . . . . 12 |- (((0[,](1 / 2)) C_ (0[,]1) /\ (0[,]1) C_ (0[,]1)) -> ((0[,](1 / 2)) X. (0[,]1)) C_ ((0[,]1) X. (0[,]1)))
202	105, 200, 201	mp2an 761	. . . . . . . . . . 11 |- ((0[,](1 / 2)) X. (0[,]1)) C_ ((0[,]1) X. (0[,]1))
203	161, 161, 152, 152	txunii 15910	. . . . . . . . . . . 12 |- (CC X. CC) = U.((Open` (abs o. - )) X.t (Open` (abs o. - )))
204	203	subspabs 15840	. . . . . . . . . . 11 |- ((((Open` (abs o. - )) X.t (Open` (abs o. - ))) e. Top /\ ((0[,]1) X. (0[,]1)) C_ (CC X. CC) /\ ((0[,](1 / 2)) X. (0[,]1)) C_ ((0[,]1) X. (0[,]1))) -> (subSp` <.((0[,](1 / 2)) X. (0[,]1)), (subSp` <.((0[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.)>.) = (subSp` <.((0[,](1 / 2)) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.))
205	197, 199, 202, 204	mp3an 1191	. . . . . . . . . 10 |- (subSp` <.((0[,](1 / 2)) X. (0[,]1)), (subSp` <.((0[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.)>.) = (subSp` <.((0[,](1 / 2)) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.)
206	196, 205	eqtri 1908	. . . . . . . . 9 |- (subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) = (subSp` <.((0[,](1 / 2)) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.)
207	206	opreq1i 4892	. . . . . . . 8 |- ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) Cn II) = ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.) Cn II)
208	190, 207	eleqtrri 1970	. . . . . . 7 |- {<.<.s, t>., v>. | ((s e. (0[,](1 / 2)) /\ t e. (0[,]1)) /\ v = (1 - ((1 - t) x. (2 x. s))))} e. ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) Cn II)
209	145, 208	eqeltri 1967	. . . . . 6 |- ({<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))} |` ((0[,](1 / 2)) X. (0[,]1))) e. ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) Cn II)
210		opreq2 4890	. . . . . . . . . . 11 |- (s = (1 / 2) -> (2 x. s) = (2 x. (1 / 2)))
211	179, 24	recidi 6916	. . . . . . . . . . 11 |- (2 x. (1 / 2)) = 1
212	210, 211	syl6eq 1944	. . . . . . . . . 10 |- (s = (1 / 2) -> (2 x. s) = 1)
213	212	opreq2d 4898	. . . . . . . . 9 |- (s = (1 / 2) -> ((1 - t) x. (2 x. s)) = ((1 - t) x. 1))
214	213	opreq2d 4898	. . . . . . . 8 |- (s = (1 / 2) -> (1 - ((1 - t) x. (2 x. s))) = (1 - ((1 - t) x. 1)))
215	212	opreq1d 4897	. . . . . . . . . . 11 |- (s = (1 / 2) -> ((2 x. s) - 1) = (1 - 1))
216	36	subidi 6551	. . . . . . . . . . 11 |- (1 - 1) = 0
217	215, 216	syl6eq 1944	. . . . . . . . . 10 |- (s = (1 / 2) -> ((2 x. s) - 1) = 0)
218	217	opreq2d 4898	. . . . . . . . 9 |- (s = (1 / 2) -> ((1 - t) x. ((2 x. s) - 1)) = ((1 - t) x. 0))
219	218	opreq2d 4898	. . . . . . . 8 |- (s = (1 / 2) -> (t + ((1 - t) x. ((2 x. s) - 1))) = (t + ((1 - t) x. 0)))
220		nncan 6635	. . . . . . . . . . 11 |- ((1 e. CC /\ t e. CC) -> (1 - (1 - t)) = t)
221	36, 220	mpan 759	. . . . . . . . . 10 |- (t e. CC -> (1 - (1 - t)) = t)
222		ax1id 6435	. . . . . . . . . . . 12 |- ((1 - t) e. CC -> ((1 - t) x. 1) = (1 - t))
223	38, 222	syl 12	. . . . . . . . . . 11 |- (t e. CC -> ((1 - t) x. 1) = (1 - t))
224	223	opreq2d 4898	. . . . . . . . . 10 |- (t e. CC -> (1 - ((1 - t) x. 1)) = (1 - (1 - t)))
225		mul01 6606	. . . . . . . . . . . . 13 |- ((1 - t) e. CC -> ((1 - t) x. 0) = 0)
226	38, 225	syl 12	. . . . . . . . . . . 12 |- (t e. CC -> ((1 - t) x. 0) = 0)
227	226	opreq2d 4898	. . . . . . . . . . 11 |- (t e. CC -> (t + ((1 - t) x. 0)) = (t + 0))
228		ax0id 6434	. . . . . . . . . . 11 |- (t e. CC -> (t + 0) = t)
229	227, 228	eqtrd 1925	. . . . . . . . . 10 |- (t e. CC -> (t + ((1 - t) x. 0)) = t)
230	221, 224, 229	3eqtr4d 1937	. . . . . . . . 9 |- (t e. CC -> (1 - ((1 - t) x. 1)) = (t + ((1 - t) x. 0)))
231	50, 230	syl 12	. . . . . . . 8 |- (t e. (0[,]1) -> (1 - ((1 - t) x. 1)) = (t + ((1 - t) x. 0)))
232		eqid 1884	. . . . . . . 8 |- {<.<.s, t>., v>. | ((s e. ((1 / 2)[,]1) /\ t e. (0[,]1)) /\ v = (t + ((1 - t) x. ((2 x. s) - 1))))} = {<.<.s, t>., v>. | ((s e. ((1 / 2)[,]1) /\ t e. (0[,]1)) /\ v = (t + ((1 - t) x. ((2 x. s) - 1))))}
233	11, 12, 53, 142, 143, 134, 79, 214, 219, 231, 232	oprpiece1res2 15881	. . . . . . 7 |- ({<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))} |` (((1 / 2)[,]1) X. (0[,]1))) = {<.<.s, t>., v>. | ((s e. ((1 / 2)[,]1) /\ t e. (0[,]1)) /\ v = (t + ((1 - t) x. ((2 x. s) - 1))))}
234		iccssre 7565	. . . . . . . . . . . 12 |- (((1 / 2) e. RR /\ 1 e. RR) -> ((1 / 2)[,]1) C_ RR)
235	25, 12, 234	mp2an 761	. . . . . . . . . . 11 |- ((1 / 2)[,]1) C_ RR
236	235, 47	sstri 2626	. . . . . . . . . 10 |- ((1 / 2)[,]1) C_ CC
237		simpr 350	. . . . . . . . . . . . . 14 |- ((s e. CC /\ t e. CC) -> t e. CC)
238		subcl 6524	. . . . . . . . . . . . . . . . 17 |- (((2 x. s) e. CC /\ 1 e. CC) -> ((2 x. s) - 1) e. CC)
239		axmulcl 6426	. . . . . . . . . . . . . . . . . 18 |- ((2 e. CC /\ s e. CC) -> (2 x. s) e. CC)
240	179, 239	mpan 759	. . . . . . . . . . . . . . . . 17 |- (s e. CC -> (2 x. s) e. CC)
241	238, 240, 36	sylancl 525	. . . . . . . . . . . . . . . 16 |- (s e. CC -> ((2 x. s) - 1) e. CC)
242	35, 38, 241	syl2an 503	. . . . . . . . . . . . . . 15 |- ((t e. CC /\ s e. CC) -> ((1 - t) x. ((2 x. s) - 1)) e. CC)
243	242	ancoms 484	. . . . . . . . . . . . . 14 |- ((s e. CC /\ t e. CC) -> ((1 - t) x. ((2 x. s) - 1)) e. CC)
244	237, 243, 41	syl11anc 524	. . . . . . . . . . . . 13 |- ((s e. CC /\ t e. CC) -> (t + ((1 - t) x. ((2 x. s) - 1))) = (((1 - t) x. ((2 x. s) - 1)) + t))
245	43	adantl 424	. . . . . . . . . . . . . 14 |- ((s e. CC /\ t e. CC) -> (t x. 1) = t)
246	245	opreq2d 4898	. . . . . . . . . . . . 13 |- ((s e. CC /\ t e. CC) -> (((1 - t) x. ((2 x. s) - 1)) + (t x. 1)) = (((1 - t) x. ((2 x. s) - 1)) + t))
247	244, 246	eqtr4d 1928	. . . . . . . . . . . 12 |- ((s e. CC /\ t e. CC) -> (t + ((1 - t) x. ((2 x. s) - 1))) = (((1 - t) x. ((2 x. s) - 1)) + (t x. 1)))
248	236	sseli 2617	. . . . . . . . . . . 12 |- (s e. ((1 / 2)[,]1) -> s e. CC)
249	247, 248, 50	syl2an 503	. . . . . . . . . . 11 |- ((s e. ((1 / 2)[,]1) /\ t e. (0[,]1)) -> (t + ((1 - t) x. ((2 x. s) - 1))) = (((1 - t) x. ((2 x. s) - 1)) + (t x. 1)))
250	58, 74	sylan 497	. . . . . . . . . . 11 |- ((s e. ((1 / 2)[,]1) /\ t e. (0[,]1)) -> (((1 - t) x. ((2 x. s) - 1)) + (t x. 1)) e. (0[,]1))
251	249, 250	eqeltrd 1971	. . . . . . . . . 10 |- ((s e. ((1 / 2)[,]1) /\ t e. (0[,]1)) -> (t + ((1 - t) x. ((2 x. s) - 1))) e. (0[,]1))
252		eqid 1884	. . . . . . . . . 10 |- {<.<.s, t>., v>. | ((s e. CC /\ t e. CC) /\ v = (t + ((1 - t) x. ((2 x. s) - 1))))} = {<.<.s, t>., v>. | ((s e. CC /\ t e. CC) /\ v = (t + ((1 - t) x. ((2 x. s) - 1))))}
253		visset 2295	. . . . . . . . . . 11 |- t e. _V
254		oprex 4907	. . . . . . . . . . 11 |- ((1 - t) x. ((2 x. s) - 1)) e. _V
255		eqid 1884	. . . . . . . . . . 11 |- {<.<.s, t>., u>. | ((s e. CC /\ t e. CC) /\ u = t)} = {<.<.s, t>., u>. | ((s e. CC /\ t e. CC) /\ u = t)}
256		eqid 1884	. . . . . . . . . . 11 |- {<.<.s, t>., w>. | ((s e. CC /\ t e. CC) /\ w = ((1 - t) x. ((2 x. s) - 1)))} = {<.<.s, t>., w>. | ((s e. CC /\ t e. CC) /\ w = ((1 - t) x. ((2 x. s) - 1)))}
257		eqid 1884	. . . . . . . . . . . 12 |- {<.t, v>. | (t e. CC /\ v = t)} = {<.t, v>. | (t e. CC /\ v = t)}
258	257	idcncf 15884	. . . . . . . . . . . 12 |- {<.t, v>. | (t e. CC /\ v = t)} e. (CC-cn->CC)
259	149, 253, 257, 258, 255	cnoprab2c 15924	. . . . . . . . . . 11 |- {<.<.s, t>., u>. | ((s e. CC /\ t e. CC) /\ u = t)} e. (((Open` (abs o. - )) X.t (Open` (abs o. - ))) Cn (Open` (abs o. - )))
260		oprex 4907	. . . . . . . . . . . 12 |- ((2 x. s) - 1) e. _V
261		eqid 1884	. . . . . . . . . . . 12 |- {<.<.s, t>., v>. | ((s e. CC /\ t e. CC) /\ v = ((2 x. s) - 1))} = {<.<.s, t>., v>. | ((s e. CC /\ t e. CC) /\ v = ((2 x. s) - 1))}
262		eqid 1884	. . . . . . . . . . . . 13 |- {<.s, u>. | (s e. CC /\ u = ((2 x. s) - 1))} = {<.s, u>. | (s e. CC /\ u = ((2 x. s) - 1))}
263		eqid 1884	. . . . . . . . . . . . . 14 |- {<.s, v>. | (s e. CC /\ v = (2 x. s))} = {<.s, v>. | (s e. CC /\ v = (2 x. s))}
264		eqid 1884	. . . . . . . . . . . . . 14 |- {<.s, w>. | (s e. CC /\ w = 1)} = {<.s, w>. | (s e. CC /\ w = 1)}
265	263	mulc1cncf 8541	. . . . . . . . . . . . . . 15 |- (2 e. CC -> {<.s, v>. | (s e. CC /\ v = (2 x. s))} e. (CC-cn->CC))
266	179, 265	ax-mp 7	. . . . . . . . . . . . . 14 |- {<.s, v>. | (s e. CC /\ v = (2 x. s))} e. (CC-cn->CC)
267	264	constcncf 15882	. . . . . . . . . . . . . . 15 |- (1 e. CC -> {<.s, w>. | (s e. CC /\ w = 1)} e. (CC-cn->CC))
268	36, 267	ax-mp 7	. . . . . . . . . . . . . 14 |- {<.s, w>. | (s e. CC /\ w = 1)} e. (CC-cn->CC)
269	149, 171, 162, 263, 264, 262, 266, 268, 185	cnopropabcoc 15918	. . . . . . . . . . . . 13 |- {<.s, u>. | (s e. CC /\ u = ((2 x. s) - 1))} e. (CC-cn->CC)
270	149, 260, 262, 269, 261	cnoprab1c 15923	. . . . . . . . . . . 12 |- {<.<.s, t>., v>. | ((s e. CC /\ t e. CC) /\ v = ((2 x. s) - 1))} e. (((Open` (abs o. - )) X.t (Open` (abs o. - ))) Cn (Open` (abs o. - )))
271	149, 170, 260, 172, 261, 256, 177, 270, 183	cnoproprabcoc 15920	. . . . . . . . . . 11 |- {<.<.s, t>., w>. | ((s e. CC /\ t e. CC) /\ w = ((1 - t) x. ((2 x. s) - 1)))} e. (((Open` (abs o. - )) X.t (Open` (abs o. - ))) Cn (Open` (abs o. - )))
272	146, 149	addcntx 15927	. . . . . . . . . . 11 |- + e. (((Open` (abs o. - )) X.t (Open` (abs o. - ))) Cn (Open` (abs o. - )))
273	149, 253, 254, 255, 256, 252, 259, 271, 272	cnoproprabcoc 15920	. . . . . . . . . 10 |- {<.<.s, t>., v>. | ((s e. CC /\ t e. CC) /\ v = (t + ((1 - t) x. ((2 x. s) - 1))))} e. (((Open` (abs o. - )) X.t (Open` (abs o. - ))) Cn (Open` (abs o. - )))
274	152, 152, 152, 236, 48, 48, 251, 252, 232, 161, 161, 161, 273	cnresoprab 15915	. . . . . . . . 9 |- {<.<.s, t>., v>. | ((s e. ((1 / 2)[,]1) /\ t e. (0[,]1)) /\ v = (t + ((1 - t) x. ((2 x. s) - 1))))} e. ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.) Cn (subSp` <.(0[,]1), (Open` (abs o. - ))>.))
275	188	opreq2i 4893	. . . . . . . . 9 |- ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.) Cn II) = ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.) Cn (subSp` <.(0[,]1), (Open` (abs o. - ))>.))
276	274, 275	eleqtrri 1970	. . . . . . . 8 |- {<.<.s, t>., v>. | ((s e. ((1 / 2)[,]1) /\ t e. (0[,]1)) /\ v = (t + ((1 - t) x. ((2 x. s) - 1))))} e. ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.) Cn II)
277	194	opeq2i 3162	. . . . . . . . . . 11 |- <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>. = <.(((1 / 2)[,]1) X. (0[,]1)), (subSp` <.((0[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.)>.
278	277	fveq2i 4684	. . . . . . . . . 10 |- (subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) = (subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (subSp` <.((0[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.)>.)
279		xpss12 4089	. . . . . . . . . . . 12 |- ((((1 / 2)[,]1) C_ (0[,]1) /\ (0[,]1) C_ (0[,]1)) -> (((1 / 2)[,]1) X. (0[,]1)) C_ ((0[,]1) X. (0[,]1)))
280	126, 200, 279	mp2an 761	. . . . . . . . . . 11 |- (((1 / 2)[,]1) X. (0[,]1)) C_ ((0[,]1) X. (0[,]1))
281	203	subspabs 15840	. . . . . . . . . . 11 |- ((((Open` (abs o. - )) X.t (Open` (abs o. - ))) e. Top /\ ((0[,]1) X. (0[,]1)) C_ (CC X. CC) /\ (((1 / 2)[,]1) X. (0[,]1)) C_ ((0[,]1) X. (0[,]1))) -> (subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (subSp` <.((0[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.)>.) = (subSp` <.(((1 / 2)[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.))
282	197, 199, 280, 281	mp3an 1191	. . . . . . . . . 10 |- (subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (subSp` <.((0[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.)>.) = (subSp` <.(((1 / 2)[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.)
283	278, 282	eqtri 1908	. . . . . . . . 9 |- (subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) = (subSp` <.(((1 / 2)[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.)
284	283	opreq1i 4892	. . . . . . . 8 |- ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) Cn II) = ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), ((Open` (abs o. - )) X.t (Open` (abs o. - )))>.) Cn II)
285	276, 284	eleqtrri 1970	. . . . . . 7 |- {<.<.s, t>., v>. | ((s e. ((1 / 2)[,]1) /\ t e. (0[,]1)) /\ v = (t + ((1 - t) x. ((2 x. s) - 1))))} e. ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) Cn II)
286	233, 285	eqeltri 1967	. . . . . 6 |- ({<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))} |` (((1 / 2)[,]1) X. (0[,]1))) e. ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) Cn II)
287	141, 209, 286	3pm3.2i 1048	. . . . 5 |- ({<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))}:((0[,]1) X. (0[,]1))-->(0[,]1) /\ ({<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))} |` ((0[,](1 / 2)) X. (0[,]1))) e. ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) Cn II) /\ ({<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))} |` (((1 / 2)[,]1) X. (0[,]1))) e. ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) Cn II))
288	1, 1, 2, 2	txunii 15910	. . . . . 6 |- ((0[,]1) X. (0[,]1)) = U.(II X.t II)
289	288, 2	paste 15893	. . . . 5 |- ((((II X.t II) e. Top /\ II 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))) /\ ({<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))}:((0[,]1) X. (0[,]1))-->(0[,]1) /\ ({<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))} |` ((0[,](1 / 2)) X. (0[,]1))) e. ((subSp` <.((0[,](1 / 2)) X. (0[,]1)), (II X.t II)>.) Cn II) /\ ({<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))} |` (((1 / 2)[,]1) X. (0[,]1))) e. ((subSp` <.(((1 / 2)[,]1) X. (0[,]1)), (II X.t II)>.) Cn II))) -> {<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))} e. ((II X.t II) Cn II))
290	94, 140, 287, 289	mp3an 1191	. . . 4 |- {<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))} e. ((II X.t II) Cn II)
291	93, 290	jctil 316	. . 3 |- ((J e. Top /\ F e. (II Cn J)) -> ({<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))} e. ((II X.t II) Cn II) /\ F e. (II Cn J)))
292		cnco 9045	. . 3 |- ((((II X.t II) e. Top /\ II e. Top /\ J e. Top) /\ ({<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))} e. ((II X.t II) Cn II) /\ F e. (II Cn J))) -> (F o. {<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))}) e. ((II X.t II) Cn J))
293	90, 91, 92, 291, 292	syl31anc 1103	. 2 |- ((J e. Top /\ F e. (II Cn J)) -> (F o. {<.<.s, t>., v>. | ((s e. (0[,]1) /\ t e. (0[,]1)) /\ v = if(s <_ (1 / 2), (1 - ((1 - t) x. (2 x. s))), (t + ((1 - t) x. ((2 x. s) - 1)))))}) e. ((II X.t II) Cn J))
294	88, 293	eqeltrd 1971	1 |- ((J e. Top /\ F e. (II Cn J)) -> H e. ((II X.t II) Cn J))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   u. cun 2591   C_ wss 2593  ifcif 2982  <.cop 3046  U.cuni 3177   class class class wbr 3338  {copab 3395   X. cxp 3984  ran crn 3987   |` cres 3988   o. ccom 3990   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445   / cdiv 6447   <_ cle 6448   < clt 6653  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:  pcorev 16087

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-r1 5750  df-rank 5751  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-rp 7232  df-n0 7309  df-z 7345  df-q 7436  df-fl 7463  df-ioo 7528  df-icc 7531  df-uz 7587  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-cncf 8525  df-top 8861  df-topsp 8862  df-bases 8863  df-topgen 8864  df-tx 8931  df-cld 8939  df-cn 9030  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200  df-subsp 10243  df-ii 15866

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