Theorem rrntotbnd 16022

Description: A set in Euclidean space is totally bounded iff its is bounded.

Hypotheses

Ref	Expression
rrntotbnd.1	|- X = (RR ^m (1...N))
rrntotbnd.2	|- M = ((RRn` N) |` (Y X. Y))

Assertion

Ref	Expression
rrntotbnd	|- ((N e. NN /\ Y C_ X) -> (M e. TotBnd <-> M e. Bnd))

Proof of Theorem rrntotbnd

Step	Hyp	Ref	Expression
1		eqid 1884	. . 3 |- dom dom M = dom dom M
2	1	totbndbnd 15944	. 2 |- (M e. TotBnd -> M e. Bnd)
3		rrntotbnd.2	. . . . . . . . 9 |- M = ((RRn` N) |` (Y X. Y))
4	3	a1i 8	. . . . . . . 8 |- ((N e. NN /\ Y C_ X) -> M = ((RRn` N) |` (Y X. Y)))
5	4	dmeqd 4159	. . . . . . 7 |- ((N e. NN /\ Y C_ X) -> dom M = dom ((RRn` N) |` (Y X. Y)))
6	5	dmeqd 4159	. . . . . 6 |- ((N e. NN /\ Y C_ X) -> dom dom M = dom dom ((RRn` N) |` (Y X. Y)))
7		rrnmet 16016	. . . . . . . 8 |- (N e. NN -> (RRn` N) e. Met)
8	7	adantr 425	. . . . . . 7 |- ((N e. NN /\ Y C_ X) -> (RRn` N) e. Met)
9		rrndm 16015	. . . . . . . . . 10 |- (N e. NN -> dom dom (RRn` N) = (RR ^m (1...N)))
10		rrntotbnd.1	. . . . . . . . . 10 |- X = (RR ^m (1...N))
11	9, 10	syl6eqr 1946	. . . . . . . . 9 |- (N e. NN -> dom dom (RRn` N) = X)
12	11	sseq2d 2645	. . . . . . . 8 |- (N e. NN -> (Y C_ dom dom (RRn` N) <-> Y C_ X))
13	12	biimpar 461	. . . . . . 7 |- ((N e. NN /\ Y C_ X) -> Y C_ dom dom (RRn` N))
14		eqid 1884	. . . . . . . 8 |- dom dom (RRn` N) = dom dom (RRn` N)
15	14	metssba2 9087	. . . . . . 7 |- (((RRn` N) e. Met /\ Y C_ dom dom (RRn` N)) -> Y = dom dom ((RRn` N) |` (Y X. Y)))
16	8, 13, 15	syl11anc 524	. . . . . 6 |- ((N e. NN /\ Y C_ X) -> Y = dom dom ((RRn` N) |` (Y X. Y)))
17	6, 16	eqtr4d 1928	. . . . 5 |- ((N e. NN /\ Y C_ X) -> dom dom M = Y)
18	17	raleqdv 2269	. . . 4 |- ((N e. NN /\ Y C_ X) -> (A.x e. dom dom ME.r e. RR+ dom dom M = (x( ball ` M)r) <-> A.x e. Y E.r e. RR+ dom dom M = (x( ball ` M)r)))
19		unieq 3185	. . . . . . . . . . . 12 |- (v = (/) -> U.v = U.(/))
20	19	sseq2d 2645	. . . . . . . . . . 11 |- (v = (/) -> (Y C_ U.v <-> Y C_ U.(/)))
21		raleq 2266	. . . . . . . . . . 11 |- (v = (/) -> (A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d) <-> A.b e. (/) E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d)))
22	20, 21	anbi12d 690	. . . . . . . . . 10 |- (v = (/) -> ((Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d)) <-> (Y C_ U.(/) /\ A.b e. (/) E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))))
23	22	rcla4ev 2381	. . . . . . . . 9 |- (((/) e. Fin /\ (Y C_ U.(/) /\ A.b e. (/) E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))) -> E.v e. Fin (Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d)))
24		emfin 10165	. . . . . . . . 9 |- (/) e. Fin
25		id 73	. . . . . . . . . . . 12 |- (Y = (/) -> Y = (/))
26		0ss 2900	. . . . . . . . . . . . 13 |- (/) C_ U.(/)
27	26	a1i 8	. . . . . . . . . . . 12 |- (Y = (/) -> (/) C_ U.(/))
28	25, 27	eqsstrd 2651	. . . . . . . . . . 11 |- (Y = (/) -> Y C_ U.(/))
29	28	adantl 424	. . . . . . . . . 10 |- (((N e. NN /\ Y C_ X) /\ Y = (/)) -> Y C_ U.(/))
30		ral0 2974	. . . . . . . . . 10 |- A.b e. (/) E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d)
31	29, 30	jctir 317	. . . . . . . . 9 |- (((N e. NN /\ Y C_ X) /\ Y = (/)) -> (Y C_ U.(/) /\ A.b e. (/) E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d)))
32	23, 24, 31	sylancr 526	. . . . . . . 8 |- (((N e. NN /\ Y C_ X) /\ Y = (/)) -> E.v e. Fin (Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d)))
33	32	adantr 425	. . . . . . 7 |- ((((N e. NN /\ Y C_ X) /\ Y = (/)) /\ d e. RR+) -> E.v e. Fin (Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d)))
34	33	r19.21aiva 2176	. . . . . 6 |- (((N e. NN /\ Y C_ X) /\ Y = (/)) -> A.d e. RR+ E.v e. Fin (Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d)))
35	34	a1d 15	. . . . 5 |- (((N e. NN /\ Y C_ X) /\ Y = (/)) -> (A.x e. Y E.r e. RR+ dom dom M = (x( ball ` M)r) -> A.d e. RR+ E.v e. Fin (Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))))
36	14	elbl 9115	. . . . . . . . . . . . . . . . . . . . . 22 |- ((((RRn` N) e. Met /\ x e. dom dom (RRn` N)) /\ (r e. RR /\ 0 < r)) -> (y e. (x( ball ` (RRn` N))r) <-> (y e. dom dom (RRn` N) /\ (x(RRn` N)y) < r)))
37	7	adantr 425	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((N e. NN /\ x e. X) -> (RRn` N) e. Met)
38	9, 10	syl6reqr 1947	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (N e. NN -> X = dom dom (RRn` N))
39	38	eleq2d 1964	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (N e. NN -> (x e. X <-> x e. dom dom (RRn` N)))
40	39	biimpa 460	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((N e. NN /\ x e. X) -> x e. dom dom (RRn` N))
41	37, 40	jca 310	. . . . . . . . . . . . . . . . . . . . . 22 |- ((N e. NN /\ x e. X) -> ((RRn` N) e. Met /\ x e. dom dom (RRn` N)))
42		rpregt0 7242	. . . . . . . . . . . . . . . . . . . . . 22 |- (r e. RR+ -> (r e. RR /\ 0 < r))
43	36, 41, 42	syl2an 503	. . . . . . . . . . . . . . . . . . . . 21 |- (((N e. NN /\ x e. X) /\ r e. RR+) -> (y e. (x( ball ` (RRn` N))r) <-> (y e. dom dom (RRn` N) /\ (x(RRn` N)y) < r)))
44	43	adantrl 430	. . . . . . . . . . . . . . . . . . . 20 |- (((N e. NN /\ x e. X) /\ (d e. RR+ /\ r e. RR+)) -> (y e. (x( ball ` (RRn` N))r) <-> (y e. dom dom (RRn` N) /\ (x(RRn` N)y) < r)))
45	10, 3	rrntotbndlem1 16020	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((((N e. NN /\ x e. X) /\ (d e. RR+ /\ r e. RR+)) /\ (y e. dom dom (RRn` N) /\ (x(RRn` N)y) < r)) -> ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} ( ball ` (RRn` N))d) e. {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))})
46	9	eleq2d 1964	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (N e. NN -> (y e. dom dom (RRn` N) <-> y e. (RR ^m (1...N))))
47	46	biimpd 170	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (N e. NN -> (y e. dom dom (RRn` N) -> y e. (RR ^m (1...N))))
48	47	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (((N e. NN /\ x e. X) /\ d e. RR+) -> (y e. dom dom (RRn` N) -> y e. (RR ^m (1...N))))
49	48	imdistani 491	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((((N e. NN /\ x e. X) /\ d e. RR+) /\ y e. dom dom (RRn` N)) -> (((N e. NN /\ x e. X) /\ d e. RR+) /\ y e. (RR ^m (1...N))))
50	10, 3	rrntotbndlem2 16021	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} (RRn` N)y) < d)
51	7	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> (RRn` N) e. Met)
52		ffvelrn 4787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 |- ((x:(1...N)-->RR /\ m e. (1...N)) -> (x` m) e. RR)
53		reex 6465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |- RR e. _V
54		oprex 4907	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |- (1...N) e. _V
55	53, 54	elmap 5393	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 |- (x e. (RR ^m (1...N)) <-> x:(1...N)-->RR)
56	52, 55	sylanb 498	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- ((x e. (RR ^m (1...N)) /\ m e. (1...N)) -> (x` m) e. RR)
57	56	adantll 428	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- ((((N e. NN /\ d e. RR+) /\ x e. (RR ^m (1...N))) /\ m e. (1...N)) -> (x` m) e. RR)
58	57	adantlrr 435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- ((((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) /\ m e. (1...N)) -> (x` m) e. RR)
59		rpdivcl 7249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 |- ((d e. RR+ /\ (sqr` N) e. RR+) -> (d / (sqr` N)) e. RR+)
60		nnrp 7238	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 |- (N e. NN -> N e. RR+)
61		rpsqrcl 7965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 |- (N e. RR+ -> (sqr` N) e. RR+)
62	60, 61	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 |- (N e. NN -> (sqr` N) e. RR+)
63	59, 62	sylan2 500	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |- ((d e. RR+ /\ N e. NN) -> (d / (sqr` N)) e. RR+)
64		rpre 7236	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |- ((d / (sqr` N)) e. RR+ -> (d / (sqr` N)) e. RR)
65	63, 64	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 |- ((d e. RR+ /\ N e. NN) -> (d / (sqr` N)) e. RR)
66	65	ancoms 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- ((N e. NN /\ d e. RR+) -> (d / (sqr` N)) e. RR)
67	66	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- ((((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) /\ m e. (1...N)) -> (d / (sqr` N)) e. RR)
68		ffvelrn 4787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- (({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}:(1...N)-->RR /\ m e. (1...N)) -> ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m) e. RR)
69		readdcl 6455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 |- (((((y` n) - (x` n)) / (d / (sqr` N))) e. RR /\ (1 / 2) e. RR) -> ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2)) e. RR)
70		ffvelrn 4787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 |- ((y:(1...N)-->RR /\ n e. (1...N)) -> (y` n) e. RR)
71	53, 54	elmap 5393	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 |- (y e. (RR ^m (1...N)) <-> y:(1...N)-->RR)
72	70, 71	sylanb 498	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 |- ((y e. (RR ^m (1...N)) /\ n e. (1...N)) -> (y` n) e. RR)
73	72	adantll 428	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 |- (((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ n e. (1...N)) -> (y` n) e. RR)
74		ffvelrn 4787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 |- ((x:(1...N)-->RR /\ n e. (1...N)) -> (x` n) e. RR)
75	74, 55	sylanb 498	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 |- ((x e. (RR ^m (1...N)) /\ n e. (1...N)) -> (x` n) e. RR)
76	75	adantlr 429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 |- (((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ n e. (1...N)) -> (x` n) e. RR)
77		resubcl 6601	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 |- (((y` n) e. RR /\ (x` n) e. RR) -> ((y` n) - (x` n)) e. RR)
78	73, 76, 77	syl11anc 524	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 |- (((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ n e. (1...N)) -> ((y` n) - (x` n)) e. RR)
79	78	adantll 428	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 |- ((((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) /\ n e. (1...N)) -> ((y` n) - (x` n)) e. RR)
80	66	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 |- ((((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) /\ n e. (1...N)) -> (d / (sqr` N)) e. RR)
81		rpne0 7243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 |- ((d / (sqr` N)) e. RR+ -> (d / (sqr` N)) =/= 0)
82	63, 81	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 |- ((d e. RR+ /\ N e. NN) -> (d / (sqr` N)) =/= 0)
83	82	ancoms 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 |- ((N e. NN /\ d e. RR+) -> (d / (sqr` N)) =/= 0)
84	83	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 |- ((((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) /\ n e. (1...N)) -> (d / (sqr` N)) =/= 0)
85		redivcl 6978	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 |- ((((y` n) - (x` n)) e. RR /\ (d / (sqr` N)) e. RR /\ (d / (sqr` N)) =/= 0) -> (((y` n) - (x` n)) / (d / (sqr` N))) e. RR)
86	79, 80, 84, 85	syl111anc 1100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 |- ((((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) /\ n e. (1...N)) -> (((y` n) - (x` n)) / (d / (sqr` N))) e. RR)
87		2re 7163	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 |- 2 e. RR
88		2ne0 7174	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 |- 2 =/= 0
89	87, 88	rereccli 6979	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 |- (1 / 2) e. RR
90	69, 86, 89	sylancl 525	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 |- ((((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) /\ n e. (1...N)) -> ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2)) e. RR)
91		flcl 7465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 |- (((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2)) e. RR -> (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))) e. ZZ)
92		zre 7348	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 |- ((|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))) e. ZZ -> (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))) e. RR)
93	90, 91, 92	3syl 24	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |- ((((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) /\ n e. (1...N)) -> (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))) e. RR)
94	93	r19.21aiva 2176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> A.n e. (1...N)(|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))) e. RR)
95		eqid 1884	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 |- {<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))} = {<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}
96	95	fopab2 4796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 |- (A.n e. (1...N)(|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))) e. RR <-> {<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}:(1...N)-->RR)
97	94, 96	sylib 215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> {<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}:(1...N)-->RR)
98	68, 97	sylan 497	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- ((((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) /\ m e. (1...N)) -> ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m) e. RR)
99		remulcl 6457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- (((d / (sqr` N)) e. RR /\ ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m) e. RR) -> ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m)) e. RR)
100	67, 98, 99	syl11anc 524	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- ((((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) /\ m e. (1...N)) -> ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m)) e. RR)
101		readdcl 6455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- (((x` m) e. RR /\ ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m)) e. RR) -> ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))) e. RR)
102	58, 100, 101	syl11anc 524	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ((((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) /\ m e. (1...N)) -> ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))) e. RR)
103	102	r19.21aiva 2176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> A.m e. (1...N)((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))) e. RR)
104		eqid 1884	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} = {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))}
105	104	fopab2 4796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (A.m e. (1...N)((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))) e. RR <-> {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))}:(1...N)-->RR)
106	103, 105	sylib 215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))}:(1...N)-->RR)
107	9	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> dom dom (RRn` N) = (RR ^m (1...N)))
108	107	eleq2d 1964	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} e. dom dom (RRn` N) <-> {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} e. (RR ^m (1...N))))
109	53, 54	elmap 5393	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} e. (RR ^m (1...N)) <-> {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))}:(1...N)-->RR)
110	108, 109	syl6bb 595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} e. dom dom (RRn` N) <-> {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))}:(1...N)-->RR))
111	106, 110	mpbird 213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} e. dom dom (RRn` N))
112	46	biimpar 461	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ((N e. NN /\ y e. (RR ^m (1...N))) -> y e. dom dom (RRn` N))
113	112	ad2ant2rl 447	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> y e. dom dom (RRn` N))
114		rpregt0 7242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- (d e. RR+ -> (d e. RR /\ 0 < d))
115	114	ad2antlr 441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> (d e. RR /\ 0 < d))
116	14	elbl2 9116	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ((((RRn` N) e. Met /\ {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} e. dom dom (RRn` N) /\ y e. dom dom (RRn` N)) /\ (d e. RR /\ 0 < d)) -> (y e. ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} ( ball ` (RRn` N))d) <-> ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} (RRn` N)y) < d))
117	51, 111, 113, 115, 116	syl31anc 1103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> (y e. ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} ( ball ` (RRn` N))d) <-> ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} (RRn` N)y) < d))
118	50, 117	mpbird 213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> y e. ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} ( ball ` (RRn` N))d))
119	118	expr 418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (((N e. NN /\ d e. RR+) /\ x e. (RR ^m (1...N))) -> (y e. (RR ^m (1...N)) -> y e. ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} ( ball ` (RRn` N))d)))
120	119	an1rs 547	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (((N e. NN /\ x e. (RR ^m (1...N))) /\ d e. RR+) -> (y e. (RR ^m (1...N)) -> y e. ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} ( ball ` (RRn` N))d)))
121	120	imp 377	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((((N e. NN /\ x e. (RR ^m (1...N))) /\ d e. RR+) /\ y e. (RR ^m (1...N))) -> y e. ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} ( ball ` (RRn` N))d))
122	10	eleq2i 1961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (x e. X <-> x e. (RR ^m (1...N)))
123	122	biimpi 168	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (x e. X -> x e. (RR ^m (1...N)))
124	123	anim2i 362	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((N e. NN /\ x e. X) -> (N e. NN /\ x e. (RR ^m (1...N))))
125	121, 124	sylanl1 509	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((((N e. NN /\ x e. X) /\ d e. RR+) /\ y e. (RR ^m (1...N))) -> y e. ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} ( ball ` (RRn` N))d))
126	49, 125	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((((N e. NN /\ x e. X) /\ d e. RR+) /\ y e. dom dom (RRn` N)) -> y e. ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} ( ball ` (RRn` N))d))
127	126	adantlrr 435	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((((N e. NN /\ x e. X) /\ (d e. RR+ /\ r e. RR+)) /\ y e. dom dom (RRn` N)) -> y e. ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} ( ball ` (RRn` N))d))
128	127	adantrr 431	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((((N e. NN /\ x e. X) /\ (d e. RR+ /\ r e. RR+)) /\ (y e. dom dom (RRn` N) /\ (x(RRn` N)y) < r)) -> y e. ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} ( ball ` (RRn` N))d))
129		eleq2 1958	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (z = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} ( ball ` (RRn` N))d) -> (y e. z <-> y e. ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} ( ball ` (RRn` N))d)))
130	129	rcla4ev 2381	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} ( ball ` (RRn` N))d) e. {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} /\ y e. ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))))} ( ball ` (RRn` N))d)) -> E.z e. {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}y e. z)
131	45, 128, 130	syl11anc 524	. . . . . . . . . . . . . . . . . . . . . 22 |- ((((N e. NN /\ x e. X) /\ (d e. RR+ /\ r e. RR+)) /\ (y e. dom dom (RRn` N) /\ (x(RRn` N)y) < r)) -> E.z e. {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}y e. z)
132		eluni2 3181	. . . . . . . . . . . . . . . . . . . . . 22 |- (y e. U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} <-> E.z e. {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}y e. z)
133	131, 132	sylibr 217	. . . . . . . . . . . . . . . . . . . . 21 |- ((((N e. NN /\ x e. X) /\ (d e. RR+ /\ r e. RR+)) /\ (y e. dom dom (RRn` N) /\ (x(RRn` N)y) < r)) -> y e. U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))})
134	133	ex 402	. . . . . . . . . . . . . . . . . . . 20 |- (((N e. NN /\ x e. X) /\ (d e. RR+ /\ r e. RR+)) -> ((y e. dom dom (RRn` N) /\ (x(RRn` N)y) < r) -> y e. U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}))
135	44, 134	sylbid 220	. . . . . . . . . . . . . . . . . . 19 |- (((N e. NN /\ x e. X) /\ (d e. RR+ /\ r e. RR+)) -> (y e. (x( ball ` (RRn` N))r) -> y e. U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}))
136	135	an4s 566	. . . . . . . . . . . . . . . . . 18 |- (((N e. NN /\ d e. RR+) /\ (x e. X /\ r e. RR+)) -> (y e. (x( ball ` (RRn` N))r) -> y e. U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}))
137	136	an1rs 547	. . . . . . . . . . . . . . . . 17 |- (((N e. NN /\ (x e. X /\ r e. RR+)) /\ d e. RR+) -> (y e. (x( ball ` (RRn` N))r) -> y e. U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}))
138		ssel2 2616	. . . . . . . . . . . . . . . . . . . . 21 |- ((Y C_ X /\ x e. Y) -> x e. X)
139	138	anim1i 361	. . . . . . . . . . . . . . . . . . . 20 |- (((Y C_ X /\ x e. Y) /\ r e. RR+) -> (x e. X /\ r e. RR+))
140	139	anasss 488	. . . . . . . . . . . . . . . . . . 19 |- ((Y C_ X /\ (x e. Y /\ r e. RR+)) -> (x e. X /\ r e. RR+))
141	140	anim2i 362	. . . . . . . . . . . . . . . . . 18 |- ((N e. NN /\ (Y C_ X /\ (x e. Y /\ r e. RR+))) -> (N e. NN /\ (x e. X /\ r e. RR+)))
142	141	anassrs 489	. . . . . . . . . . . . . . . . 17 |- (((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) -> (N e. NN /\ (x e. X /\ r e. RR+)))
143	137, 142	sylan 497	. . . . . . . . . . . . . . . 16 |- ((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) -> (y e. (x( ball ` (RRn` N))r) -> y e. U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}))
144	143	ssrdv 2622	. . . . . . . . . . . . . . 15 |- ((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) -> (x( ball ` (RRn` N))r) C_ U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))})
145	16, 6	eqtr4d 1928	. . . . . . . . . . . . . . . . . . 19 |- ((N e. NN /\ Y C_ X) -> Y = dom dom M)
146	145	adantr 425	. . . . . . . . . . . . . . . . . 18 |- (((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) -> Y = dom dom M)
147	146	eqeq1d 1892	. . . . . . . . . . . . . . . . 17 |- (((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) -> (Y = (x( ball ` M)r) <-> dom dom M = (x( ball ` M)r)))
148		sseq1 2637	. . . . . . . . . . . . . . . . . 18 |- (Y = (x( ball ` M)r) -> (Y C_ (x( ball ` (RRn` N))r) <-> (x( ball ` M)r) C_ (x( ball ` (RRn` N))r)))
149	14, 3	blssp 15844	. . . . . . . . . . . . . . . . . . . 20 |- ((((RRn` N) e. Met /\ Y C_ dom dom (RRn` N)) /\ (x e. Y /\ r e. RR+)) -> (x( ball ` M)r) = ((x( ball ` (RRn` N))r) i^i Y))
150		inss1 2812	. . . . . . . . . . . . . . . . . . . . 21 |- ((x( ball ` (RRn` N))r) i^i Y) C_ (x( ball ` (RRn` N))r)
151	150	a1i 8	. . . . . . . . . . . . . . . . . . . 20 |- ((((RRn` N) e. Met /\ Y C_ dom dom (RRn` N)) /\ (x e. Y /\ r e. RR+)) -> ((x( ball ` (RRn` N))r) i^i Y) C_ (x( ball ` (RRn` N))r))
152	149, 151	eqsstrd 2651	. . . . . . . . . . . . . . . . . . 19 |- ((((RRn` N) e. Met /\ Y C_ dom dom (RRn` N)) /\ (x e. Y /\ r e. RR+)) -> (x( ball ` M)r) C_ (x( ball ` (RRn` N))r))
153	8, 13	jca 310	. . . . . . . . . . . . . . . . . . 19 |- ((N e. NN /\ Y C_ X) -> ((RRn` N) e. Met /\ Y C_ dom dom (RRn` N)))
154	152, 153	sylan 497	. . . . . . . . . . . . . . . . . 18 |- (((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) -> (x( ball ` M)r) C_ (x( ball ` (RRn` N))r))
155	148, 154	syl5cbir 228	. . . . . . . . . . . . . . . . 17 |- (((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) -> (Y = (x( ball ` M)r) -> Y C_ (x( ball ` (RRn` N))r)))
156	147, 155	sylbird 222	. . . . . . . . . . . . . . . 16 |- (((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) -> (dom dom M = (x( ball ` M)r) -> Y C_ (x( ball ` (RRn` N))r)))
157	156	adantr 425	. . . . . . . . . . . . . . 15 |- ((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) -> (dom dom M = (x( ball ` M)r) -> Y C_ (x( ball ` (RRn` N))r)))
158		sstr 2625	. . . . . . . . . . . . . . . 16 |- ((Y C_ (x( ball ` (RRn` N))r) /\ (x( ball ` (RRn` N))r) C_ U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}) -> Y C_ U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))})
159	158	expcom 403	. . . . . . . . . . . . . . 15 |- ((x( ball ` (RRn` N))r) C_ U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} -> (Y C_ (x( ball ` (RRn` N))r) -> Y C_ U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}))
160	144, 157, 159	sylsyld 32	. . . . . . . . . . . . . 14 |- ((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) -> (dom dom M = (x( ball ` M)r) -> Y C_ U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}))
161	122, 55	bitri 190	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (x e. X <-> x:(1...N)-->RR)
162	161	biimpi 168	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (x e. X -> x:(1...N)-->RR)
163	162	adantl 424	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ((N e. NN /\ x e. X) -> x:(1...N)-->RR)
164	163, 138	sylan2 500	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ((N e. NN /\ (Y C_ X /\ x e. Y)) -> x:(1...N)-->RR)
165	164	anassrs 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (((N e. NN /\ Y C_ X) /\ x e. Y) -> x:(1...N)-->RR)
166	52, 165	sylan 497	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((((N e. NN /\ Y C_ X) /\ x e. Y) /\ m e. (1...N)) -> (x` m) e. RR)
167	166	adantlrr 435	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ m e. (1...N)) -> (x` m) e. RR)
168	167	ad2ant2rl 447	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) /\ (f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ m e. (1...N))) -> (x` m) e. RR)
169		remulcl 6457	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (((d / (sqr` N)) e. RR /\ (f` m) e. RR) -> ((d / (sqr` N)) x. (f` m)) e. RR)
170	66	adantlr 429	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (((N e. NN /\ Y C_ X) /\ d e. RR+) -> (d / (sqr` N)) e. RR)
171	170	adantlr 429	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) -> (d / (sqr` N)) e. RR)
172		ffvelrn 4787	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((f:(1...N)-->RR /\ m e. (1...N)) -> (f` m) e. RR)
173		fzssuz 7677	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) C_ (ZZ>=` -u((|_` (r / (d / (sqr` N)))) + 1))
174		uzssz 7599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (ZZ>=` -u((|_` (r / (d / (sqr` N)))) + 1)) C_ ZZ
175	173, 174	sstri 2626	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) C_ ZZ
176		zssre 7351	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ZZ C_ RR
177	175, 176	sstri 2626	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) C_ RR
178		fss 4571	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) C_ RR) -> f:(1...N)-->RR)
179	177, 178	mpan2 760	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) -> f:(1...N)-->RR)
180	172, 179	sylan 497	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ m e. (1...N)) -> (f` m) e. RR)
181	169, 171, 180	syl2an 503	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) /\ (f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ m e. (1...N))) -> ((d / (sqr` N)) x. (f` m)) e. RR)
182		readdcl 6455	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((x` m) e. RR /\ ((d / (sqr` N)) x. (f` m)) e. RR) -> ((x` m) + ((d / (sqr` N)) x. (f` m))) e. RR)
183	168, 181, 182	syl11anc 524	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) /\ (f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ m e. (1...N))) -> ((x` m) + ((d / (sqr` N)) x. (f` m))) e. RR)
184	183	anassrs 489	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) /\ f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1))) /\ m e. (1...N)) -> ((x` m) + ((d / (sqr` N)) x. (f` m))) e. RR)
185	184	r19.21aiva 2176	. . . . . . . . . . . . . . . . . . . . . 22 |- (((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) /\ f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1))) -> A.m e. (1...N)((x` m) + ((d / (sqr` N)) x. (f` m))) e. RR)
186		eqid 1884	. . . . . . . . . . . . . . . . . . . . . . 23 |- {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} = {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))}
187	186	fopab2 4796	. . . . . . . . . . . . . . . . . . . . . 22 |- (A.m e. (1...N)((x` m) + ((d / (sqr` N)) x. (f` m))) e. RR <-> {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))}:(1...N)-->RR)
188	185, 187	sylib 215	. . . . . . . . . . . . . . . . . . . . 21 |- (((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) /\ f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1))) -> {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))}:(1...N)-->RR)
189	9	eleq2d 1964	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (N e. NN -> ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} e. dom dom (RRn` N) <-> {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} e. (RR ^m (1...N))))
190	53, 54	elmap 5393	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} e. (RR ^m (1...N)) <-> {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))}:(1...N)-->RR)
191	189, 190	syl6bb 595	. . . . . . . . . . . . . . . . . . . . . . 23 |- (N e. NN -> ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} e. dom dom (RRn` N) <-> {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))}:(1...N)-->RR))
192	191	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . . 22 |- (((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) -> ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} e. dom dom (RRn` N) <-> {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))}:(1...N)-->RR))
193	192	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . 21 |- (((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) /\ f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1))) -> ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} e. dom dom (RRn` N) <-> {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))}:(1...N)-->RR))
194	188, 193	mpbird 213	. . . . . . . . . . . . . . . . . . . 20 |- (((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) /\ f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1))) -> {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} e. dom dom (RRn` N))
195	194	adantrr 431	. . . . . . . . . . . . . . . . . . 19 |- (((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) /\ (f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ b = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))) -> {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} e. dom dom (RRn` N))
196		simprr 451	. . . . . . . . . . . . . . . . . . 19 |- (((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) /\ (f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ b = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))) -> b = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))
197		opreq1 4889	. . . . . . . . . . . . . . . . . . . . 21 |- (y = {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} -> (y( ball ` (RRn` N))d) = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))
198	197	eqeq2d 1895	. . . . . . . . . . . . . . . . . . . 20 |- (y = {<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} -> (b = (y( ball ` (RRn` N))d) <-> b = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d)))
199	198	rcla4ev 2381	. . . . . . . . . . . . . . . . . . 19 |- (({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} e. dom dom (RRn` N) /\ b = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d)) -> E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))
200	195, 196, 199	syl11anc 524	. . . . . . . . . . . . . . . . . 18 |- (((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) /\ (f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ b = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))) -> E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))
201	200	ex 402	. . . . . . . . . . . . . . . . 17 |- ((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) -> ((f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ b = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d)) -> E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d)))
202	201	19.23adv 1584	. . . . . . . . . . . . . . . 16 |- ((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) -> (E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ b = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d)) -> E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d)))
203		visset 2295	. . . . . . . . . . . . . . . . 17 |- b e. _V
204		eqeq1 1890	. . . . . . . . . . . . . . . . . . 19 |- (a = b -> (a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d) <-> b = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d)))
205	204	anbi2d 678	. . . . . . . . . . . . . . . . . 18 |- (a = b -> ((f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d)) <-> (f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ b = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))))
206	205	exbidv 1657	. . . . . . . . . . . . . . . . 17 |- (a = b -> (E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d)) <-> E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ b = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))))
207	203, 206	elab 2403	. . . . . . . . . . . . . . . 16 |- (b e. {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} <-> E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ b = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d)))
208	202, 207	syl5ib 223	. . . . . . . . . . . . . . 15 |- ((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) -> (b e. {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} -> E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d)))
209	208	r19.21aiv 2175	. . . . . . . . . . . . . 14 |- ((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) -> A.b e. {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))
210	160, 209	jctird 663	. . . . . . . . . . . . 13 |- ((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) -> (dom dom M = (x( ball ` M)r) -> (Y C_ U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} /\ A.b e. {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))))
211		elnnuz 7609	. . . . . . . . . . . . . . . . . . . 20 |- (N e. NN <-> N e. (ZZ>=` 1))
212		fzfi 15786	. . . . . . . . . . . . . . . . . . . 20 |- (N e. (ZZ>=` 1) -> (1...N) e. Fin)
213	211, 212	sylbi 216	. . . . . . . . . . . . . . . . . . 19 |- (N e. NN -> (1...N) e. Fin)
214	213	ad2antrr 440	. . . . . . . . . . . . . . . . . 18 |- (((N e. NN /\ r e. RR+) /\ d e. RR+) -> (1...N) e. Fin)
215		simplr 449	. . . . . . . . . . . . . . . . . . . . 21 |- (((N e. NN /\ r e. RR+) /\ d e. RR+) -> r e. RR+)
216	63	ancoms 484	. . . . . . . . . . . . . . . . . . . . . 22 |- ((N e. NN /\ d e. RR+) -> (d / (sqr` N)) e. RR+)
217	216	adantlr 429	. . . . . . . . . . . . . . . . . . . . 21 |- (((N e. NN /\ r e. RR+) /\ d e. RR+) -> (d / (sqr` N)) e. RR+)
218		rpdivcl 7249	. . . . . . . . . . . . . . . . . . . . 21 |- ((r e. RR+ /\ (d / (sqr` N)) e. RR+) -> (r / (d / (sqr` N))) e. RR+)
219	215, 217, 218	syl11anc 524	. . . . . . . . . . . . . . . . . . . 20 |- (((N e. NN /\ r e. RR+) /\ d e. RR+) -> (r / (d / (sqr` N))) e. RR+)
220		rpre 7236	. . . . . . . . . . . . . . . . . . . . 21 |- ((r / (d / (sqr` N))) e. RR+ -> (r / (d / (sqr` N))) e. RR)
221		rpge0 7241	. . . . . . . . . . . . . . . . . . . . 21 |- ((r / (d / (sqr` N))) e. RR+ -> 0 <_ (r / (d / (sqr` N))))
222		flge0nn0 7482	. . . . . . . . . . . . . . . . . . . . 21 |- (((r / (d / (sqr` N))) e. RR /\ 0 <_ (r / (d / (sqr` N)))) -> (|_` (r / (d / (sqr` N)))) e. NN0)
223	220, 221, 222	syl11anc 524	. . . . . . . . . . . . . . . . . . . 20 |- ((r / (d / (sqr` N))) e. RR+ -> (|_` (r / (d / (sqr` N)))) e. NN0)
224		nn0p1nn 7384	. . . . . . . . . . . . . . . . . . . 20 |- ((|_` (r / (d / (sqr` N)))) e. NN0 -> ((|_` (r / (d / (sqr` N)))) + 1) e. NN)
225	219, 223, 224	3syl 24	. . . . . . . . . . . . . . . . . . 19 |- (((N e. NN /\ r e. RR+) /\ d e. RR+) -> ((|_` (r / (d / (sqr` N)))) + 1) e. NN)
226		elnnuz 7609	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((|_` (r / (d / (sqr` N)))) + 1) e. NN <-> ((|_` (r / (d / (sqr` N)))) + 1) e. (ZZ>=` 1))
227		1z 7368	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- 1 e. ZZ
228		znegcl 7372	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (1 e. ZZ -> -u1 e. ZZ)
229	227, 228	ax-mp 7	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- -u1 e. ZZ
230		0re 6603	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- 0 e. RR
231		1re 6598	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- 1 e. RR
232		lt01 6871	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- 0 < 1
233	230, 231, 232	ltleii 6756	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- 0 <_ 1
234		le0neg2 6860	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (1 e. RR -> (0 <_ 1 <-> -u1 <_ 0))
235	231, 234	ax-mp 7	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (0 <_ 1 <-> -u1 <_ 0)
236	233, 235	mpbi 206	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- -u1 <_ 0
237	231	renegcli 6576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- -u1 e. RR
238	237, 230, 231	letri 6763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ((-u1 <_ 0 /\ 0 <_ 1) -> -u1 <_ 1)
239	236, 233, 238	mp2an 761	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- -u1 <_ 1
240		eluz 7595	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((-u1 e. ZZ /\ 1 e. ZZ) -> (1 e. (ZZ>=` -u1) <-> -u1 <_ 1))
241	239, 240	mpbiri 211	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((-u1 e. ZZ /\ 1 e. ZZ) -> 1 e. (ZZ>=` -u1))
242	229, 227, 241	mp2an 761	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 1 e. (ZZ>=` -u1)
243		uzss 7600	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (1 e. (ZZ>=` -u1) -> (ZZ>=` 1) C_ (ZZ>=` -u1))
244	242, 243	ax-mp 7	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (ZZ>=` 1) C_ (ZZ>=` -u1)
245	244	sseli 2617	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((|_` (r / (d / (sqr` N)))) + 1) e. (ZZ>=` 1) -> ((|_` (r / (d / (sqr` N)))) + 1) e. (ZZ>=` -u1))
246	226, 245	sylbi 216	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((|_` (r / (d / (sqr` N)))) + 1) e. NN -> ((|_` (r / (d / (sqr` N)))) + 1) e. (ZZ>=` -u1))
247		uzneg 7598	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((|_` (r / (d / (sqr` N)))) + 1) e. (ZZ>=` -u1) -> -u-u1 e. (ZZ>=` -u((|_` (r / (d / (sqr` N)))) + 1)))
248	246, 247	syl 12	. . . . . . . . . . . . . . . . . . . . . 22 |- (((|_` (r / (d / (sqr` N)))) + 1) e. NN -> -u-u1 e. (ZZ>=` -u((|_` (r / (d / (sqr` N)))) + 1)))
249		ax1cn 6422	. . . . . . . . . . . . . . . . . . . . . . 23 |- 1 e. CC
250	249	negnegi 6549	. . . . . . . . . . . . . . . . . . . . . 22 |- -u-u1 = 1
251	248, 250	syl5eqelr 1976	. . . . . . . . . . . . . . . . . . . . 21 |- (((|_` (r / (d / (sqr` N)))) + 1) e. NN -> 1 e. (ZZ>=` -u((|_` (r / (d / (sqr` N)))) + 1)))
252		uzss 7600	. . . . . . . . . . . . . . . . . . . . 21 |- (1 e. (ZZ>=` -u((|_` (r / (d / (sqr` N)))) + 1)) -> (ZZ>=` 1) C_ (ZZ>=` -u((|_` (r / (d / (sqr` N)))) + 1)))
253	251, 252	syl 12	. . . . . . . . . . . . . . . . . . . 20 |- (((|_` (r / (d / (sqr` N)))) + 1) e. NN -> (ZZ>=` 1) C_ (ZZ>=` -u((|_` (r / (d / (sqr` N)))) + 1)))
254	226	biimpi 168	. . . . . . . . . . . . . . . . . . . 20 |- (((|_` (r / (d / (sqr` N)))) + 1) e. NN -> ((|_` (r / (d / (sqr` N)))) + 1) e. (ZZ>=` 1))
255	253, 254	sseldd 2620	. . . . . . . . . . . . . . . . . . 19 |- (((|_` (r / (d / (sqr` N)))) + 1) e. NN -> ((|_` (r / (d / (sqr` N)))) + 1) e. (ZZ>=` -u((|_` (r / (d / (sqr` N)))) + 1)))
256		fzfi 15786	. . . . . . . . . . . . . . . . . . 19 |- (((|_` (r / (d / (sqr` N)))) + 1) e. (ZZ>=` -u((|_` (r / (d / (sqr` N)))) + 1)) -> (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) e. Fin)
257	225, 255, 256	3syl 24	. . . . . . . . . . . . . . . . . 18 |- (((N e. NN /\ r e. RR+) /\ d e. RR+) -> (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) e. Fin)
258		fixp 10180	. . . . . . . . . . . . . . . . . 18 |- (((1...N) e. Fin /\ (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) e. Fin) -> ((1...N) X. (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1))) e. Fin)
259	214, 257, 258	syl11anc 524	. . . . . . . . . . . . . . . . 17 |- (((N e. NN /\ r e. RR+) /\ d e. RR+) -> ((1...N) X. (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1))) e. Fin)
260		pwfi 5661	. . . . . . . . . . . . . . . . 17 |- (((1...N) X. (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1))) e. Fin <-> ~P((1...N) X. (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1))) e. Fin)
261	259, 260	sylib 215	. . . . . . . . . . . . . . . 16 |- (((N e. NN /\ r e. RR+) /\ d e. RR+) -> ~P((1...N) X. (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1))) e. Fin)
262		mapsspw 5400	. . . . . . . . . . . . . . . . . . 19 |- ((1...N) e. _V -> ((-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) ^m (1...N)) C_ ~P((1...N) X. (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1))))
263	54, 262	ax-mp 7	. . . . . . . . . . . . . . . . . 18 |- ((-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) ^m (1...N)) C_ ~P((1...N) X. (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)))
264		ssfi 5630	. . . . . . . . . . . . . . . . . 18 |- ((~P((1...N) X. (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1))) e. Fin /\ ((-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) ^m (1...N)) C_ ~P((1...N) X. (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)))) -> ((-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) ^m (1...N)) e. Fin)
265	263, 264	mpan2 760	. . . . . . . . . . . . . . . . 17 |- (~P((1...N) X. (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1))) e. Fin -> ((-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) ^m (1...N)) e. Fin)
266		oprex 4907	. . . . . . . . . . . . . . . . . 18 |- (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) e. _V
267	266, 54	mapval 5391	. . . . . . . . . . . . . . . . 17 |- ((-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) ^m (1...N)) = {f | f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1))}
268	265, 267	syl5eqelr 1976	. . . . . . . . . . . . . . . 16 |- (~P((1...N) X. (-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1))) e. Fin -> {f | f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1))} e. Fin)
269		firnfi4 15744	. . . . . . . . . . . . . . . 16 |- ({f | f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1))} e. Fin -> {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} e. Fin)
270	261, 268, 269	3syl 24	. . . . . . . . . . . . . . 15 |- (((N e. NN /\ r e. RR+) /\ d e. RR+) -> {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} e. Fin)
271	270	adantllr 433	. . . . . . . . . . . . . 14 |- ((((N e. NN /\ Y C_ X) /\ r e. RR+) /\ d e. RR+) -> {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} e. Fin)
272	271	adantlrl 434	. . . . . . . . . . . . 13 |- ((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) -> {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} e. Fin)
273	210, 272	jctild 662	. . . . . . . . . . . 12 |- ((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) -> (dom dom M = (x( ball ` M)r) -> ({a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} e. Fin /\ (Y C_ U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} /\ A.b e. {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d)))))
274		unieq 3185	. . . . . . . . . . . . . . 15 |- (v = {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} -> U.v = U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))})
275	274	sseq2d 2645	. . . . . . . . . . . . . 14 |- (v = {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} -> (Y C_ U.v <-> Y C_ U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}))
276		raleq 2266	. . . . . . . . . . . . . 14 |- (v = {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} -> (A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d) <-> A.b e. {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d)))
277	275, 276	anbi12d 690	. . . . . . . . . . . . 13 |- (v = {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} -> ((Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d)) <-> (Y C_ U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} /\ A.b e. {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))))
278	277	rcla4ev 2381	. . . . . . . . . . . 12 |- (({a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} e. Fin /\ (Y C_ U.{a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))} /\ A.b e. {a | E.f(f:(1...N)-->(-u((|_` (r / (d / (sqr` N)))) + 1)...((|_` (r / (d / (sqr` N)))) + 1)) /\ a = ({<.m, u>. | (m e. (1...N) /\ u = ((x` m) + ((d / (sqr` N)) x. (f` m))))} ( ball ` (RRn` N))d))}E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))) -> E.v e. Fin (Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d)))
279	273, 278	syl6 25	. . . . . . . . . . 11 |- ((((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) /\ d e. RR+) -> (dom dom M = (x( ball ` M)r) -> E.v e. Fin (Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))))
280	279	r19.21adva 2182	. . . . . . . . . 10 |- (((N e. NN /\ Y C_ X) /\ (x e. Y /\ r e. RR+)) -> (dom dom M = (x( ball ` M)r) -> A.d e. RR+ E.v e. Fin (Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))))
281	280	ex 402	. . . . . . . . 9 |- ((N e. NN /\ Y C_ X) -> ((x e. Y /\ r e. RR+) -> (dom dom M = (x( ball ` M)r) -> A.d e. RR+ E.v e. Fin (Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d)))))
282	281	r19.23advv 2218	. . . . . . . 8 |- ((N e. NN /\ Y C_ X) -> (E.x e. Y E.r e. RR+ dom dom M = (x( ball ` M)r) -> A.d e. RR+ E.v e. Fin (Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))))
283		r19.2z 2958	. . . . . . . 8 |- ((Y =/= (/) /\ A.x e. Y E.r e. RR+ dom dom M = (x( ball ` M)r)) -> E.x e. Y E.r e. RR+ dom dom M = (x( ball ` M)r))
284	282, 283	syl5 20	. . . . . . 7 |- ((N e. NN /\ Y C_ X) -> ((Y =/= (/) /\ A.x e. Y E.r e. RR+ dom dom M = (x( ball ` M)r)) -> A.d e. RR+ E.v e. Fin (Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))))
285	284	expdimp 406	. . . . . 6 |- (((N e. NN /\ Y C_ X) /\ Y =/= (/)) -> (A.x e. Y E.r e. RR+ dom dom M = (x( ball ` M)r) -> A.d e. RR+ E.v e. Fin (Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))))
286		df-ne 2019	. . . . . 6 |- (Y =/= (/) <-> -. Y = (/))
287	285, 286	sylan2br 502	. . . . 5 |- (((N e. NN /\ Y C_ X) /\ -. Y = (/)) -> (A.x e. Y E.r e. RR+ dom dom M = (x( ball ` M)r) -> A.d e. RR+ E.v e. Fin (Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))))
288	35, 287	pm2.61dan 535	. . . 4 |- ((N e. NN /\ Y C_ X) -> (A.x e. Y E.r e. RR+ dom dom M = (x( ball ` M)r) -> A.d e. RR+ E.v e. Fin (Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))))
289	18, 288	sylbid 220	. . 3 |- ((N e. NN /\ Y C_ X) -> (A.x e. dom dom ME.r e. RR+ dom dom M = (x( ball ` M)r) -> A.d e. RR+ E.v e. Fin (Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))))
290		metres 9100	. . . . 5 |- ((RRn` N) e. Met -> ((RRn` N) |` (Y X. Y)) e. Met)
291	290, 3	syl5eqel 1975	. . . 4 |- ((RRn` N) e. Met -> M e. Met)
292	1	isbnd2 15940	. . . 4 |- (M e. Met -> (M e. Bnd <-> A.x e. dom dom ME.r e. RR+ dom dom M = (x( ball ` M)r)))
293	8, 291, 292	3syl 24	. . 3 |- ((N e. NN /\ Y C_ X) -> (M e. Bnd <-> A.x e. dom dom ME.r e. RR+ dom dom M = (x( ball ` M)r)))
294	14, 3	sstotbnd 15936	. . . 4 |- (((RRn` N) e. Met /\ Y C_ dom dom (RRn` N)) -> (M e. TotBnd <-> A.d e. RR+ E.v e. Fin (Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))))
295	8, 13, 294	syl11anc 524	. . 3 |- ((N e. NN /\ Y C_ X) -> (M e. TotBnd <-> A.d e. RR+ E.v e. Fin (Y C_ U.v /\ A.b e. v E.y e. dom dom (RRn` N)b = (y( ball ` (RRn` N))d))))
296	289, 293, 295	3imtr4d 602	. 2 |- ((N e. NN /\ Y C_ X) -> (M e. Bnd -> M e. TotBnd))
297	2, 296	impbid2 576	1 |- ((N e. NN /\ Y C_ X) -> (M e. TotBnd <-> M e. Bnd))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292   i^i cin 2592   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  U.cuni 3177   class class class wbr 3338  {copab 3395   X. cxp 3984  dom cdm 3986   |` cres 3988  -->wf 3994  ` cfv 3998  (class class class)co 4884   ^m cmap 5381  Fincfn 5426  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447   <_ cle 6448  NNcn 6449  NN0cn0 6450  ZZcz 6451  RR+crp 6453   < clt 6653  2c2 7145  |_cfl 7462  ZZ>=cuz 7586  ...cfz 7637  sqrcsqr 7919  Metcme 9066   ball cbl 9068  TotBndctotbnd 15930  Bndcbnd 15931  RRncrrn 16011

This theorem is referenced by:  rrnheibor 16023

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-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-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-2o 5178  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-fin 5430  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-3 7155  df-4 7156  df-rp 7232  df-n0 7309  df-z 7345  df-fl 7463  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-sum 8240  df-met 9070  df-bl 9072  df-totbnd 15932  df-bnd 15938  df-rrn 16012

Copyright terms: Public domain