Theorem rrndstprj2 16018

Description: Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 16017 can be used to show that the supremum norm and Euclidean norm are equivalent.

Hypotheses

Ref	Expression
rrndstprj1.1	|- M = ((abs o. - ) |` (RR X. RR))
rrndstprj1.2	|- X = (RR ^m (1...N))

Assertion

Ref	Expression
rrndstprj2	|- (((N e. NN /\ F e. X /\ G e. X) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) -> (F(RRn` N)G) < (R x. (sqr` N)))

Distinct variable groups:   n,M   n,X   n,N   n,F   n,G   R,n

Proof of Theorem rrndstprj2

Step	Hyp	Ref	Expression
1		rrnmval 16014	. . . 4 |- ((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) -> (F(RRn` N)G) = (sqr` sum_k e. (1...N)(((F` k) - (G` k))^2)))
2	1	adantr 425	. . 3 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) -> (F(RRn` N)G) = (sqr` sum_k e. (1...N)(((F` k) - (G` k))^2)))
3		elnnuz 7609	. . . . . . . . . 10 |- (N e. NN <-> N e. (ZZ>=` 1))
4	3	biimpi 168	. . . . . . . . 9 |- (N e. NN -> N e. (ZZ>=` 1))
5	4	3ad2ant1 897	. . . . . . . 8 |- ((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) -> N e. (ZZ>=` 1))
6	5	adantr 425	. . . . . . 7 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) -> N e. (ZZ>=` 1))
7		ffvelrn 4787	. . . . . . . . . . . . . 14 |- ((F:(1...N)-->RR /\ k e. (1...N)) -> (F` k) e. RR)
8		reex 6465	. . . . . . . . . . . . . . 15 |- RR e. _V
9		oprex 4907	. . . . . . . . . . . . . . 15 |- (1...N) e. _V
10	8, 9	elmap 5393	. . . . . . . . . . . . . 14 |- (F e. (RR ^m (1...N)) <-> F:(1...N)-->RR)
11	7, 10	sylanb 498	. . . . . . . . . . . . 13 |- ((F e. (RR ^m (1...N)) /\ k e. (1...N)) -> (F` k) e. RR)
12	11	3ad2antl2 1039	. . . . . . . . . . . 12 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ k e. (1...N)) -> (F` k) e. RR)
13		ffvelrn 4787	. . . . . . . . . . . . . 14 |- ((G:(1...N)-->RR /\ k e. (1...N)) -> (G` k) e. RR)
14	8, 9	elmap 5393	. . . . . . . . . . . . . 14 |- (G e. (RR ^m (1...N)) <-> G:(1...N)-->RR)
15	13, 14	sylanb 498	. . . . . . . . . . . . 13 |- ((G e. (RR ^m (1...N)) /\ k e. (1...N)) -> (G` k) e. RR)
16	15	3ad2antl3 1040	. . . . . . . . . . . 12 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ k e. (1...N)) -> (G` k) e. RR)
17		resubcl 6601	. . . . . . . . . . . 12 |- (((F` k) e. RR /\ (G` k) e. RR) -> ((F` k) - (G` k)) e. RR)
18	12, 16, 17	syl11anc 524	. . . . . . . . . . 11 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ k e. (1...N)) -> ((F` k) - (G` k)) e. RR)
19		resqcl 7866	. . . . . . . . . . 11 |- (((F` k) - (G` k)) e. RR -> (((F` k) - (G` k))^2) e. RR)
20	18, 19	syl 12	. . . . . . . . . 10 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ k e. (1...N)) -> (((F` k) - (G` k))^2) e. RR)
21	20	adantlr 429	. . . . . . . . 9 |- ((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) /\ k e. (1...N)) -> (((F` k) - (G` k))^2) e. RR)
22		rpre 7236	. . . . . . . . . . . 12 |- (R e. RR+ -> R e. RR)
23		resqcl 7866	. . . . . . . . . . . 12 |- (R e. RR -> (R^2) e. RR)
24	22, 23	syl 12	. . . . . . . . . . 11 |- (R e. RR+ -> (R^2) e. RR)
25	24	adantr 425	. . . . . . . . . 10 |- ((R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R) -> (R^2) e. RR)
26	25	ad2antlr 441	. . . . . . . . 9 |- ((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) /\ k e. (1...N)) -> (R^2) e. RR)
27		rrndstprj1.1	. . . . . . . . . . . . . . . . . . 19 |- M = ((abs o. - ) |` (RR X. RR))
28	27	remetdval 9186	. . . . . . . . . . . . . . . . . 18 |- (((F` k) e. RR /\ (G` k) e. RR) -> ((F` k)M(G` k)) = (abs` ((F` k) - (G` k))))
29	12, 16, 28	syl11anc 524	. . . . . . . . . . . . . . . . 17 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ k e. (1...N)) -> ((F` k)M(G` k)) = (abs` ((F` k) - (G` k))))
30	29	opreq1d 4897	. . . . . . . . . . . . . . . 16 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ k e. (1...N)) -> (((F` k)M(G` k))^2) = ((abs` ((F` k) - (G` k)))^2))
31		absresq 8118	. . . . . . . . . . . . . . . . 17 |- (((F` k) - (G` k)) e. RR -> ((abs` ((F` k) - (G` k)))^2) = (((F` k) - (G` k))^2))
32	18, 31	syl 12	. . . . . . . . . . . . . . . 16 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ k e. (1...N)) -> ((abs` ((F` k) - (G` k)))^2) = (((F` k) - (G` k))^2))
33	30, 32	eqtrd 1925	. . . . . . . . . . . . . . 15 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ k e. (1...N)) -> (((F` k)M(G` k))^2) = (((F` k) - (G` k))^2))
34	33	ad2ant2r 445	. . . . . . . . . . . . . 14 |- ((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) /\ (k e. (1...N) /\ ((F` k)M(G` k)) < R)) -> (((F` k)M(G` k))^2) = (((F` k) - (G` k))^2))
35	27	remet 9188	. . . . . . . . . . . . . . . . . . 19 |- M e. Met
36	35	a1i 8	. . . . . . . . . . . . . . . . . 18 |- ((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) /\ k e. (1...N)) -> M e. Met)
37	12	adantlr 429	. . . . . . . . . . . . . . . . . 18 |- ((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) /\ k e. (1...N)) -> (F` k) e. RR)
38	16	adantlr 429	. . . . . . . . . . . . . . . . . 18 |- ((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) /\ k e. (1...N)) -> (G` k) e. RR)
39	27	remetba 9187	. . . . . . . . . . . . . . . . . . 19 |- RR = dom dom M
40	39	metcl 9088	. . . . . . . . . . . . . . . . . 18 |- ((M e. Met /\ (F` k) e. RR /\ (G` k) e. RR) -> ((F` k)M(G` k)) e. RR)
41	36, 37, 38, 40	syl111anc 1100	. . . . . . . . . . . . . . . . 17 |- ((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) /\ k e. (1...N)) -> ((F` k)M(G` k)) e. RR)
42	39	metge0 9096	. . . . . . . . . . . . . . . . . 18 |- ((M e. Met /\ (F` k) e. RR /\ (G` k) e. RR) -> 0 <_ ((F` k)M(G` k)))
43	36, 37, 38, 42	syl111anc 1100	. . . . . . . . . . . . . . . . 17 |- ((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) /\ k e. (1...N)) -> 0 <_ ((F` k)M(G` k)))
44	22	ad2antlr 441	. . . . . . . . . . . . . . . . 17 |- ((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) /\ k e. (1...N)) -> R e. RR)
45		rpge0 7241	. . . . . . . . . . . . . . . . . 18 |- (R e. RR+ -> 0 <_ R)
46	45	ad2antlr 441	. . . . . . . . . . . . . . . . 17 |- ((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) /\ k e. (1...N)) -> 0 <_ R)
47		lt2sq 7875	. . . . . . . . . . . . . . . . 17 |- (((((F` k)M(G` k)) e. RR /\ 0 <_ ((F` k)M(G` k))) /\ (R e. RR /\ 0 <_ R)) -> (((F` k)M(G` k)) < R <-> (((F` k)M(G` k))^2) < (R^2)))
48	41, 43, 44, 46, 47	syl22anc 1101	. . . . . . . . . . . . . . . 16 |- ((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) /\ k e. (1...N)) -> (((F` k)M(G` k)) < R <-> (((F` k)M(G` k))^2) < (R^2)))
49	48	biimpa 460	. . . . . . . . . . . . . . 15 |- (((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) /\ k e. (1...N)) /\ ((F` k)M(G` k)) < R) -> (((F` k)M(G` k))^2) < (R^2))
50	49	anasss 488	. . . . . . . . . . . . . 14 |- ((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) /\ (k e. (1...N) /\ ((F` k)M(G` k)) < R)) -> (((F` k)M(G` k))^2) < (R^2))
51	34, 50	eqbrtrrd 3359	. . . . . . . . . . . . 13 |- ((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) /\ (k e. (1...N) /\ ((F` k)M(G` k)) < R)) -> (((F` k) - (G` k))^2) < (R^2))
52		fveq2 4681	. . . . . . . . . . . . . . . . . 18 |- (n = k -> (F` n) = (F` k))
53		fveq2 4681	. . . . . . . . . . . . . . . . . 18 |- (n = k -> (G` n) = (G` k))
54	52, 53	opreq12d 4900	. . . . . . . . . . . . . . . . 17 |- (n = k -> ((F` n)M(G` n)) = ((F` k)M(G` k)))
55	54	breq1d 3348	. . . . . . . . . . . . . . . 16 |- (n = k -> (((F` n)M(G` n)) < R <-> ((F` k)M(G` k)) < R))
56	55	rcla4v 2376	. . . . . . . . . . . . . . 15 |- (k e. (1...N) -> (A.n e. (1...N)((F` n)M(G` n)) < R -> ((F` k)M(G` k)) < R))
57	56	imdistani 491	. . . . . . . . . . . . . 14 |- ((k e. (1...N) /\ A.n e. (1...N)((F` n)M(G` n)) < R) -> (k e. (1...N) /\ ((F` k)M(G` k)) < R))
58	57	ancoms 484	. . . . . . . . . . . . 13 |- ((A.n e. (1...N)((F` n)M(G` n)) < R /\ k e. (1...N)) -> (k e. (1...N) /\ ((F` k)M(G` k)) < R))
59	51, 58	sylan2 500	. . . . . . . . . . . 12 |- ((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) /\ (A.n e. (1...N)((F` n)M(G` n)) < R /\ k e. (1...N))) -> (((F` k) - (G` k))^2) < (R^2))
60	59	expr 418	. . . . . . . . . . 11 |- ((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) /\ A.n e. (1...N)((F` n)M(G` n)) < R) -> (k e. (1...N) -> (((F` k) - (G` k))^2) < (R^2)))
61	60	anasss 488	. . . . . . . . . 10 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) -> (k e. (1...N) -> (((F` k) - (G` k))^2) < (R^2)))
62	61	imp 377	. . . . . . . . 9 |- ((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) /\ k e. (1...N)) -> (((F` k) - (G` k))^2) < (R^2))
63	21, 26, 62	3jca 1050	. . . . . . . 8 |- ((((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) /\ k e. (1...N)) -> ((((F` k) - (G` k))^2) e. RR /\ (R^2) e. RR /\ (((F` k) - (G` k))^2) < (R^2)))
64	63	r19.21aiva 2176	. . . . . . 7 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) -> A.k e. (1...N)((((F` k) - (G` k))^2) e. RR /\ (R^2) e. RR /\ (((F` k) - (G` k))^2) < (R^2)))
65		fsumlt 15821	. . . . . . 7 |- ((N e. (ZZ>=` 1) /\ A.k e. (1...N)((((F` k) - (G` k))^2) e. RR /\ (R^2) e. RR /\ (((F` k) - (G` k))^2) < (R^2))) -> sum_k e. (1...N)(((F` k) - (G` k))^2) < sum_k e. (1...N)(R^2))
66	6, 64, 65	syl11anc 524	. . . . . 6 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) -> sum_k e. (1...N)(((F` k) - (G` k))^2) < sum_k e. (1...N)(R^2))
67		fsumconst 8298	. . . . . . . . . 10 |- ((N e. (ZZ>=` 1) /\ (R^2) e. CC) -> sum_k e. (1...N)(R^2) = (((N - 1) + 1) x. (R^2)))
68		rpcn 7237	. . . . . . . . . . 11 |- (R e. RR+ -> R e. CC)
69		sqcl 7856	. . . . . . . . . . 11 |- (R e. CC -> (R^2) e. CC)
70	68, 69	syl 12	. . . . . . . . . 10 |- (R e. RR+ -> (R^2) e. CC)
71	67, 4, 70	syl2an 503	. . . . . . . . 9 |- ((N e. NN /\ R e. RR+) -> sum_k e. (1...N)(R^2) = (((N - 1) + 1) x. (R^2)))
72		npcan 6559	. . . . . . . . . . . 12 |- ((N e. CC /\ 1 e. CC) -> ((N - 1) + 1) = N)
73		nncn 7113	. . . . . . . . . . . 12 |- (N e. NN -> N e. CC)
74		ax1cn 6422	. . . . . . . . . . . 12 |- 1 e. CC
75	72, 73, 74	sylancl 525	. . . . . . . . . . 11 |- (N e. NN -> ((N - 1) + 1) = N)
76	75	adantr 425	. . . . . . . . . 10 |- ((N e. NN /\ R e. RR+) -> ((N - 1) + 1) = N)
77	76	opreq1d 4897	. . . . . . . . 9 |- ((N e. NN /\ R e. RR+) -> (((N - 1) + 1) x. (R^2)) = (N x. (R^2)))
78	71, 77	eqtrd 1925	. . . . . . . 8 |- ((N e. NN /\ R e. RR+) -> sum_k e. (1...N)(R^2) = (N x. (R^2)))
79	78	3ad2antl1 1038	. . . . . . 7 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) -> sum_k e. (1...N)(R^2) = (N x. (R^2)))
80	79	adantrr 431	. . . . . 6 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) -> sum_k e. (1...N)(R^2) = (N x. (R^2)))
81	66, 80	breqtrd 3361	. . . . 5 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) -> sum_k e. (1...N)(((F` k) - (G` k))^2) < (N x. (R^2)))
82	20	r19.21aiva 2176	. . . . . . . 8 |- ((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) -> A.k e. (1...N)(((F` k) - (G` k))^2) e. RR)
83		fsumrecl 8277	. . . . . . . 8 |- ((N e. (ZZ>=` 1) /\ A.k e. (1...N)(((F` k) - (G` k))^2) e. RR) -> sum_k e. (1...N)(((F` k) - (G` k))^2) e. RR)
84	5, 82, 83	syl11anc 524	. . . . . . 7 |- ((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) -> sum_k e. (1...N)(((F` k) - (G` k))^2) e. RR)
85		sqge0 7878	. . . . . . . . . . 11 |- (((F` k) - (G` k)) e. RR -> 0 <_ (((F` k) - (G` k))^2))
86	18, 85	syl 12	. . . . . . . . . 10 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ k e. (1...N)) -> 0 <_ (((F` k) - (G` k))^2))
87	20, 86	jca 310	. . . . . . . . 9 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ k e. (1...N)) -> ((((F` k) - (G` k))^2) e. RR /\ 0 <_ (((F` k) - (G` k))^2)))
88	87	r19.21aiva 2176	. . . . . . . 8 |- ((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) -> A.k e. (1...N)((((F` k) - (G` k))^2) e. RR /\ 0 <_ (((F` k) - (G` k))^2)))
89		fsumcmp0 8301	. . . . . . . 8 |- ((N e. (ZZ>=` 1) /\ A.k e. (1...N)((((F` k) - (G` k))^2) e. RR /\ 0 <_ (((F` k) - (G` k))^2))) -> 0 <_ sum_k e. (1...N)(((F` k) - (G` k))^2))
90	5, 88, 89	syl11anc 524	. . . . . . 7 |- ((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) -> 0 <_ sum_k e. (1...N)(((F` k) - (G` k))^2))
91		sqsqr 7973	. . . . . . 7 |- ((sum_k e. (1...N)(((F` k) - (G` k))^2) e. RR /\ 0 <_ sum_k e. (1...N)(((F` k) - (G` k))^2)) -> ((sqr` sum_k e. (1...N)(((F` k) - (G` k))^2))^2) = sum_k e. (1...N)(((F` k) - (G` k))^2))
92	84, 90, 91	syl11anc 524	. . . . . 6 |- ((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) -> ((sqr` sum_k e. (1...N)(((F` k) - (G` k))^2))^2) = sum_k e. (1...N)(((F` k) - (G` k))^2))
93	92	adantr 425	. . . . 5 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) -> ((sqr` sum_k e. (1...N)(((F` k) - (G` k))^2))^2) = sum_k e. (1...N)(((F` k) - (G` k))^2))
94		sqmul 7857	. . . . . . . . 9 |- (((sqr` N) e. CC /\ R e. CC) -> (((sqr` N) x. R)^2) = (((sqr` N)^2) x. (R^2)))
95		nnre 7112	. . . . . . . . . . 11 |- (N e. NN -> N e. RR)
96		nnnn0 7315	. . . . . . . . . . . 12 |- (N e. NN -> N e. NN0)
97		nn0ge0 7326	. . . . . . . . . . . 12 |- (N e. NN0 -> 0 <_ N)
98	96, 97	syl 12	. . . . . . . . . . 11 |- (N e. NN -> 0 <_ N)
99		sqrcl 7960	. . . . . . . . . . 11 |- ((N e. RR /\ 0 <_ N) -> (sqr` N) e. RR)
100	95, 98, 99	syl11anc 524	. . . . . . . . . 10 |- (N e. NN -> (sqr` N) e. RR)
101	100	recnd 6468	. . . . . . . . 9 |- (N e. NN -> (sqr` N) e. CC)
102	94, 101, 68	syl2an 503	. . . . . . . 8 |- ((N e. NN /\ R e. RR+) -> (((sqr` N) x. R)^2) = (((sqr` N)^2) x. (R^2)))
103		mulcom 6459	. . . . . . . . . 10 |- (((sqr` N) e. CC /\ R e. CC) -> ((sqr` N) x. R) = (R x. (sqr` N)))
104	103, 101, 68	syl2an 503	. . . . . . . . 9 |- ((N e. NN /\ R e. RR+) -> ((sqr` N) x. R) = (R x. (sqr` N)))
105	104	opreq1d 4897	. . . . . . . 8 |- ((N e. NN /\ R e. RR+) -> (((sqr` N) x. R)^2) = ((R x. (sqr` N))^2))
106		sqsqr 7973	. . . . . . . . . . 11 |- ((N e. RR /\ 0 <_ N) -> ((sqr` N)^2) = N)
107	95, 98, 106	syl11anc 524	. . . . . . . . . 10 |- (N e. NN -> ((sqr` N)^2) = N)
108	107	adantr 425	. . . . . . . . 9 |- ((N e. NN /\ R e. RR+) -> ((sqr` N)^2) = N)
109	108	opreq1d 4897	. . . . . . . 8 |- ((N e. NN /\ R e. RR+) -> (((sqr` N)^2) x. (R^2)) = (N x. (R^2)))
110	102, 105, 109	3eqtr3d 1934	. . . . . . 7 |- ((N e. NN /\ R e. RR+) -> ((R x. (sqr` N))^2) = (N x. (R^2)))
111	110	3ad2antl1 1038	. . . . . 6 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) -> ((R x. (sqr` N))^2) = (N x. (R^2)))
112	111	adantrr 431	. . . . 5 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) -> ((R x. (sqr` N))^2) = (N x. (R^2)))
113	81, 93, 112	3brtr4d 3367	. . . 4 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) -> ((sqr` sum_k e. (1...N)(((F` k) - (G` k))^2))^2) < ((R x. (sqr` N))^2))
114		sqrcl 7960	. . . . . . . 8 |- ((sum_k e. (1...N)(((F` k) - (G` k))^2) e. RR /\ 0 <_ sum_k e. (1...N)(((F` k) - (G` k))^2)) -> (sqr` sum_k e. (1...N)(((F` k) - (G` k))^2)) e. RR)
115	84, 90, 114	syl11anc 524	. . . . . . 7 |- ((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) -> (sqr` sum_k e. (1...N)(((F` k) - (G` k))^2)) e. RR)
116	115	adantr 425	. . . . . 6 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) -> (sqr` sum_k e. (1...N)(((F` k) - (G` k))^2)) e. RR)
117		sqrge0 7962	. . . . . . . 8 |- ((sum_k e. (1...N)(((F` k) - (G` k))^2) e. RR /\ 0 <_ sum_k e. (1...N)(((F` k) - (G` k))^2)) -> 0 <_ (sqr` sum_k e. (1...N)(((F` k) - (G` k))^2)))
118	84, 90, 117	syl11anc 524	. . . . . . 7 |- ((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) -> 0 <_ (sqr` sum_k e. (1...N)(((F` k) - (G` k))^2)))
119	118	adantr 425	. . . . . 6 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) -> 0 <_ (sqr` sum_k e. (1...N)(((F` k) - (G` k))^2)))
120		remulcl 6457	. . . . . . . . 9 |- ((R e. RR /\ (sqr` N) e. RR) -> (R x. (sqr` N)) e. RR)
121	120, 22, 100	syl2an 503	. . . . . . . 8 |- ((R e. RR+ /\ N e. NN) -> (R x. (sqr` N)) e. RR)
122	121	ancoms 484	. . . . . . 7 |- ((N e. NN /\ R e. RR+) -> (R x. (sqr` N)) e. RR)
123	122	3ad2antl1 1038	. . . . . 6 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) -> (R x. (sqr` N)) e. RR)
124		mulge0 6868	. . . . . . . . 9 |- (((R e. RR /\ 0 <_ R) /\ ((sqr` N) e. RR /\ 0 <_ (sqr` N))) -> 0 <_ (R x. (sqr` N)))
125	22, 45	jca 310	. . . . . . . . 9 |- (R e. RR+ -> (R e. RR /\ 0 <_ R))
126		sqrge0 7962	. . . . . . . . . . 11 |- ((N e. RR /\ 0 <_ N) -> 0 <_ (sqr` N))
127	95, 98, 126	syl11anc 524	. . . . . . . . . 10 |- (N e. NN -> 0 <_ (sqr` N))
128	100, 127	jca 310	. . . . . . . . 9 |- (N e. NN -> ((sqr` N) e. RR /\ 0 <_ (sqr` N)))
129	124, 125, 128	syl2an 503	. . . . . . . 8 |- ((R e. RR+ /\ N e. NN) -> 0 <_ (R x. (sqr` N)))
130	129	ancoms 484	. . . . . . 7 |- ((N e. NN /\ R e. RR+) -> 0 <_ (R x. (sqr` N)))
131	130	3ad2antl1 1038	. . . . . 6 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) -> 0 <_ (R x. (sqr` N)))
132		lt2sq 7875	. . . . . 6 |- ((((sqr` sum_k e. (1...N)(((F` k) - (G` k))^2)) e. RR /\ 0 <_ (sqr` sum_k e. (1...N)(((F` k) - (G` k))^2))) /\ ((R x. (sqr` N)) e. RR /\ 0 <_ (R x. (sqr` N)))) -> ((sqr` sum_k e. (1...N)(((F` k) - (G` k))^2)) < (R x. (sqr` N)) <-> ((sqr` sum_k e. (1...N)(((F` k) - (G` k))^2))^2) < ((R x. (sqr` N))^2)))
133	116, 119, 123, 131, 132	syl22anc 1101	. . . . 5 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ R e. RR+) -> ((sqr` sum_k e. (1...N)(((F` k) - (G` k))^2)) < (R x. (sqr` N)) <-> ((sqr` sum_k e. (1...N)(((F` k) - (G` k))^2))^2) < ((R x. (sqr` N))^2)))
134	133	adantrr 431	. . . 4 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) -> ((sqr` sum_k e. (1...N)(((F` k) - (G` k))^2)) < (R x. (sqr` N)) <-> ((sqr` sum_k e. (1...N)(((F` k) - (G` k))^2))^2) < ((R x. (sqr` N))^2)))
135	113, 134	mpbird 213	. . 3 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) -> (sqr` sum_k e. (1...N)(((F` k) - (G` k))^2)) < (R x. (sqr` N)))
136	2, 135	eqbrtrd 3357	. 2 |- (((N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) -> (F(RRn` N)G) < (R x. (sqr` N)))
137		biid 187	. . 3 |- (N e. NN <-> N e. NN)
138		rrndstprj1.2	. . . 4 |- X = (RR ^m (1...N))
139	138	eleq2i 1961	. . 3 |- (F e. X <-> F e. (RR ^m (1...N)))
140	138	eleq2i 1961	. . 3 |- (G e. X <-> G e. (RR ^m (1...N)))
141	137, 139, 140	3anbi123i 1056	. 2 |- ((N e. NN /\ F e. X /\ G e. X) <-> (N e. NN /\ F e. (RR ^m (1...N)) /\ G e. (RR ^m (1...N))))
142	136, 141	sylanb 498	1 |- (((N e. NN /\ F e. X /\ G e. X) /\ (R e. RR+ /\ A.n e. (1...N)((F` n)M(G` n)) < R)) -> (F(RRn` N)G) < (R x. (sqr` N)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   class class class wbr 3338   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   <_ cle 6448  NNcn 6449  NN0cn0 6450  RR+crp 6453   < clt 6653  2c2 7145  ZZ>=cuz 7586  ...cfz 7637  ^cexp 7811  sqrcsqr 7919  abscabs 8000  sum_csu 8239  Metcme 9066  RRncrrn 16011

This theorem is referenced by:  rrncms 16019  rrntotbndlem2 16021

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-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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