Theorem opnin 9146

Description: The intersection of two open sets of a metric space is open.

Hypothesis

Ref	Expression
opni.1	|- J = (Open` D)

Assertion

Ref	Expression
opnin	|- ((D e. Met /\ A e. J /\ B e. J) -> (A i^i B) e. J)

Proof of Theorem opnin

Step	Hyp	Ref	Expression
1		simpr 350	. . . . . . . . . . 11 |- ((D e. Met /\ (A i^i B) C_ dom dom D) -> (A i^i B) C_ dom dom D)
2	1	a1i 8	. . . . . . . . . 10 |- ((A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A) /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B)) -> ((D e. Met /\ (A i^i B) C_ dom dom D) -> (A i^i B) C_ dom dom D))
3		reeanv 2249	. . . . . . . . . . . . . . . . . . 19 |- (E.v e. ran ( ball ` D)E.u e. ran ( ball ` D)((x e. v /\ v C_ A) /\ (x e. u /\ u C_ B)) <-> (E.v e. ran ( ball ` D)(x e. v /\ v C_ A) /\ E.u e. ran ( ball ` D)(x e. u /\ u C_ B)))
4		blss 9130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ((D e. Met /\ v e. ran ( ball ` D) /\ x e. v) -> E.f e. RR (0 < f /\ (x( ball ` D)f) C_ v))
5	4	3expib 1070	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (D e. Met -> ((v e. ran ( ball ` D) /\ x e. v) -> E.f e. RR (0 < f /\ (x( ball ` D)f) C_ v)))
6		blss 9130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ((D e. Met /\ u e. ran ( ball ` D) /\ x e. u) -> E.g e. RR (0 < g /\ (x( ball ` D)g) C_ u))
7	6	3expib 1070	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (D e. Met -> ((u e. ran ( ball ` D) /\ x e. u) -> E.g e. RR (0 < g /\ (x( ball ` D)g) C_ u)))
8	5, 7	anim12d 617	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (D e. Met -> (((v e. ran ( ball ` D) /\ x e. v) /\ (u e. ran ( ball ` D) /\ x e. u)) -> (E.f e. RR (0 < f /\ (x( ball ` D)f) C_ v) /\ E.g e. RR (0 < g /\ (x( ball ` D)g) C_ u))))
9	8	adantr 425	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((D e. Met /\ x e. dom dom D) -> (((v e. ran ( ball ` D) /\ x e. v) /\ (u e. ran ( ball ` D) /\ x e. u)) -> (E.f e. RR (0 < f /\ (x( ball ` D)f) C_ v) /\ E.g e. RR (0 < g /\ (x( ball ` D)g) C_ u))))
10		eqid 1884	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 |- dom dom D = dom dom D
11	10	blin 9129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 |- (((D e. Met /\ x e. dom dom D) /\ (f e. RR /\ 0 < f) /\ (g e. RR /\ 0 < g)) -> ((x( ball ` D)f) i^i (x( ball ` D)g)) = (x( ball ` D)if(f <_ g, f, g)))
12	11	3expb 1068	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |- (((D e. Met /\ x e. dom dom D) /\ ((f e. RR /\ 0 < f) /\ (g e. RR /\ 0 < g))) -> ((x( ball ` D)f) i^i (x( ball ` D)g)) = (x( ball ` D)if(f <_ g, f, g)))
13	12	sseq1d 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 |- (((D e. Met /\ x e. dom dom D) /\ ((f e. RR /\ 0 < f) /\ (g e. RR /\ 0 < g))) -> (((x( ball ` D)f) i^i (x( ball ` D)g)) C_ (A i^i B) <-> (x( ball ` D)if(f <_ g, f, g)) C_ (A i^i B)))
14	10	blelrn 9125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 |- (((D e. Met /\ x e. dom dom D) /\ (if(f <_ g, f, g) e. RR /\ 0 < if(f <_ g, f, g))) -> (x( ball ` D)if(f <_ g, f, g)) e. ran ( ball ` D))
15		eleq1 1957	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 |- (f = if(f <_ g, f, g) -> (f e. RR <-> if(f <_ g, f, g) e. RR))
16		breq2 3342	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 |- (f = if(f <_ g, f, g) -> (0 < f <-> 0 < if(f <_ g, f, g)))
17	15, 16	anbi12d 690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 |- (f = if(f <_ g, f, g) -> ((f e. RR /\ 0 < f) <-> (if(f <_ g, f, g) e. RR /\ 0 < if(f <_ g, f, g))))
18		eleq1 1957	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 |- (g = if(f <_ g, f, g) -> (g e. RR <-> if(f <_ g, f, g) e. RR))
19		breq2 3342	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 |- (g = if(f <_ g, f, g) -> (0 < g <-> 0 < if(f <_ g, f, g)))
20	18, 19	anbi12d 690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 |- (g = if(f <_ g, f, g) -> ((g e. RR /\ 0 < g) <-> (if(f <_ g, f, g) e. RR /\ 0 < if(f <_ g, f, g))))
21	17, 20	ifboth 3002	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 |- (((f e. RR /\ 0 < f) /\ (g e. RR /\ 0 < g)) -> (if(f <_ g, f, g) e. RR /\ 0 < if(f <_ g, f, g)))
22	14, 21	sylan2 500	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 |- (((D e. Met /\ x e. dom dom D) /\ ((f e. RR /\ 0 < f) /\ (g e. RR /\ 0 < g))) -> (x( ball ` D)if(f <_ g, f, g)) e. ran ( ball ` D))
23	22	adantr 425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 |- ((((D e. Met /\ x e. dom dom D) /\ ((f e. RR /\ 0 < f) /\ (g e. RR /\ 0 < g))) /\ (x( ball ` D)if(f <_ g, f, g)) C_ (A i^i B)) -> (x( ball ` D)if(f <_ g, f, g)) e. ran ( ball ` D))
24	10	blcntr 9122	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 |- (((D e. Met /\ x e. dom dom D) /\ (if(f <_ g, f, g) e. RR /\ 0 < if(f <_ g, f, g))) -> x e. (x( ball ` D)if(f <_ g, f, g)))
25	24, 21	sylan2 500	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 |- (((D e. Met /\ x e. dom dom D) /\ ((f e. RR /\ 0 < f) /\ (g e. RR /\ 0 < g))) -> x e. (x( ball ` D)if(f <_ g, f, g)))
26	25	anim1i 361	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 |- ((((D e. Met /\ x e. dom dom D) /\ ((f e. RR /\ 0 < f) /\ (g e. RR /\ 0 < g))) /\ (x( ball ` D)if(f <_ g, f, g)) C_ (A i^i B)) -> (x e. (x( ball ` D)if(f <_ g, f, g)) /\ (x( ball ` D)if(f <_ g, f, g)) C_ (A i^i B)))
27		eleq2 1958	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 |- (y = (x( ball ` D)if(f <_ g, f, g)) -> (x e. y <-> x e. (x( ball ` D)if(f <_ g, f, g))))
28		sseq1 2637	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 |- (y = (x( ball ` D)if(f <_ g, f, g)) -> (y C_ (A i^i B) <-> (x( ball ` D)if(f <_ g, f, g)) C_ (A i^i B)))
29	27, 28	anbi12d 690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 |- (y = (x( ball ` D)if(f <_ g, f, g)) -> ((x e. y /\ y C_ (A i^i B)) <-> (x e. (x( ball ` D)if(f <_ g, f, g)) /\ (x( ball ` D)if(f <_ g, f, g)) C_ (A i^i B))))
30	29	rcla4ev 2381	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 |- (((x( ball ` D)if(f <_ g, f, g)) e. ran ( ball ` D) /\ (x e. (x( ball ` D)if(f <_ g, f, g)) /\ (x( ball ` D)if(f <_ g, f, g)) C_ (A i^i B))) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))
31	23, 26, 30	syl11anc 524	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |- ((((D e. Met /\ x e. dom dom D) /\ ((f e. RR /\ 0 < f) /\ (g e. RR /\ 0 < g))) /\ (x( ball ` D)if(f <_ g, f, g)) C_ (A i^i B)) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))
32	31	ex 402	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 |- (((D e. Met /\ x e. dom dom D) /\ ((f e. RR /\ 0 < f) /\ (g e. RR /\ 0 < g))) -> ((x( ball ` D)if(f <_ g, f, g)) C_ (A i^i B) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))
33	13, 32	sylbid 220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- (((D e. Met /\ x e. dom dom D) /\ ((f e. RR /\ 0 < f) /\ (g e. RR /\ 0 < g))) -> (((x( ball ` D)f) i^i (x( ball ` D)g)) C_ (A i^i B) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))
34	33	ex 402	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- ((D e. Met /\ x e. dom dom D) -> (((f e. RR /\ 0 < f) /\ (g e. RR /\ 0 < g)) -> (((x( ball ` D)f) i^i (x( ball ` D)g)) C_ (A i^i B) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
35	34	com3l 38	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- (((f e. RR /\ 0 < f) /\ (g e. RR /\ 0 < g)) -> (((x( ball ` D)f) i^i (x( ball ` D)g)) C_ (A i^i B) -> ((D e. Met /\ x e. dom dom D) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
36	35	an4s 566	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- (((f e. RR /\ g e. RR) /\ (0 < f /\ 0 < g)) -> (((x( ball ` D)f) i^i (x( ball ` D)g)) C_ (A i^i B) -> ((D e. Met /\ x e. dom dom D) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
37	36	ex 402	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ((f e. RR /\ g e. RR) -> ((0 < f /\ 0 < g) -> (((x( ball ` D)f) i^i (x( ball ` D)g)) C_ (A i^i B) -> ((D e. Met /\ x e. dom dom D) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))))
38	37	com4l 43	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ((0 < f /\ 0 < g) -> (((x( ball ` D)f) i^i (x( ball ` D)g)) C_ (A i^i B) -> ((D e. Met /\ x e. dom dom D) -> ((f e. RR /\ g e. RR) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))))
39		ss2in 2820	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- (((x( ball ` D)f) C_ A /\ (x( ball ` D)g) C_ B) -> ((x( ball ` D)f) i^i (x( ball ` D)g)) C_ (A i^i B))
40		sstr 2625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- (((x( ball ` D)f) C_ v /\ v C_ A) -> (x( ball ` D)f) C_ A)
41		sstr 2625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- (((x( ball ` D)g) C_ u /\ u C_ B) -> (x( ball ` D)g) C_ B)
42	39, 40, 41	syl2an 503	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ((((x( ball ` D)f) C_ v /\ v C_ A) /\ ((x( ball ` D)g) C_ u /\ u C_ B)) -> ((x( ball ` D)f) i^i (x( ball ` D)g)) C_ (A i^i B))
43	42	an4s 566	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ((((x( ball ` D)f) C_ v /\ (x( ball ` D)g) C_ u) /\ (v C_ A /\ u C_ B)) -> ((x( ball ` D)f) i^i (x( ball ` D)g)) C_ (A i^i B))
44	38, 43	syl5 20	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ((0 < f /\ 0 < g) -> ((((x( ball ` D)f) C_ v /\ (x( ball ` D)g) C_ u) /\ (v C_ A /\ u C_ B)) -> ((D e. Met /\ x e. dom dom D) -> ((f e. RR /\ g e. RR) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))))
45	44	expdimp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (((0 < f /\ 0 < g) /\ ((x( ball ` D)f) C_ v /\ (x( ball ` D)g) C_ u)) -> ((v C_ A /\ u C_ B) -> ((D e. Met /\ x e. dom dom D) -> ((f e. RR /\ g e. RR) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))))
46	45	an4s 566	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (((0 < f /\ (x( ball ` D)f) C_ v) /\ (0 < g /\ (x( ball ` D)g) C_ u)) -> ((v C_ A /\ u C_ B) -> ((D e. Met /\ x e. dom dom D) -> ((f e. RR /\ g e. RR) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))))
47	46	com4t 44	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ((D e. Met /\ x e. dom dom D) -> ((f e. RR /\ g e. RR) -> (((0 < f /\ (x( ball ` D)f) C_ v) /\ (0 < g /\ (x( ball ` D)g) C_ u)) -> ((v C_ A /\ u C_ B) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))))
48	47	r19.23advv 2218	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((D e. Met /\ x e. dom dom D) -> (E.f e. RR E.g e. RR ((0 < f /\ (x( ball ` D)f) C_ v) /\ (0 < g /\ (x( ball ` D)g) C_ u)) -> ((v C_ A /\ u C_ B) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
49		reeanv 2249	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (E.f e. RR E.g e. RR ((0 < f /\ (x( ball ` D)f) C_ v) /\ (0 < g /\ (x( ball ` D)g) C_ u)) <-> (E.f e. RR (0 < f /\ (x( ball ` D)f) C_ v) /\ E.g e. RR (0 < g /\ (x( ball ` D)g) C_ u)))
50	48, 49	syl5ibr 224	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((D e. Met /\ x e. dom dom D) -> ((E.f e. RR (0 < f /\ (x( ball ` D)f) C_ v) /\ E.g e. RR (0 < g /\ (x( ball ` D)g) C_ u)) -> ((v C_ A /\ u C_ B) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
51	9, 50	syld 30	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((D e. Met /\ x e. dom dom D) -> (((v e. ran ( ball ` D) /\ x e. v) /\ (u e. ran ( ball ` D) /\ x e. u)) -> ((v C_ A /\ u C_ B) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
52	51	com3l 38	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((v e. ran ( ball ` D) /\ x e. v) /\ (u e. ran ( ball ` D) /\ x e. u)) -> ((v C_ A /\ u C_ B) -> ((D e. Met /\ x e. dom dom D) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
53	52	imp 377	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((((v e. ran ( ball ` D) /\ x e. v) /\ (u e. ran ( ball ` D) /\ x e. u)) /\ (v C_ A /\ u C_ B)) -> ((D e. Met /\ x e. dom dom D) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))
54	53	an4s 566	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((((v e. ran ( ball ` D) /\ x e. v) /\ v C_ A) /\ ((u e. ran ( ball ` D) /\ x e. u) /\ u C_ B)) -> ((D e. Met /\ x e. dom dom D) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))
55		anass 487	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((v e. ran ( ball ` D) /\ x e. v) /\ v C_ A) <-> (v e. ran ( ball ` D) /\ (x e. v /\ v C_ A)))
56		anass 487	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((u e. ran ( ball ` D) /\ x e. u) /\ u C_ B) <-> (u e. ran ( ball ` D) /\ (x e. u /\ u C_ B)))
57	54, 55, 56	syl2anbr 505	. . . . . . . . . . . . . . . . . . . . . 22 |- (((v e. ran ( ball ` D) /\ (x e. v /\ v C_ A)) /\ (u e. ran ( ball ` D) /\ (x e. u /\ u C_ B))) -> ((D e. Met /\ x e. dom dom D) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))
58	57	an4s 566	. . . . . . . . . . . . . . . . . . . . 21 |- (((v e. ran ( ball ` D) /\ u e. ran ( ball ` D)) /\ ((x e. v /\ v C_ A) /\ (x e. u /\ u C_ B))) -> ((D e. Met /\ x e. dom dom D) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))
59	58	ex 402	. . . . . . . . . . . . . . . . . . . 20 |- ((v e. ran ( ball ` D) /\ u e. ran ( ball ` D)) -> (((x e. v /\ v C_ A) /\ (x e. u /\ u C_ B)) -> ((D e. Met /\ x e. dom dom D) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
60	59	r19.23aivv 2217	. . . . . . . . . . . . . . . . . . 19 |- (E.v e. ran ( ball ` D)E.u e. ran ( ball ` D)((x e. v /\ v C_ A) /\ (x e. u /\ u C_ B)) -> ((D e. Met /\ x e. dom dom D) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))
61	3, 60	sylbir 218	. . . . . . . . . . . . . . . . . 18 |- ((E.v e. ran ( ball ` D)(x e. v /\ v C_ A) /\ E.u e. ran ( ball ` D)(x e. u /\ u C_ B)) -> ((D e. Met /\ x e. dom dom D) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))
62		eleq1 1957	. . . . . . . . . . . . . . . . . . . . 21 |- (z = x -> (z e. v <-> x e. v))
63	62	anbi1d 679	. . . . . . . . . . . . . . . . . . . 20 |- (z = x -> ((z e. v /\ v C_ A) <-> (x e. v /\ v C_ A)))
64	63	rexbidv 2124	. . . . . . . . . . . . . . . . . . 19 |- (z = x -> (E.v e. ran ( ball ` D)(z e. v /\ v C_ A) <-> E.v e. ran ( ball ` D)(x e. v /\ v C_ A)))
65	64	rcla4va 2378	. . . . . . . . . . . . . . . . . 18 |- ((x e. A /\ A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A)) -> E.v e. ran ( ball ` D)(x e. v /\ v C_ A))
66		eleq1 1957	. . . . . . . . . . . . . . . . . . . . 21 |- (w = x -> (w e. u <-> x e. u))
67	66	anbi1d 679	. . . . . . . . . . . . . . . . . . . 20 |- (w = x -> ((w e. u /\ u C_ B) <-> (x e. u /\ u C_ B)))
68	67	rexbidv 2124	. . . . . . . . . . . . . . . . . . 19 |- (w = x -> (E.u e. ran ( ball ` D)(w e. u /\ u C_ B) <-> E.u e. ran ( ball ` D)(x e. u /\ u C_ B)))
69	68	rcla4va 2378	. . . . . . . . . . . . . . . . . 18 |- ((x e. B /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B)) -> E.u e. ran ( ball ` D)(x e. u /\ u C_ B))
70	61, 65, 69	syl2an 503	. . . . . . . . . . . . . . . . 17 |- (((x e. A /\ A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A)) /\ (x e. B /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B))) -> ((D e. Met /\ x e. dom dom D) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))
71	70	an4s 566	. . . . . . . . . . . . . . . 16 |- (((x e. A /\ x e. B) /\ (A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A) /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B))) -> ((D e. Met /\ x e. dom dom D) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))
72		elin 2786	. . . . . . . . . . . . . . . 16 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
73	71, 72	sylanb 498	. . . . . . . . . . . . . . 15 |- ((x e. (A i^i B) /\ (A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A) /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B))) -> ((D e. Met /\ x e. dom dom D) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))
74	73	ex 402	. . . . . . . . . . . . . 14 |- (x e. (A i^i B) -> ((A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A) /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B)) -> ((D e. Met /\ x e. dom dom D) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
75	74	com23 36	. . . . . . . . . . . . 13 |- (x e. (A i^i B) -> ((D e. Met /\ x e. dom dom D) -> ((A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A) /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B)) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
76		ssel 2615	. . . . . . . . . . . . . 14 |- ((A i^i B) C_ dom dom D -> (x e. (A i^i B) -> x e. dom dom D))
77	76	com12 14	. . . . . . . . . . . . 13 |- (x e. (A i^i B) -> ((A i^i B) C_ dom dom D -> x e. dom dom D))
78	75, 77	sylan2d 507	. . . . . . . . . . . 12 |- (x e. (A i^i B) -> ((D e. Met /\ (A i^i B) C_ dom dom D) -> ((A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A) /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B)) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
79	78	com13 37	. . . . . . . . . . 11 |- ((A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A) /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B)) -> ((D e. Met /\ (A i^i B) C_ dom dom D) -> (x e. (A i^i B) -> E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
80	79	r19.21adv 2181	. . . . . . . . . 10 |- ((A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A) /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B)) -> ((D e. Met /\ (A i^i B) C_ dom dom D) -> A.x e. (A i^i B)E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))
81	2, 80	jcad 661	. . . . . . . . 9 |- ((A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A) /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B)) -> ((D e. Met /\ (A i^i B) C_ dom dom D) -> ((A i^i B) C_ dom dom D /\ A.x e. (A i^i B)E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
82	81	exp3a 405	. . . . . . . 8 |- ((A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A) /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B)) -> (D e. Met -> ((A i^i B) C_ dom dom D -> ((A i^i B) C_ dom dom D /\ A.x e. (A i^i B)E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))))
83	82	com3r 39	. . . . . . 7 |- ((A i^i B) C_ dom dom D -> ((A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A) /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B)) -> (D e. Met -> ((A i^i B) C_ dom dom D /\ A.x e. (A i^i B)E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B))))))
84	83	imp 377	. . . . . 6 |- (((A i^i B) C_ dom dom D /\ (A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A) /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B))) -> (D e. Met -> ((A i^i B) C_ dom dom D /\ A.x e. (A i^i B)E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
85		ssinss1 2821	. . . . . . 7 |- (A C_ dom dom D -> (A i^i B) C_ dom dom D)
86	85	adantr 425	. . . . . 6 |- ((A C_ dom dom D /\ B C_ dom dom D) -> (A i^i B) C_ dom dom D)
87	84, 86	sylan 497	. . . . 5 |- (((A C_ dom dom D /\ B C_ dom dom D) /\ (A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A) /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B))) -> (D e. Met -> ((A i^i B) C_ dom dom D /\ A.x e. (A i^i B)E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
88	87	an4s 566	. . . 4 |- (((A C_ dom dom D /\ A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A)) /\ (B C_ dom dom D /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B))) -> (D e. Met -> ((A i^i B) C_ dom dom D /\ A.x e. (A i^i B)E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
89	88	com12 14	. . 3 |- (D e. Met -> (((A C_ dom dom D /\ A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A)) /\ (B C_ dom dom D /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B))) -> ((A i^i B) C_ dom dom D /\ A.x e. (A i^i B)E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
90		opni.1	. . . . 5 |- J = (Open` D)
91	10, 90	isopn 9136	. . . 4 |- (D e. Met -> (A e. J <-> (A C_ dom dom D /\ A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A))))
92	10, 90	isopn 9136	. . . 4 |- (D e. Met -> (B e. J <-> (B C_ dom dom D /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B))))
93	91, 92	anbi12d 690	. . 3 |- (D e. Met -> ((A e. J /\ B e. J) <-> ((A C_ dom dom D /\ A.z e. A E.v e. ran ( ball ` D)(z e. v /\ v C_ A)) /\ (B C_ dom dom D /\ A.w e. B E.u e. ran ( ball ` D)(w e. u /\ u C_ B)))))
94	10, 90	isopn 9136	. . 3 |- (D e. Met -> ((A i^i B) e. J <-> ((A i^i B) C_ dom dom D /\ A.x e. (A i^i B)E.y e. ran ( ball ` D)(x e. y /\ y C_ (A i^i B)))))
95	89, 93, 94	3imtr4d 602	. 2 |- (D e. Met -> ((A e. J /\ B e. J) -> (A i^i B) e. J))
96	95	3impib 1065	1 |- ((D e. Met /\ A e. J /\ B e. J) -> (A i^i B) e. J)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   i^i cin 2592   C_ wss 2593  ifcif 2982   class class class wbr 3338  dom cdm 3986  ran crn 3987  ` cfv 3998  (class class class)co 4884  RRcr 6385  0cc0 6386   <_ cle 6448   < clt 6653  Metcme 9066   ball cbl 9068  Opencopn 9069

This theorem is referenced by:  opntop 9147  bcthlem9 9285

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  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-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-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-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  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-met 9070  df-bl 9072  df-opn 9073

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