Theorem rrnmet 16016

Description: Euclidean space is a metric space.

Assertion

Ref	Expression
rrnmet	|- (N e. NN -> (RRn` N) e. Met)

Proof of Theorem rrnmet

Step	Hyp	Ref	Expression
1		fvex 4689	. . 3 |- (RRn` N) e. _V
2		eqid 1884	. . . 4 |- dom dom (RRn` N) = dom dom (RRn` N)
3	2	ismet 9075	. . 3 |- ((RRn` N) e. _V -> ((RRn` N) e. Met <-> ((RRn` N):(dom dom (RRn` N) X. dom dom (RRn` N))-->RR /\ A.x e. dom dom (RRn` N)A.y e. dom dom (RRn` N)(((x(RRn` N)y) = 0 <-> x = y) /\ A.z e. dom dom (RRn` N)(x(RRn` N)y) <_ ((z(RRn` N)x) + (z(RRn` N)y))))))
4	1, 3	ax-mp 7	. 2 |- ((RRn` N) e. Met <-> ((RRn` N):(dom dom (RRn` N) X. dom dom (RRn` N))-->RR /\ A.x e. dom dom (RRn` N)A.y e. dom dom (RRn` N)(((x(RRn` N)y) = 0 <-> x = y) /\ A.z e. dom dom (RRn` N)(x(RRn` N)y) <_ ((z(RRn` N)x) + (z(RRn` N)y)))))
5		elnnuz 7609	. . . . . . . . . . 11 |- (N e. NN <-> N e. (ZZ>=` 1))
6	5	biimpi 168	. . . . . . . . . 10 |- (N e. NN -> N e. (ZZ>=` 1))
7	6	3ad2ant1 897	. . . . . . . . 9 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> N e. (ZZ>=` 1))
8		ffvelrn 4787	. . . . . . . . . . . . . 14 |- ((x:(1...N)-->RR /\ k e. (1...N)) -> (x` k) e. RR)
9		reex 6465	. . . . . . . . . . . . . . 15 |- RR e. _V
10		oprex 4907	. . . . . . . . . . . . . . 15 |- (1...N) e. _V
11	9, 10	elmap 5393	. . . . . . . . . . . . . 14 |- (x e. (RR ^m (1...N)) <-> x:(1...N)-->RR)
12	8, 11	sylanb 498	. . . . . . . . . . . . 13 |- ((x e. (RR ^m (1...N)) /\ k e. (1...N)) -> (x` k) e. RR)
13	12	3ad2antl2 1039	. . . . . . . . . . . 12 |- (((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ k e. (1...N)) -> (x` k) e. RR)
14		ffvelrn 4787	. . . . . . . . . . . . . 14 |- ((y:(1...N)-->RR /\ k e. (1...N)) -> (y` k) e. RR)
15	9, 10	elmap 5393	. . . . . . . . . . . . . 14 |- (y e. (RR ^m (1...N)) <-> y:(1...N)-->RR)
16	14, 15	sylanb 498	. . . . . . . . . . . . 13 |- ((y e. (RR ^m (1...N)) /\ k e. (1...N)) -> (y` k) e. RR)
17	16	3ad2antl3 1040	. . . . . . . . . . . 12 |- (((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ k e. (1...N)) -> (y` k) e. RR)
18		resubcl 6601	. . . . . . . . . . . 12 |- (((x` k) e. RR /\ (y` k) e. RR) -> ((x` k) - (y` k)) e. RR)
19	13, 17, 18	syl11anc 524	. . . . . . . . . . 11 |- (((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ k e. (1...N)) -> ((x` k) - (y` k)) e. RR)
20		resqcl 7866	. . . . . . . . . . 11 |- (((x` k) - (y` k)) e. RR -> (((x` k) - (y` k))^2) e. RR)
21	19, 20	syl 12	. . . . . . . . . 10 |- (((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ k e. (1...N)) -> (((x` k) - (y` k))^2) e. RR)
22	21	r19.21aiva 2176	. . . . . . . . 9 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> A.k e. (1...N)(((x` k) - (y` k))^2) e. RR)
23		fsumrecl 8277	. . . . . . . . 9 |- ((N e. (ZZ>=` 1) /\ A.k e. (1...N)(((x` k) - (y` k))^2) e. RR) -> sum_k e. (1...N)(((x` k) - (y` k))^2) e. RR)
24	7, 22, 23	syl11anc 524	. . . . . . . 8 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> sum_k e. (1...N)(((x` k) - (y` k))^2) e. RR)
25		sqge0 7878	. . . . . . . . . . . 12 |- (((x` k) - (y` k)) e. RR -> 0 <_ (((x` k) - (y` k))^2))
26	19, 25	syl 12	. . . . . . . . . . 11 |- (((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ k e. (1...N)) -> 0 <_ (((x` k) - (y` k))^2))
27	21, 26	jca 310	. . . . . . . . . 10 |- (((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ k e. (1...N)) -> ((((x` k) - (y` k))^2) e. RR /\ 0 <_ (((x` k) - (y` k))^2)))
28	27	r19.21aiva 2176	. . . . . . . . 9 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> A.k e. (1...N)((((x` k) - (y` k))^2) e. RR /\ 0 <_ (((x` k) - (y` k))^2)))
29		fsumcmp0 8301	. . . . . . . . 9 |- ((N e. (ZZ>=` 1) /\ A.k e. (1...N)((((x` k) - (y` k))^2) e. RR /\ 0 <_ (((x` k) - (y` k))^2))) -> 0 <_ sum_k e. (1...N)(((x` k) - (y` k))^2))
30	7, 28, 29	syl11anc 524	. . . . . . . 8 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> 0 <_ sum_k e. (1...N)(((x` k) - (y` k))^2))
31		sqrcl 7960	. . . . . . . 8 |- ((sum_k e. (1...N)(((x` k) - (y` k))^2) e. RR /\ 0 <_ sum_k e. (1...N)(((x` k) - (y` k))^2)) -> (sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)) e. RR)
32	24, 30, 31	syl11anc 524	. . . . . . 7 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> (sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)) e. RR)
33	32	3expib 1070	. . . . . 6 |- (N e. NN -> ((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> (sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)) e. RR))
34	33	r19.21aivv 2183	. . . . 5 |- (N e. NN -> A.x e. (RR ^m (1...N))A.y e. (RR ^m (1...N))(sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)) e. RR)
35		eqid 1884	. . . . . 6 |- {<.<.x, y>., d>. | ((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ d = (sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)))} = {<.<.x, y>., d>. | ((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ d = (sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)))}
36	35	foprab2 5061	. . . . 5 |- (A.x e. (RR ^m (1...N))A.y e. (RR ^m (1...N))(sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)) e. RR <-> {<.<.x, y>., d>. | ((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ d = (sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)))}:((RR ^m (1...N)) X. (RR ^m (1...N)))-->RR)
37	34, 36	sylib 215	. . . 4 |- (N e. NN -> {<.<.x, y>., d>. | ((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ d = (sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)))}:((RR ^m (1...N)) X. (RR ^m (1...N)))-->RR)
38		rrnval 16013	. . . . 5 |- (N e. NN -> (RRn` N) = {<.<.x, y>., d>. | ((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ d = (sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)))})
39	38	feq1d 4556	. . . 4 |- (N e. NN -> ((RRn` N):((RR ^m (1...N)) X. (RR ^m (1...N)))-->RR <-> {<.<.x, y>., d>. | ((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ d = (sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)))}:((RR ^m (1...N)) X. (RR ^m (1...N)))-->RR))
40	37, 39	mpbird 213	. . 3 |- (N e. NN -> (RRn` N):((RR ^m (1...N)) X. (RR ^m (1...N)))-->RR)
41		rrndm 16015	. . . . . 6 |- (N e. NN -> dom dom (RRn` N) = (RR ^m (1...N)))
42		xpeq1 4016	. . . . . 6 |- (dom dom (RRn` N) = (RR ^m (1...N)) -> (dom dom (RRn` N) X. dom dom (RRn` N)) = ((RR ^m (1...N)) X. dom dom (RRn` N)))
43	41, 42	syl 12	. . . . 5 |- (N e. NN -> (dom dom (RRn` N) X. dom dom (RRn` N)) = ((RR ^m (1...N)) X. dom dom (RRn` N)))
44		xpeq2 4017	. . . . . 6 |- (dom dom (RRn` N) = (RR ^m (1...N)) -> ((RR ^m (1...N)) X. dom dom (RRn` N)) = ((RR ^m (1...N)) X. (RR ^m (1...N))))
45	41, 44	syl 12	. . . . 5 |- (N e. NN -> ((RR ^m (1...N)) X. dom dom (RRn` N)) = ((RR ^m (1...N)) X. (RR ^m (1...N))))
46	43, 45	eqtrd 1925	. . . 4 |- (N e. NN -> (dom dom (RRn` N) X. dom dom (RRn` N)) = ((RR ^m (1...N)) X. (RR ^m (1...N))))
47	46	feq2d 4557	. . 3 |- (N e. NN -> ((RRn` N):(dom dom (RRn` N) X. dom dom (RRn` N))-->RR <-> (RRn` N):((RR ^m (1...N)) X. (RR ^m (1...N)))-->RR))
48	40, 47	mpbird 213	. 2 |- (N e. NN -> (RRn` N):(dom dom (RRn` N) X. dom dom (RRn` N))-->RR)
49	41	eleq2d 1964	. . . . 5 |- (N e. NN -> (x e. dom dom (RRn` N) <-> x e. (RR ^m (1...N))))
50	41	eleq2d 1964	. . . . 5 |- (N e. NN -> (y e. dom dom (RRn` N) <-> y e. (RR ^m (1...N))))
51	49, 50	anbi12d 690	. . . 4 |- (N e. NN -> ((x e. dom dom (RRn` N) /\ y e. dom dom (RRn` N)) <-> (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))))
52		rrnmval 16014	. . . . . . . 8 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> (x(RRn` N)y) = (sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)))
53	52	eqeq1d 1892	. . . . . . 7 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> ((x(RRn` N)y) = 0 <-> (sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)) = 0))
54		sqr00 7964	. . . . . . . 8 |- ((sum_k e. (1...N)(((x` k) - (y` k))^2) e. RR /\ 0 <_ sum_k e. (1...N)(((x` k) - (y` k))^2)) -> ((sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)) = 0 <-> sum_k e. (1...N)(((x` k) - (y` k))^2) = 0))
55	24, 30, 54	syl11anc 524	. . . . . . 7 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> ((sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)) = 0 <-> sum_k e. (1...N)(((x` k) - (y` k))^2) = 0))
56		fsum00 15820	. . . . . . . . 9 |- ((N e. (ZZ>=` 1) /\ A.k e. (1...N)((((x` k) - (y` k))^2) e. RR /\ 0 <_ (((x` k) - (y` k))^2))) -> (sum_k e. (1...N)(((x` k) - (y` k))^2) = 0 <-> A.k e. (1...N)(((x` k) - (y` k))^2) = 0))
57	7, 28, 56	syl11anc 524	. . . . . . . 8 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> (sum_k e. (1...N)(((x` k) - (y` k))^2) = 0 <-> A.k e. (1...N)(((x` k) - (y` k))^2) = 0))
58		recn 6466	. . . . . . . . . . 11 |- (((x` k) - (y` k)) e. RR -> ((x` k) - (y` k)) e. CC)
59		sqeq0 7858	. . . . . . . . . . 11 |- (((x` k) - (y` k)) e. CC -> ((((x` k) - (y` k))^2) = 0 <-> ((x` k) - (y` k)) = 0))
60	19, 58, 59	3syl 24	. . . . . . . . . 10 |- (((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ k e. (1...N)) -> ((((x` k) - (y` k))^2) = 0 <-> ((x` k) - (y` k)) = 0))
61	13	recnd 6468	. . . . . . . . . . 11 |- (((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ k e. (1...N)) -> (x` k) e. CC)
62	17	recnd 6468	. . . . . . . . . . 11 |- (((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ k e. (1...N)) -> (y` k) e. CC)
63		subeq0 6563	. . . . . . . . . . 11 |- (((x` k) e. CC /\ (y` k) e. CC) -> (((x` k) - (y` k)) = 0 <-> (x` k) = (y` k)))
64	61, 62, 63	syl11anc 524	. . . . . . . . . 10 |- (((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ k e. (1...N)) -> (((x` k) - (y` k)) = 0 <-> (x` k) = (y` k)))
65	60, 64	bitrd 587	. . . . . . . . 9 |- (((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) /\ k e. (1...N)) -> ((((x` k) - (y` k))^2) = 0 <-> (x` k) = (y` k)))
66	65	ralbidva 2119	. . . . . . . 8 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> (A.k e. (1...N)(((x` k) - (y` k))^2) = 0 <-> A.k e. (1...N)(x` k) = (y` k)))
67		eqidd 1885	. . . . . . . . . . . 12 |- ((x Fn (1...N) /\ y Fn (1...N)) -> (1...N) = (1...N))
68	67	biantrurd 796	. . . . . . . . . . 11 |- ((x Fn (1...N) /\ y Fn (1...N)) -> (A.k e. (1...N)(x` k) = (y` k) <-> ((1...N) = (1...N) /\ A.k e. (1...N)(x` k) = (y` k))))
69		eqfnfv 4766	. . . . . . . . . . 11 |- ((x Fn (1...N) /\ y Fn (1...N)) -> (x = y <-> ((1...N) = (1...N) /\ A.k e. (1...N)(x` k) = (y` k))))
70	68, 69	bitr4d 590	. . . . . . . . . 10 |- ((x Fn (1...N) /\ y Fn (1...N)) -> (A.k e. (1...N)(x` k) = (y` k) <-> x = y))
71		ffn 4562	. . . . . . . . . . 11 |- (x:(1...N)-->RR -> x Fn (1...N))
72	11, 71	sylbi 216	. . . . . . . . . 10 |- (x e. (RR ^m (1...N)) -> x Fn (1...N))
73		ffn 4562	. . . . . . . . . . 11 |- (y:(1...N)-->RR -> y Fn (1...N))
74	15, 73	sylbi 216	. . . . . . . . . 10 |- (y e. (RR ^m (1...N)) -> y Fn (1...N))
75	70, 72, 74	syl2an 503	. . . . . . . . 9 |- ((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> (A.k e. (1...N)(x` k) = (y` k) <-> x = y))
76	75	3adant1 894	. . . . . . . 8 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> (A.k e. (1...N)(x` k) = (y` k) <-> x = y))
77	57, 66, 76	3bitrd 603	. . . . . . 7 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> (sum_k e. (1...N)(((x` k) - (y` k))^2) = 0 <-> x = y))
78	53, 55, 77	3bitrd 603	. . . . . 6 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> ((x(RRn` N)y) = 0 <-> x = y))
79	41	eleq2d 1964	. . . . . . . . . 10 |- (N e. NN -> (z e. dom dom (RRn` N) <-> z e. (RR ^m (1...N))))
80	79	biimpd 170	. . . . . . . . 9 |- (N e. NN -> (z e. dom dom (RRn` N) -> z e. (RR ^m (1...N))))
81	80	3ad2ant1 897	. . . . . . . 8 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> (z e. dom dom (RRn` N) -> z e. (RR ^m (1...N))))
82		simpl 346	. . . . . . . . . . . . . 14 |- ((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) -> N e. NN)
83		ffvelrn 4787	. . . . . . . . . . . . . . . . . . 19 |- ((x:(1...N)-->RR /\ m e. (1...N)) -> (x` m) e. RR)
84	83	3ad2antl1 1038	. . . . . . . . . . . . . . . . . 18 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ m e. (1...N)) -> (x` m) e. RR)
85		ffvelrn 4787	. . . . . . . . . . . . . . . . . . 19 |- ((z:(1...N)-->RR /\ m e. (1...N)) -> (z` m) e. RR)
86	85	3ad2antl3 1040	. . . . . . . . . . . . . . . . . 18 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ m e. (1...N)) -> (z` m) e. RR)
87		resubcl 6601	. . . . . . . . . . . . . . . . . 18 |- (((x` m) e. RR /\ (z` m) e. RR) -> ((x` m) - (z` m)) e. RR)
88	84, 86, 87	syl11anc 524	. . . . . . . . . . . . . . . . 17 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ m e. (1...N)) -> ((x` m) - (z` m)) e. RR)
89	88	r19.21aiva 2176	. . . . . . . . . . . . . . . 16 |- ((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) -> A.m e. (1...N)((x` m) - (z` m)) e. RR)
90		eqid 1884	. . . . . . . . . . . . . . . . 17 |- {<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))} = {<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}
91	90	fopab2 4796	. . . . . . . . . . . . . . . 16 |- (A.m e. (1...N)((x` m) - (z` m)) e. RR <-> {<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}:(1...N)-->RR)
92	89, 91	sylib 215	. . . . . . . . . . . . . . 15 |- ((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) -> {<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}:(1...N)-->RR)
93	92	adantl 424	. . . . . . . . . . . . . 14 |- ((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) -> {<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}:(1...N)-->RR)
94		ffvelrn 4787	. . . . . . . . . . . . . . . . . . 19 |- ((y:(1...N)-->RR /\ m e. (1...N)) -> (y` m) e. RR)
95	94	3ad2antl2 1039	. . . . . . . . . . . . . . . . . 18 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ m e. (1...N)) -> (y` m) e. RR)
96		resubcl 6601	. . . . . . . . . . . . . . . . . 18 |- (((z` m) e. RR /\ (y` m) e. RR) -> ((z` m) - (y` m)) e. RR)
97	86, 95, 96	syl11anc 524	. . . . . . . . . . . . . . . . 17 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ m e. (1...N)) -> ((z` m) - (y` m)) e. RR)
98	97	r19.21aiva 2176	. . . . . . . . . . . . . . . 16 |- ((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) -> A.m e. (1...N)((z` m) - (y` m)) e. RR)
99		eqid 1884	. . . . . . . . . . . . . . . . 17 |- {<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))} = {<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}
100	99	fopab2 4796	. . . . . . . . . . . . . . . 16 |- (A.m e. (1...N)((z` m) - (y` m)) e. RR <-> {<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}:(1...N)-->RR)
101	98, 100	sylib 215	. . . . . . . . . . . . . . 15 |- ((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) -> {<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}:(1...N)-->RR)
102	101	adantl 424	. . . . . . . . . . . . . 14 |- ((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) -> {<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}:(1...N)-->RR)
103		trirn 15834	. . . . . . . . . . . . . 14 |- ((N e. NN /\ {<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}:(1...N)-->RR /\ {<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}:(1...N)-->RR) -> (sqr` sum_k e. (1...N)((({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k) + ({<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}` k))^2)) <_ ((sqr` sum_k e. (1...N)(({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k)^2)) + (sqr` sum_k e. (1...N)(({<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}` k)^2))))
104	82, 93, 102, 103	syl111anc 1100	. . . . . . . . . . . . 13 |- ((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) -> (sqr` sum_k e. (1...N)((({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k) + ({<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}` k))^2)) <_ ((sqr` sum_k e. (1...N)(({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k)^2)) + (sqr` sum_k e. (1...N)(({<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}` k)^2))))
105		fveq2 4681	. . . . . . . . . . . . . . . . . . . . 21 |- (m = k -> (x` m) = (x` k))
106		fveq2 4681	. . . . . . . . . . . . . . . . . . . . 21 |- (m = k -> (z` m) = (z` k))
107	105, 106	opreq12d 4900	. . . . . . . . . . . . . . . . . . . 20 |- (m = k -> ((x` m) - (z` m)) = ((x` k) - (z` k)))
108		oprex 4907	. . . . . . . . . . . . . . . . . . . 20 |- ((x` k) - (z` k)) e. _V
109	107, 90, 108	fvopab4 4743	. . . . . . . . . . . . . . . . . . 19 |- (k e. (1...N) -> ({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k) = ((x` k) - (z` k)))
110		fveq2 4681	. . . . . . . . . . . . . . . . . . . . 21 |- (m = k -> (y` m) = (y` k))
111	106, 110	opreq12d 4900	. . . . . . . . . . . . . . . . . . . 20 |- (m = k -> ((z` m) - (y` m)) = ((z` k) - (y` k)))
112		oprex 4907	. . . . . . . . . . . . . . . . . . . 20 |- ((z` k) - (y` k)) e. _V
113	111, 99, 112	fvopab4 4743	. . . . . . . . . . . . . . . . . . 19 |- (k e. (1...N) -> ({<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}` k) = ((z` k) - (y` k)))
114	109, 113	opreq12d 4900	. . . . . . . . . . . . . . . . . 18 |- (k e. (1...N) -> (({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k) + ({<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}` k)) = (((x` k) - (z` k)) + ((z` k) - (y` k))))
115	114	adantl 424	. . . . . . . . . . . . . . . . 17 |- (((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) /\ k e. (1...N)) -> (({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k) + ({<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}` k)) = (((x` k) - (z` k)) + ((z` k) - (y` k))))
116	8	3ad2antl1 1038	. . . . . . . . . . . . . . . . . . . 20 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ k e. (1...N)) -> (x` k) e. RR)
117	116	recnd 6468	. . . . . . . . . . . . . . . . . . 19 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ k e. (1...N)) -> (x` k) e. CC)
118		ffvelrn 4787	. . . . . . . . . . . . . . . . . . . . 21 |- ((z:(1...N)-->RR /\ k e. (1...N)) -> (z` k) e. RR)
119	118	3ad2antl3 1040	. . . . . . . . . . . . . . . . . . . 20 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ k e. (1...N)) -> (z` k) e. RR)
120	119	recnd 6468	. . . . . . . . . . . . . . . . . . 19 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ k e. (1...N)) -> (z` k) e. CC)
121	14	3ad2antl2 1039	. . . . . . . . . . . . . . . . . . . 20 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ k e. (1...N)) -> (y` k) e. RR)
122	121	recnd 6468	. . . . . . . . . . . . . . . . . . 19 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ k e. (1...N)) -> (y` k) e. CC)
123		npncan 6560	. . . . . . . . . . . . . . . . . . 19 |- (((x` k) e. CC /\ (z` k) e. CC /\ (y` k) e. CC) -> (((x` k) - (z` k)) + ((z` k) - (y` k))) = ((x` k) - (y` k)))
124	117, 120, 122, 123	syl111anc 1100	. . . . . . . . . . . . . . . . . 18 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ k e. (1...N)) -> (((x` k) - (z` k)) + ((z` k) - (y` k))) = ((x` k) - (y` k)))
125	124	adantll 428	. . . . . . . . . . . . . . . . 17 |- (((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) /\ k e. (1...N)) -> (((x` k) - (z` k)) + ((z` k) - (y` k))) = ((x` k) - (y` k)))
126	115, 125	eqtr2d 1926	. . . . . . . . . . . . . . . 16 |- (((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) /\ k e. (1...N)) -> ((x` k) - (y` k)) = (({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k) + ({<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}` k)))
127	126	opreq1d 4897	. . . . . . . . . . . . . . 15 |- (((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) /\ k e. (1...N)) -> (((x` k) - (y` k))^2) = ((({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k) + ({<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}` k))^2))
128	127	sumeq2dv 8252	. . . . . . . . . . . . . 14 |- ((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) -> sum_k e. (1...N)(((x` k) - (y` k))^2) = sum_k e. (1...N)((({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k) + ({<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}` k))^2))
129	128	fveq2d 4685	. . . . . . . . . . . . 13 |- ((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) -> (sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)) = (sqr` sum_k e. (1...N)((({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k) + ({<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}` k))^2)))
130		negsubdi2 6623	. . . . . . . . . . . . . . . . . . . 20 |- (((z` k) e. CC /\ (x` k) e. CC) -> -u((z` k) - (x` k)) = ((x` k) - (z` k)))
131	120, 117, 130	syl11anc 524	. . . . . . . . . . . . . . . . . . 19 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ k e. (1...N)) -> -u((z` k) - (x` k)) = ((x` k) - (z` k)))
132	131	opreq1d 4897	. . . . . . . . . . . . . . . . . 18 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ k e. (1...N)) -> (-u((z` k) - (x` k))^2) = (((x` k) - (z` k))^2))
133		subcl 6524	. . . . . . . . . . . . . . . . . . . 20 |- (((z` k) e. CC /\ (x` k) e. CC) -> ((z` k) - (x` k)) e. CC)
134	120, 117, 133	syl11anc 524	. . . . . . . . . . . . . . . . . . 19 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ k e. (1...N)) -> ((z` k) - (x` k)) e. CC)
135		sqneg 7855	. . . . . . . . . . . . . . . . . . 19 |- (((z` k) - (x` k)) e. CC -> (-u((z` k) - (x` k))^2) = (((z` k) - (x` k))^2))
136	134, 135	syl 12	. . . . . . . . . . . . . . . . . 18 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ k e. (1...N)) -> (-u((z` k) - (x` k))^2) = (((z` k) - (x` k))^2))
137	109	eqcomd 1889	. . . . . . . . . . . . . . . . . . . 20 |- (k e. (1...N) -> ((x` k) - (z` k)) = ({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k))
138	137	adantl 424	. . . . . . . . . . . . . . . . . . 19 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ k e. (1...N)) -> ((x` k) - (z` k)) = ({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k))
139	138	opreq1d 4897	. . . . . . . . . . . . . . . . . 18 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ k e. (1...N)) -> (((x` k) - (z` k))^2) = (({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k)^2))
140	132, 136, 139	3eqtr3d 1934	. . . . . . . . . . . . . . . . 17 |- (((x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR) /\ k e. (1...N)) -> (((z` k) - (x` k))^2) = (({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k)^2))
141	140	adantll 428	. . . . . . . . . . . . . . . 16 |- (((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) /\ k e. (1...N)) -> (((z` k) - (x` k))^2) = (({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k)^2))
142	141	sumeq2dv 8252	. . . . . . . . . . . . . . 15 |- ((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) -> sum_k e. (1...N)(((z` k) - (x` k))^2) = sum_k e. (1...N)(({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k)^2))
143	142	fveq2d 4685	. . . . . . . . . . . . . 14 |- ((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) -> (sqr` sum_k e. (1...N)(((z` k) - (x` k))^2)) = (sqr` sum_k e. (1...N)(({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k)^2)))
144	113	eqcomd 1889	. . . . . . . . . . . . . . . . . 18 |- (k e. (1...N) -> ((z` k) - (y` k)) = ({<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}` k))
145	144	adantl 424	. . . . . . . . . . . . . . . . 17 |- (((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) /\ k e. (1...N)) -> ((z` k) - (y` k)) = ({<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}` k))
146	145	opreq1d 4897	. . . . . . . . . . . . . . . 16 |- (((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) /\ k e. (1...N)) -> (((z` k) - (y` k))^2) = (({<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}` k)^2))
147	146	sumeq2dv 8252	. . . . . . . . . . . . . . 15 |- ((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) -> sum_k e. (1...N)(((z` k) - (y` k))^2) = sum_k e. (1...N)(({<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}` k)^2))
148	147	fveq2d 4685	. . . . . . . . . . . . . 14 |- ((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) -> (sqr` sum_k e. (1...N)(((z` k) - (y` k))^2)) = (sqr` sum_k e. (1...N)(({<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}` k)^2)))
149	143, 148	opreq12d 4900	. . . . . . . . . . . . 13 |- ((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) -> ((sqr` sum_k e. (1...N)(((z` k) - (x` k))^2)) + (sqr` sum_k e. (1...N)(((z` k) - (y` k))^2))) = ((sqr` sum_k e. (1...N)(({<.m, n>. | (m e. (1...N) /\ n = ((x` m) - (z` m)))}` k)^2)) + (sqr` sum_k e. (1...N)(({<.m, n>. | (m e. (1...N) /\ n = ((z` m) - (y` m)))}` k)^2))))
150	104, 129, 149	3brtr4d 3367	. . . . . . . . . . . 12 |- ((N e. NN /\ (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR)) -> (sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)) <_ ((sqr` sum_k e. (1...N)(((z` k) - (x` k))^2)) + (sqr` sum_k e. (1...N)(((z` k) - (y` k))^2))))
151	9, 10	elmap 5393	. . . . . . . . . . . . 13 |- (z e. (RR ^m (1...N)) <-> z:(1...N)-->RR)
152	11, 15, 151	3anbi123i 1056	. . . . . . . . . . . 12 |- ((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)) /\ z e. (RR ^m (1...N))) <-> (x:(1...N)-->RR /\ y:(1...N)-->RR /\ z:(1...N)-->RR))
153	150, 152	sylan2b 501	. . . . . . . . . . 11 |- ((N e. NN /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)) /\ z e. (RR ^m (1...N)))) -> (sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)) <_ ((sqr` sum_k e. (1...N)(((z` k) - (x` k))^2)) + (sqr` sum_k e. (1...N)(((z` k) - (y` k))^2))))
154	52	3adant3r3 1079	. . . . . . . . . . 11 |- ((N e. NN /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)) /\ z e. (RR ^m (1...N)))) -> (x(RRn` N)y) = (sqr` sum_k e. (1...N)(((x` k) - (y` k))^2)))
155		rrnmval 16014	. . . . . . . . . . . . . . 15 |- ((N e. NN /\ z e. (RR ^m (1...N)) /\ x e. (RR ^m (1...N))) -> (z(RRn` N)x) = (sqr` sum_k e. (1...N)(((z` k) - (x` k))^2)))
156	155	3expb 1068	. . . . . . . . . . . . . 14 |- ((N e. NN /\ (z e. (RR ^m (1...N)) /\ x e. (RR ^m (1...N)))) -> (z(RRn` N)x) = (sqr` sum_k e. (1...N)(((z` k) - (x` k))^2)))
157	156	ancom2s 545	. . . . . . . . . . . . 13 |- ((N e. NN /\ (x e. (RR ^m (1...N)) /\ z e. (RR ^m (1...N)))) -> (z(RRn` N)x) = (sqr` sum_k e. (1...N)(((z` k) - (x` k))^2)))
158	157	3adantr2 1036	. . . . . . . . . . . 12 |- ((N e. NN /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)) /\ z e. (RR ^m (1...N)))) -> (z(RRn` N)x) = (sqr` sum_k e. (1...N)(((z` k) - (x` k))^2)))
159		rrnmval 16014	. . . . . . . . . . . . . . 15 |- ((N e. NN /\ z e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> (z(RRn` N)y) = (sqr` sum_k e. (1...N)(((z` k) - (y` k))^2)))
160	159	3expb 1068	. . . . . . . . . . . . . 14 |- ((N e. NN /\ (z e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)))) -> (z(RRn` N)y) = (sqr` sum_k e. (1...N)(((z` k) - (y` k))^2)))
161	160	ancom2s 545	. . . . . . . . . . . . 13 |- ((N e. NN /\ (y e. (RR ^m (1...N)) /\ z e. (RR ^m (1...N)))) -> (z(RRn` N)y) = (sqr` sum_k e. (1...N)(((z` k) - (y` k))^2)))
162	161	3adantr1 1035	. . . . . . . . . . . 12 |- ((N e. NN /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)) /\ z e. (RR ^m (1...N)))) -> (z(RRn` N)y) = (sqr` sum_k e. (1...N)(((z` k) - (y` k))^2)))
163	158, 162	opreq12d 4900	. . . . . . . . . . 11 |- ((N e. NN /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)) /\ z e. (RR ^m (1...N)))) -> ((z(RRn` N)x) + (z(RRn` N)y)) = ((sqr` sum_k e. (1...N)(((z` k) - (x` k))^2)) + (sqr` sum_k e. (1...N)(((z` k) - (y` k))^2))))
164	153, 154, 163	3brtr4d 3367	. . . . . . . . . 10 |- ((N e. NN /\ (x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N)) /\ z e. (RR ^m (1...N)))) -> (x(RRn` N)y) <_ ((z(RRn` N)x) + (z(RRn` N)y)))
165	164	3exp2 1086	. . . . . . . . 9 |- (N e. NN -> (x e. (RR ^m (1...N)) -> (y e. (RR ^m (1...N)) -> (z e. (RR ^m (1...N)) -> (x(RRn` N)y) <_ ((z(RRn` N)x) + (z(RRn` N)y))))))
166	165	3imp 1061	. . . . . . . 8 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> (z e. (RR ^m (1...N)) -> (x(RRn` N)y) <_ ((z(RRn` N)x) + (z(RRn` N)y))))
167	81, 166	syld 30	. . . . . . 7 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> (z e. dom dom (RRn` N) -> (x(RRn` N)y) <_ ((z(RRn` N)x) + (z(RRn` N)y))))
168	167	r19.21aiv 2175	. . . . . 6 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> A.z e. dom dom (RRn` N)(x(RRn` N)y) <_ ((z(RRn` N)x) + (z(RRn` N)y)))
169	78, 168	jca 310	. . . . 5 |- ((N e. NN /\ x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> (((x(RRn` N)y) = 0 <-> x = y) /\ A.z e. dom dom (RRn` N)(x(RRn` N)y) <_ ((z(RRn` N)x) + (z(RRn` N)y))))
170	169	3expib 1070	. . . 4 |- (N e. NN -> ((x e. (RR ^m (1...N)) /\ y e. (RR ^m (1...N))) -> (((x(RRn` N)y) = 0 <-> x = y) /\ A.z e. dom dom (RRn` N)(x(RRn` N)y) <_ ((z(RRn` N)x) + (z(RRn` N)y)))))
171	51, 170	sylbid 220	. . 3 |- (N e. NN -> ((x e. dom dom (RRn` N) /\ y e. dom dom (RRn` N)) -> (((x(RRn` N)y) = 0 <-> x = y) /\ A.z e. dom dom (RRn` N)(x(RRn` N)y) <_ ((z(RRn` N)x) + (z(RRn` N)y)))))
172	171	r19.21aivv 2183	. 2 |- (N e. NN -> A.x e. dom dom (RRn` N)A.y e. dom dom (RRn` N)(((x(RRn` N)y) = 0 <-> x = y) /\ A.z e. dom dom (RRn` N)(x(RRn` N)y) <_ ((z(RRn` N)x) + (z(RRn` N)y))))
173	4, 48, 172	sylanbrc 527	1 |- (N e. NN -> (RRn` N) e. Met)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   class class class wbr 3338  {copab 3395   X. cxp 3984  dom cdm 3986   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885   ^m cmap 5381  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   - cmin 6445  -ucneg 6446   <_ cle 6448  NNcn 6449  2c2 7145  ZZ>=cuz 7586  ...cfz 7637  ^cexp 7811  sqrcsqr 7919  sum_csu 8239  Metcme 9066  RRncrrn 16011

This theorem is referenced by:  rrncms 16019  rrntotbndlem1 16020  rrntotbnd 16022  rrnheibor 16023  ismrer1 16024  reheibor 16025

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

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-3 7155  df-4 7156  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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