Theorem rrntotbndlem2 16021

Description: Lemma for rrntotbnd 16022.

Hypotheses

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

Assertion

Ref	Expression
rrntotbndlem2	|- (((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)

Distinct variable groups:   X,d,m,n,u,v,x,y   M,d,m,n,u,v,x,y   Y,d,m,n,u,v,x,y   N,d,m,n,u,v,x,y

Proof of Theorem rrntotbndlem2

Step	Hyp	Ref	Expression
1		simpll 448	. . 3 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> N e. NN)
2		ffvelrn 4787	. . . . . . . . . 10 |- ((x:(1...N)-->RR /\ m e. (1...N)) -> (x` m) e. RR)
3		reex 6465	. . . . . . . . . . 11 |- RR e. _V
4		oprex 4907	. . . . . . . . . . 11 |- (1...N) e. _V
5	3, 4	elmap 5393	. . . . . . . . . 10 |- (x e. (RR ^m (1...N)) <-> x:(1...N)-->RR)
6	2, 5	sylanb 498	. . . . . . . . 9 |- ((x e. (RR ^m (1...N)) /\ m e. (1...N)) -> (x` m) e. RR)
7	6	adantll 428	. . . . . . . 8 |- ((((N e. NN /\ d e. RR+) /\ x e. (RR ^m (1...N))) /\ m e. (1...N)) -> (x` m) e. RR)
8	7	adantlrr 435	. . . . . . 7 |- ((((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)
9		rpdivcl 7249	. . . . . . . . . . . 12 |- ((d e. RR+ /\ (sqr` N) e. RR+) -> (d / (sqr` N)) e. RR+)
10		nnrp 7238	. . . . . . . . . . . . 13 |- (N e. NN -> N e. RR+)
11		rpsqrcl 7965	. . . . . . . . . . . . 13 |- (N e. RR+ -> (sqr` N) e. RR+)
12	10, 11	syl 12	. . . . . . . . . . . 12 |- (N e. NN -> (sqr` N) e. RR+)
13	9, 12	sylan2 500	. . . . . . . . . . 11 |- ((d e. RR+ /\ N e. NN) -> (d / (sqr` N)) e. RR+)
14		rpre 7236	. . . . . . . . . . 11 |- ((d / (sqr` N)) e. RR+ -> (d / (sqr` N)) e. RR)
15	13, 14	syl 12	. . . . . . . . . 10 |- ((d e. RR+ /\ N e. NN) -> (d / (sqr` N)) e. RR)
16	15	ancoms 484	. . . . . . . . 9 |- ((N e. NN /\ d e. RR+) -> (d / (sqr` N)) e. RR)
17	16	ad2antrr 440	. . . . . . . 8 |- ((((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)
18		ffvelrn 4787	. . . . . . . . 9 |- (({<.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)
19		readdcl 6455	. . . . . . . . . . . . 13 |- (((((y` n) - (x` n)) / (d / (sqr` N))) e. RR /\ (1 / 2) e. RR) -> ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2)) e. RR)
20		ffvelrn 4787	. . . . . . . . . . . . . . . . . 18 |- ((y:(1...N)-->RR /\ n e. (1...N)) -> (y` n) e. RR)
21	3, 4	elmap 5393	. . . . . . . . . . . . . . . . . 18 |- (y e. (RR ^m (1...N)) <-> y:(1...N)-->RR)
22	20, 21	sylanb 498	. . . . . . . . . . . . . . . . 17 |- ((y e. (RR ^m (1...N)) /\ n e. (1...N)) -> (y` n) e. RR)
23	22	adantll 428	. . . . . . . . . . . . . . . 16 |- (((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ n e. (1...N)) -> (y` n) e. RR)
24		ffvelrn 4787	. . . . . . . . . . . . . . . . . 18 |- ((x:(1...N)-->RR /\ n e. (1...N)) -> (x` n) e. RR)
25	24, 5	sylanb 498	. . . . . . . . . . . . . . . . 17 |- ((x e. (RR ^m (1...N)) /\ n e. (1...N)) -> (x` n) e. RR)
26	25	adantlr 429	. . . . . . . . . . . . . . . 16 |- (((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ n e. (1...N)) -> (x` n) e. RR)
27		resubcl 6601	. . . . . . . . . . . . . . . 16 |- (((y` n) e. RR /\ (x` n) e. RR) -> ((y` n) - (x` n)) e. RR)
28	23, 26, 27	syl11anc 524	. . . . . . . . . . . . . . 15 |- (((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ n e. (1...N)) -> ((y` n) - (x` n)) e. RR)
29	28	adantll 428	. . . . . . . . . . . . . 14 |- ((((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)
30	16	ad2antrr 440	. . . . . . . . . . . . . 14 |- ((((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)
31		rpne0 7243	. . . . . . . . . . . . . . . . 17 |- ((d / (sqr` N)) e. RR+ -> (d / (sqr` N)) =/= 0)
32	13, 31	syl 12	. . . . . . . . . . . . . . . 16 |- ((d e. RR+ /\ N e. NN) -> (d / (sqr` N)) =/= 0)
33	32	ancoms 484	. . . . . . . . . . . . . . 15 |- ((N e. NN /\ d e. RR+) -> (d / (sqr` N)) =/= 0)
34	33	ad2antrr 440	. . . . . . . . . . . . . 14 |- ((((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)
35		redivcl 6978	. . . . . . . . . . . . . 14 |- ((((y` n) - (x` n)) e. RR /\ (d / (sqr` N)) e. RR /\ (d / (sqr` N)) =/= 0) -> (((y` n) - (x` n)) / (d / (sqr` N))) e. RR)
36	29, 30, 34, 35	syl111anc 1100	. . . . . . . . . . . . 13 |- ((((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)
37		2re 7163	. . . . . . . . . . . . . 14 |- 2 e. RR
38		2ne0 7174	. . . . . . . . . . . . . 14 |- 2 =/= 0
39	37, 38	rereccli 6979	. . . . . . . . . . . . 13 |- (1 / 2) e. RR
40	19, 36, 39	sylancl 525	. . . . . . . . . . . 12 |- ((((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)
41		flcl 7465	. . . . . . . . . . . 12 |- (((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2)) e. RR -> (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))) e. ZZ)
42		zre 7348	. . . . . . . . . . . 12 |- ((|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))) e. ZZ -> (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))) e. RR)
43	40, 41, 42	3syl 24	. . . . . . . . . . 11 |- ((((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)
44	43	r19.21aiva 2176	. . . . . . . . . 10 |- (((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)
45		eqid 1884	. . . . . . . . . . 11 |- {<.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))))}
46	45	fopab2 4796	. . . . . . . . . 10 |- (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)
47	44, 46	sylib 215	. . . . . . . . 9 |- (((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)
48	18, 47	sylan 497	. . . . . . . 8 |- ((((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)
49		remulcl 6457	. . . . . . . 8 |- (((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)
50	17, 48, 49	syl11anc 524	. . . . . . 7 |- ((((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)
51		readdcl 6455	. . . . . . 7 |- (((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)
52	8, 50, 51	syl11anc 524	. . . . . 6 |- ((((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)
53	52	r19.21aiva 2176	. . . . 5 |- (((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)
54		eqid 1884	. . . . . 6 |- {<.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))))}
55	54	fopab2 4796	. . . . 5 |- (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)
56	53, 55	sylib 215	. . . 4 |- (((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)
57	3, 4	elmap 5393	. . . 4 |- ({<.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)
58	56, 57	sylibr 217	. . 3 |- (((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. (RR ^m (1...N)))
59		simprr 451	. . 3 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> y e. (RR ^m (1...N)))
60	13	ancoms 484	. . . 4 |- ((N e. NN /\ d e. RR+) -> (d / (sqr` N)) e. RR+)
61	60	adantr 425	. . 3 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> (d / (sqr` N)) e. RR+)
62		fveq2 4681	. . . . . . . . 9 |- (m = a -> (x` m) = (x` a))
63		fveq2 4681	. . . . . . . . . 10 |- (m = a -> ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m) = ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` a))
64	63	opreq2d 4898	. . . . . . . . 9 |- (m = a -> ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m)) = ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` a)))
65	62, 64	opreq12d 4900	. . . . . . . 8 |- (m = a -> ((x` m) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` m))) = ((x` a) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` a))))
66		oprex 4907	. . . . . . . 8 |- ((x` a) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` a))) e. _V
67	65, 54, 66	fvopab4 4743	. . . . . . 7 |- (a e. (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))))}` a) = ((x` a) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` a))))
68	67	opreq1d 4897	. . . . . 6 |- (a e. (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))))}` a)((abs o. - ) |` (RR X. RR))(y` a)) = (((x` a) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` a)))((abs o. - ) |` (RR X. RR))(y` a)))
69	68	adantl 424	. . . . 5 |- ((((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) /\ a e. (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))))}` a)((abs o. - ) |` (RR X. RR))(y` a)) = (((x` a) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` a)))((abs o. - ) |` (RR X. RR))(y` a)))
70		fveq2 4681	. . . . . . . . . . . . . . 15 |- (n = a -> (y` n) = (y` a))
71		fveq2 4681	. . . . . . . . . . . . . . 15 |- (n = a -> (x` n) = (x` a))
72	70, 71	opreq12d 4900	. . . . . . . . . . . . . 14 |- (n = a -> ((y` n) - (x` n)) = ((y` a) - (x` a)))
73	72	opreq1d 4897	. . . . . . . . . . . . 13 |- (n = a -> (((y` n) - (x` n)) / (d / (sqr` N))) = (((y` a) - (x` a)) / (d / (sqr` N))))
74	73	opreq1d 4897	. . . . . . . . . . . 12 |- (n = a -> ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2)) = ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))
75	74	fveq2d 4685	. . . . . . . . . . 11 |- (n = a -> (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))) = (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))))
76		fvex 4689	. . . . . . . . . . 11 |- (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) e. _V
77	75, 45, 76	fvopab4 4743	. . . . . . . . . 10 |- (a e. (1...N) -> ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` a) = (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))))
78	77	opreq2d 4898	. . . . . . . . 9 |- (a e. (1...N) -> ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` a)) = ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))))
79	78	opreq2d 4898	. . . . . . . 8 |- (a e. (1...N) -> ((x` a) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` a))) = ((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))))))
80	79	opreq1d 4897	. . . . . . 7 |- (a e. (1...N) -> (((x` a) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` a)))((abs o. - ) |` (RR X. RR))(y` a)) = (((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))))((abs o. - ) |` (RR X. RR))(y` a)))
81	80	adantl 424	. . . . . 6 |- ((((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) /\ a e. (1...N)) -> (((x` a) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` a)))((abs o. - ) |` (RR X. RR))(y` a)) = (((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))))((abs o. - ) |` (RR X. RR))(y` a)))
82		simprl 450	. . . . . . . . . . 11 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (x` a) e. RR)
83	14	adantr 425	. . . . . . . . . . . 12 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (d / (sqr` N)) e. RR)
84		readdcl 6455	. . . . . . . . . . . . . 14 |- (((((y` a) - (x` a)) / (d / (sqr` N))) e. RR /\ (1 / 2) e. RR) -> ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)) e. RR)
85		resubcl 6601	. . . . . . . . . . . . . . . . 17 |- (((y` a) e. RR /\ (x` a) e. RR) -> ((y` a) - (x` a)) e. RR)
86	85	ancoms 484	. . . . . . . . . . . . . . . 16 |- (((x` a) e. RR /\ (y` a) e. RR) -> ((y` a) - (x` a)) e. RR)
87	86	adantl 424	. . . . . . . . . . . . . . 15 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> ((y` a) - (x` a)) e. RR)
88	31	adantr 425	. . . . . . . . . . . . . . 15 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (d / (sqr` N)) =/= 0)
89		redivcl 6978	. . . . . . . . . . . . . . 15 |- ((((y` a) - (x` a)) e. RR /\ (d / (sqr` N)) e. RR /\ (d / (sqr` N)) =/= 0) -> (((y` a) - (x` a)) / (d / (sqr` N))) e. RR)
90	87, 83, 88, 89	syl111anc 1100	. . . . . . . . . . . . . 14 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (((y` a) - (x` a)) / (d / (sqr` N))) e. RR)
91	84, 90, 39	sylancl 525	. . . . . . . . . . . . 13 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)) e. RR)
92		flcl 7465	. . . . . . . . . . . . 13 |- (((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)) e. RR -> (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) e. ZZ)
93		zre 7348	. . . . . . . . . . . . 13 |- ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) e. ZZ -> (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) e. RR)
94	91, 92, 93	3syl 24	. . . . . . . . . . . 12 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) e. RR)
95		remulcl 6457	. . . . . . . . . . . 12 |- (((d / (sqr` N)) e. RR /\ (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) e. RR) -> ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) e. RR)
96	83, 94, 95	syl11anc 524	. . . . . . . . . . 11 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) e. RR)
97		readdcl 6455	. . . . . . . . . . 11 |- (((x` a) e. RR /\ ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) e. RR) -> ((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))))) e. RR)
98	82, 96, 97	syl11anc 524	. . . . . . . . . 10 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> ((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))))) e. RR)
99		simprr 451	. . . . . . . . . 10 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (y` a) e. RR)
100		eqid 1884	. . . . . . . . . . 11 |- ((abs o. - ) |` (RR X. RR)) = ((abs o. - ) |` (RR X. RR))
101	100	remetdval 9186	. . . . . . . . . 10 |- ((((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))))) e. RR /\ (y` a) e. RR) -> (((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))))((abs o. - ) |` (RR X. RR))(y` a)) = (abs` (((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))))) - (y` a))))
102	98, 99, 101	syl11anc 524	. . . . . . . . 9 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))))((abs o. - ) |` (RR X. RR))(y` a)) = (abs` (((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))))) - (y` a))))
103	94	recnd 6468	. . . . . . . . . . . . . 14 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) e. CC)
104	90	recnd 6468	. . . . . . . . . . . . . 14 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (((y` a) - (x` a)) / (d / (sqr` N))) e. CC)
105		subcl 6524	. . . . . . . . . . . . . 14 |- (((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) e. CC /\ (((y` a) - (x` a)) / (d / (sqr` N))) e. CC) -> ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N)))) e. CC)
106	103, 104, 105	syl11anc 524	. . . . . . . . . . . . 13 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N)))) e. CC)
107		abscl 8084	. . . . . . . . . . . . 13 |- (((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N)))) e. CC -> (abs` ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N))))) e. RR)
108	106, 107	syl 12	. . . . . . . . . . . 12 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (abs` ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N))))) e. RR)
109	39	a1i 8	. . . . . . . . . . . 12 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (1 / 2) e. RR)
110		1re 6598	. . . . . . . . . . . . 13 |- 1 e. RR
111	110	a1i 8	. . . . . . . . . . . 12 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> 1 e. RR)
112		rddif 15798	. . . . . . . . . . . . 13 |- ((((y` a) - (x` a)) / (d / (sqr` N))) e. RR -> (abs` ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N))))) <_ (1 / 2))
113	90, 112	syl 12	. . . . . . . . . . . 12 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (abs` ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N))))) <_ (1 / 2))
114		halflt1 7216	. . . . . . . . . . . . 13 |- (1 / 2) < 1
115	114	a1i 8	. . . . . . . . . . . 12 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (1 / 2) < 1)
116	108, 109, 111, 113, 115	lelttrd 6697	. . . . . . . . . . 11 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (abs` ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N))))) < 1)
117		rpgt0 7240	. . . . . . . . . . . . 13 |- ((d / (sqr` N)) e. RR+ -> 0 < (d / (sqr` N)))
118	117	adantr 425	. . . . . . . . . . . 12 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> 0 < (d / (sqr` N)))
119		ltmul2 7009	. . . . . . . . . . . 12 |- (((abs` ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N))))) e. RR /\ 1 e. RR /\ ((d / (sqr` N)) e. RR /\ 0 < (d / (sqr` N)))) -> ((abs` ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N))))) < 1 <-> ((d / (sqr` N)) x. (abs` ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N)))))) < ((d / (sqr` N)) x. 1)))
120	108, 111, 83, 118, 119	syl112anc 1104	. . . . . . . . . . 11 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> ((abs` ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N))))) < 1 <-> ((d / (sqr` N)) x. (abs` ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N)))))) < ((d / (sqr` N)) x. 1)))
121	116, 120	mpbid 212	. . . . . . . . . 10 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> ((d / (sqr` N)) x. (abs` ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N)))))) < ((d / (sqr` N)) x. 1))
122		recn 6466	. . . . . . . . . . . . . . . 16 |- ((x` a) e. RR -> (x` a) e. CC)
123	122	ad2antrl 442	. . . . . . . . . . . . . . 15 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (x` a) e. CC)
124	96	recnd 6468	. . . . . . . . . . . . . . 15 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) e. CC)
125		addcom 6458	. . . . . . . . . . . . . . 15 |- (((x` a) e. CC /\ ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) e. CC) -> ((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))))) = (((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) + (x` a)))
126	123, 124, 125	syl11anc 524	. . . . . . . . . . . . . 14 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> ((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))))) = (((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) + (x` a)))
127	126	opreq1d 4897	. . . . . . . . . . . . 13 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))))) - (y` a)) = ((((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) + (x` a)) - (y` a)))
128		recn 6466	. . . . . . . . . . . . . . 15 |- ((y` a) e. RR -> (y` a) e. CC)
129	128	ad2antll 443	. . . . . . . . . . . . . 14 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (y` a) e. CC)
130		subsub3 6628	. . . . . . . . . . . . . 14 |- ((((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) e. CC /\ (y` a) e. CC /\ (x` a) e. CC) -> (((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) - ((y` a) - (x` a))) = ((((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) + (x` a)) - (y` a)))
131	124, 129, 123, 130	syl111anc 1100	. . . . . . . . . . . . 13 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) - ((y` a) - (x` a))) = ((((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) + (x` a)) - (y` a)))
132	87	recnd 6468	. . . . . . . . . . . . . . . . 17 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> ((y` a) - (x` a)) e. CC)
133		rpcn 7237	. . . . . . . . . . . . . . . . . 18 |- ((d / (sqr` N)) e. RR+ -> (d / (sqr` N)) e. CC)
134	133	adantr 425	. . . . . . . . . . . . . . . . 17 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (d / (sqr` N)) e. CC)
135		divcan2 6910	. . . . . . . . . . . . . . . . 17 |- ((((y` a) - (x` a)) e. CC /\ (d / (sqr` N)) e. CC /\ (d / (sqr` N)) =/= 0) -> ((d / (sqr` N)) x. (((y` a) - (x` a)) / (d / (sqr` N)))) = ((y` a) - (x` a)))
136	132, 134, 88, 135	syl111anc 1100	. . . . . . . . . . . . . . . 16 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> ((d / (sqr` N)) x. (((y` a) - (x` a)) / (d / (sqr` N)))) = ((y` a) - (x` a)))
137	136	eqcomd 1889	. . . . . . . . . . . . . . 15 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> ((y` a) - (x` a)) = ((d / (sqr` N)) x. (((y` a) - (x` a)) / (d / (sqr` N)))))
138	137	opreq2d 4898	. . . . . . . . . . . . . 14 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) - ((y` a) - (x` a))) = (((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) - ((d / (sqr` N)) x. (((y` a) - (x` a)) / (d / (sqr` N))))))
139		subdi 6590	. . . . . . . . . . . . . . 15 |- (((d / (sqr` N)) e. CC /\ (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) e. CC /\ (((y` a) - (x` a)) / (d / (sqr` N))) e. CC) -> ((d / (sqr` N)) x. ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N))))) = (((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) - ((d / (sqr` N)) x. (((y` a) - (x` a)) / (d / (sqr` N))))))
140	134, 103, 104, 139	syl111anc 1100	. . . . . . . . . . . . . 14 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> ((d / (sqr` N)) x. ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N))))) = (((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) - ((d / (sqr` N)) x. (((y` a) - (x` a)) / (d / (sqr` N))))))
141	138, 140	eqtr4d 1928	. . . . . . . . . . . . 13 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))) - ((y` a) - (x` a))) = ((d / (sqr` N)) x. ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N))))))
142	127, 131, 141	3eqtr2d 1932	. . . . . . . . . . . 12 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))))) - (y` a)) = ((d / (sqr` N)) x. ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N))))))
143	142	fveq2d 4685	. . . . . . . . . . 11 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (abs` (((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))))) - (y` a))) = (abs` ((d / (sqr` N)) x. ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N)))))))
144		absmul 8109	. . . . . . . . . . . 12 |- (((d / (sqr` N)) e. CC /\ ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N)))) e. CC) -> (abs` ((d / (sqr` N)) x. ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N)))))) = ((abs` (d / (sqr` N))) x. (abs` ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N)))))))
145	134, 106, 144	syl11anc 524	. . . . . . . . . . 11 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (abs` ((d / (sqr` N)) x. ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N)))))) = ((abs` (d / (sqr` N))) x. (abs` ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N)))))))
146		rpge0 7241	. . . . . . . . . . . . . 14 |- ((d / (sqr` N)) e. RR+ -> 0 <_ (d / (sqr` N)))
147	146	adantr 425	. . . . . . . . . . . . 13 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> 0 <_ (d / (sqr` N)))
148		absid 8113	. . . . . . . . . . . . 13 |- (((d / (sqr` N)) e. RR /\ 0 <_ (d / (sqr` N))) -> (abs` (d / (sqr` N))) = (d / (sqr` N)))
149	83, 147, 148	syl11anc 524	. . . . . . . . . . . 12 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (abs` (d / (sqr` N))) = (d / (sqr` N)))
150	149	opreq1d 4897	. . . . . . . . . . 11 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> ((abs` (d / (sqr` N))) x. (abs` ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N)))))) = ((d / (sqr` N)) x. (abs` ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N)))))))
151	143, 145, 150	3eqtrd 1929	. . . . . . . . . 10 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (abs` (((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))))) - (y` a))) = ((d / (sqr` N)) x. (abs` ((|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))) - (((y` a) - (x` a)) / (d / (sqr` N)))))))
152		mulid1 6464	. . . . . . . . . . . . 13 |- ((d / (sqr` N)) e. CC -> ((d / (sqr` N)) x. 1) = (d / (sqr` N)))
153	133, 152	syl 12	. . . . . . . . . . . 12 |- ((d / (sqr` N)) e. RR+ -> ((d / (sqr` N)) x. 1) = (d / (sqr` N)))
154	153	adantr 425	. . . . . . . . . . 11 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> ((d / (sqr` N)) x. 1) = (d / (sqr` N)))
155	154	eqcomd 1889	. . . . . . . . . 10 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (d / (sqr` N)) = ((d / (sqr` N)) x. 1))
156	121, 151, 155	3brtr4d 3367	. . . . . . . . 9 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (abs` (((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2))))) - (y` a))) < (d / (sqr` N)))
157	102, 156	eqbrtrd 3357	. . . . . . . 8 |- (((d / (sqr` N)) e. RR+ /\ ((x` a) e. RR /\ (y` a) e. RR)) -> (((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))))((abs o. - ) |` (RR X. RR))(y` a)) < (d / (sqr` N)))
158		ffvelrn 4787	. . . . . . . . . . 11 |- ((x:(1...N)-->RR /\ a e. (1...N)) -> (x` a) e. RR)
159	158, 5	sylanb 498	. . . . . . . . . 10 |- ((x e. (RR ^m (1...N)) /\ a e. (1...N)) -> (x` a) e. RR)
160	159	adantlr 429	. . . . . . . . 9 |- (((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ a e. (1...N)) -> (x` a) e. RR)
161		ffvelrn 4787	. . . . . . . . . . 11 |- ((y:(1...N)-->RR /\ a e. (1...N)) -> (y` a) e. RR)
162	161, 21	sylanb 498	. . . . . . . . . 10 |- ((y e. (RR ^m (1...N)) /\ a e. (1...N)) -> (y` a) e. RR)
163	162	adantll 428	. . . . . . . . 9 |- (((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ a e. (1...N)) -> (y` a) e. RR)
164	160, 163	jca 310	. . . . . . . 8 |- (((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ a e. (1...N)) -> ((x` a) e. RR /\ (y` a) e. RR))
165	157, 60, 164	syl2an 503	. . . . . . 7 |- (((N e. NN /\ d e. RR+) /\ ((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ a e. (1...N))) -> (((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))))((abs o. - ) |` (RR X. RR))(y` a)) < (d / (sqr` N)))
166	165	anassrs 489	. . . . . 6 |- ((((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) /\ a e. (1...N)) -> (((x` a) + ((d / (sqr` N)) x. (|_` ((((y` a) - (x` a)) / (d / (sqr` N))) + (1 / 2)))))((abs o. - ) |` (RR X. RR))(y` a)) < (d / (sqr` N)))
167	81, 166	eqbrtrd 3357	. . . . 5 |- ((((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) /\ a e. (1...N)) -> (((x` a) + ((d / (sqr` N)) x. ({<.n, v>. | (n e. (1...N) /\ v = (|_` ((((y` n) - (x` n)) / (d / (sqr` N))) + (1 / 2))))}` a)))((abs o. - ) |` (RR X. RR))(y` a)) < (d / (sqr` N)))
168	69, 167	eqbrtrd 3357	. . . 4 |- ((((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) /\ a e. (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))))}` a)((abs o. - ) |` (RR X. RR))(y` a)) < (d / (sqr` N)))
169	168	r19.21aiva 2176	. . 3 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> A.a e. (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))))}` a)((abs o. - ) |` (RR X. RR))(y` a)) < (d / (sqr` N)))
170		eqid 1884	. . . 4 |- (RR ^m (1...N)) = (RR ^m (1...N))
171	100, 170	rrndstprj2 16018	. . 3 |- (((N e. NN /\ {<.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)) /\ y e. (RR ^m (1...N))) /\ ((d / (sqr` N)) e. RR+ /\ A.a e. (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))))}` a)((abs o. - ) |` (RR X. RR))(y` a)) < (d / (sqr` 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 / (sqr` N)) x. (sqr` N)))
172	1, 58, 59, 61, 169, 171	syl32anc 1108	. 2 |- (((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 / (sqr` N)) x. (sqr` N)))
173		rpcn 7237	. . . . 5 |- (d e. RR+ -> d e. CC)
174	173	adantl 424	. . . 4 |- ((N e. NN /\ d e. RR+) -> d e. CC)
175		rpcn 7237	. . . . . 6 |- ((sqr` N) e. RR+ -> (sqr` N) e. CC)
176	10, 11, 175	3syl 24	. . . . 5 |- (N e. NN -> (sqr` N) e. CC)
177	176	adantr 425	. . . 4 |- ((N e. NN /\ d e. RR+) -> (sqr` N) e. CC)
178		rpne0 7243	. . . . . 6 |- ((sqr` N) e. RR+ -> (sqr` N) =/= 0)
179	10, 11, 178	3syl 24	. . . . 5 |- (N e. NN -> (sqr` N) =/= 0)
180	179	adantr 425	. . . 4 |- ((N e. NN /\ d e. RR+) -> (sqr` N) =/= 0)
181		divcan1 6909	. . . 4 |- ((d e. CC /\ (sqr` N) e. CC /\ (sqr` N) =/= 0) -> ((d / (sqr` N)) x. (sqr` N)) = d)
182	174, 177, 180, 181	syl111anc 1100	. . 3 |- ((N e. NN /\ d e. RR+) -> ((d / (sqr` N)) x. (sqr` N)) = d)
183	182	adantr 425	. 2 |- (((N e. NN /\ d e. RR+) /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> ((d / (sqr` N)) x. (sqr` N)) = d)
184	172, 183	breqtrd 3361	1 |- (((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)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105   class class class wbr 3338  {copab 3395   X. cxp 3984   |` cres 3988   o. ccom 3990  -->wf 3994  ` cfv 3998  (class class class)co 4884   ^m cmap 5381  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445   / cdiv 6447   <_ cle 6448  NNcn 6449  ZZcz 6451  RR+crp 6453   < clt 6653  2c2 7145  |_cfl 7462  ...cfz 7637  sqrcsqr 7919  abscabs 8000  RRncrrn 16011

This theorem is referenced by:  rrntotbnd 16022

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-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-rp 7232  df-n0 7309  df-z 7345  df-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-rrn 16012

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