Theorem fsum0diag4 8523

Description: Two ways to express "the sum of A(j) x. B(k) where j + k <_ N."

Assertion

Ref	Expression
fsum0diag4	|- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> sum_j e. (0...N)sum_k e. (0...(N - j))(A x. B) = sum_k e. (0...N)(B x. sum_j e. (0...(N - k))A))

Distinct variable groups:   A,k   B,j   j,k,N

Proof of Theorem fsum0diag4

Step	Hyp	Ref	Expression
1		fzss2 7676	. . . . . . . . . . . . . . . . 17 |- ((N e. (ZZ>=` m) /\ 0 e. ZZ) -> (0...m) C_ (0...N))
2		elfzuz3 7648	. . . . . . . . . . . . . . . . 17 |- ((N e. NN0 /\ m e. (0...N)) -> N e. (ZZ>=` m))
3		0z 7355	. . . . . . . . . . . . . . . . 17 |- 0 e. ZZ
4	1, 2, 3	sylancl 525	. . . . . . . . . . . . . . . 16 |- ((N e. NN0 /\ m e. (0...N)) -> (0...m) C_ (0...N))
5	4	sseld 2619	. . . . . . . . . . . . . . 15 |- ((N e. NN0 /\ m e. (0...N)) -> (k e. (0...m) -> k e. (0...N)))
6	5	imim1d 33	. . . . . . . . . . . . . 14 |- ((N e. NN0 /\ m e. (0...N)) -> ((k e. (0...N) -> B e. CC) -> (k e. (0...m) -> B e. CC)))
7	6	adantlr 429	. . . . . . . . . . . . 13 |- (((N e. NN0 /\ A.j e. (0...N)A e. CC) /\ m e. (0...N)) -> ((k e. (0...N) -> B e. CC) -> (k e. (0...m) -> B e. CC)))
8		mulcom 6459	. . . . . . . . . . . . . . . 16 |- (([_(m - k) / j]_A e. CC /\ B e. CC) -> ([_(m - k) / j]_A x. B) = (B x. [_(m - k) / j]_A))
9		ra4csbela 2587	. . . . . . . . . . . . . . . . . . . . 21 |- (((m - k) e. (0...N) /\ A.j e. (0...N)A e. CC) -> [_(m - k) / j]_A e. CC)
10		elfz 7641	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (((m - k) e. ZZ /\ 0 e. ZZ /\ N e. ZZ) -> ((m - k) e. (0...N) <-> (0 <_ (m - k) /\ (m - k) <_ N)))
11	3, 10	mp3an2 1179	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((m - k) e. ZZ /\ N e. ZZ) -> ((m - k) e. (0...N) <-> (0 <_ (m - k) /\ (m - k) <_ N)))
12		zsubcl 7377	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((m e. ZZ /\ k e. ZZ) -> (m - k) e. ZZ)
13		elfzelz 7652	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (m e. (0...N) -> m e. ZZ)
14		elfzelz 7652	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (k e. (0...m) -> k e. ZZ)
15	12, 13, 14	syl2an 503	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((m e. (0...N) /\ k e. (0...m)) -> (m - k) e. ZZ)
16		nn0z 7363	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (N e. NN0 -> N e. ZZ)
17	11, 15, 16	syl2an 503	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((m e. (0...N) /\ k e. (0...m)) /\ N e. NN0) -> ((m - k) e. (0...N) <-> (0 <_ (m - k) /\ (m - k) <_ N)))
18	17	ancoms 484	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((N e. NN0 /\ (m e. (0...N) /\ k e. (0...m))) -> ((m - k) e. (0...N) <-> (0 <_ (m - k) /\ (m - k) <_ N)))
19	18	anassrs 489	. . . . . . . . . . . . . . . . . . . . . 22 |- (((N e. NN0 /\ m e. (0...N)) /\ k e. (0...m)) -> ((m - k) e. (0...N) <-> (0 <_ (m - k) /\ (m - k) <_ N)))
20		elfzle2 7654	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (k e. (0...m) -> k <_ m)
21	20	adantl 424	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((m e. (0...N) /\ k e. (0...m)) -> k <_ m)
22		subge0 6863	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((m e. RR /\ k e. RR) -> (0 <_ (m - k) <-> k <_ m))
23		zre 7348	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (m e. ZZ -> m e. RR)
24	13, 23	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (m e. (0...N) -> m e. RR)
25		zre 7348	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (k e. ZZ -> k e. RR)
26	14, 25	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (k e. (0...m) -> k e. RR)
27	22, 24, 26	syl2an 503	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((m e. (0...N) /\ k e. (0...m)) -> (0 <_ (m - k) <-> k <_ m))
28	21, 27	mpbird 213	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((m e. (0...N) /\ k e. (0...m)) -> 0 <_ (m - k))
29	28	adantll 428	. . . . . . . . . . . . . . . . . . . . . 22 |- (((N e. NN0 /\ m e. (0...N)) /\ k e. (0...m)) -> 0 <_ (m - k))
30		zre 7348	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((m - k) e. ZZ -> (m - k) e. RR)
31	15, 30	syl 12	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((m e. (0...N) /\ k e. (0...m)) -> (m - k) e. RR)
32	31	adantll 428	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((N e. NN0 /\ m e. (0...N)) /\ k e. (0...m)) -> (m - k) e. RR)
33	24	ad2antlr 441	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((N e. NN0 /\ m e. (0...N)) /\ k e. (0...m)) -> m e. RR)
34		nn0re 7317	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (N e. NN0 -> N e. RR)
35	34	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((N e. NN0 /\ m e. (0...N)) /\ k e. (0...m)) -> N e. RR)
36		elfzle1 7653	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (k e. (0...m) -> 0 <_ k)
37	36	adantl 424	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((m e. (0...N) /\ k e. (0...m)) -> 0 <_ k)
38		subge02 6866	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((m e. RR /\ k e. RR) -> (0 <_ k <-> (m - k) <_ m))
39	38, 24, 26	syl2an 503	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((m e. (0...N) /\ k e. (0...m)) -> (0 <_ k <-> (m - k) <_ m))
40	37, 39	mpbid 212	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((m e. (0...N) /\ k e. (0...m)) -> (m - k) <_ m)
41	40	adantll 428	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((N e. NN0 /\ m e. (0...N)) /\ k e. (0...m)) -> (m - k) <_ m)
42		elfzle2 7654	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (m e. (0...N) -> m <_ N)
43	42	ad2antlr 441	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((N e. NN0 /\ m e. (0...N)) /\ k e. (0...m)) -> m <_ N)
44	32, 33, 35, 41, 43	letrd 6696	. . . . . . . . . . . . . . . . . . . . . 22 |- (((N e. NN0 /\ m e. (0...N)) /\ k e. (0...m)) -> (m - k) <_ N)
45	19, 29, 44	mpbir2and 802	. . . . . . . . . . . . . . . . . . . . 21 |- (((N e. NN0 /\ m e. (0...N)) /\ k e. (0...m)) -> (m - k) e. (0...N))
46	9, 45	sylan 497	. . . . . . . . . . . . . . . . . . . 20 |- ((((N e. NN0 /\ m e. (0...N)) /\ k e. (0...m)) /\ A.j e. (0...N)A e. CC) -> [_(m - k) / j]_A e. CC)
47	46	exp41 413	. . . . . . . . . . . . . . . . . . 19 |- (N e. NN0 -> (m e. (0...N) -> (k e. (0...m) -> (A.j e. (0...N)A e. CC -> [_(m - k) / j]_A e. CC))))
48	47	com23 36	. . . . . . . . . . . . . . . . . 18 |- (N e. NN0 -> (k e. (0...m) -> (m e. (0...N) -> (A.j e. (0...N)A e. CC -> [_(m - k) / j]_A e. CC))))
49	48	com24 41	. . . . . . . . . . . . . . . . 17 |- (N e. NN0 -> (A.j e. (0...N)A e. CC -> (m e. (0...N) -> (k e. (0...m) -> [_(m - k) / j]_A e. CC))))
50	49	imp41 395	. . . . . . . . . . . . . . . 16 |- ((((N e. NN0 /\ A.j e. (0...N)A e. CC) /\ m e. (0...N)) /\ k e. (0...m)) -> [_(m - k) / j]_A e. CC)
51	8, 50	sylan 497	. . . . . . . . . . . . . . 15 |- (((((N e. NN0 /\ A.j e. (0...N)A e. CC) /\ m e. (0...N)) /\ k e. (0...m)) /\ B e. CC) -> ([_(m - k) / j]_A x. B) = (B x. [_(m - k) / j]_A))
52	51	exp31 407	. . . . . . . . . . . . . 14 |- (((N e. NN0 /\ A.j e. (0...N)A e. CC) /\ m e. (0...N)) -> (k e. (0...m) -> (B e. CC -> ([_(m - k) / j]_A x. B) = (B x. [_(m - k) / j]_A))))
53	52	a2d 16	. . . . . . . . . . . . 13 |- (((N e. NN0 /\ A.j e. (0...N)A e. CC) /\ m e. (0...N)) -> ((k e. (0...m) -> B e. CC) -> (k e. (0...m) -> ([_(m - k) / j]_A x. B) = (B x. [_(m - k) / j]_A))))
54	7, 53	syld 30	. . . . . . . . . . . 12 |- (((N e. NN0 /\ A.j e. (0...N)A e. CC) /\ m e. (0...N)) -> ((k e. (0...N) -> B e. CC) -> (k e. (0...m) -> ([_(m - k) / j]_A x. B) = (B x. [_(m - k) / j]_A))))
55	54	ralimdv2 2173	. . . . . . . . . . 11 |- (((N e. NN0 /\ A.j e. (0...N)A e. CC) /\ m e. (0...N)) -> (A.k e. (0...N)B e. CC -> A.k e. (0...m)([_(m - k) / j]_A x. B) = (B x. [_(m - k) / j]_A)))
56	55	imp 377	. . . . . . . . . 10 |- ((((N e. NN0 /\ A.j e. (0...N)A e. CC) /\ m e. (0...N)) /\ A.k e. (0...N)B e. CC) -> A.k e. (0...m)([_(m - k) / j]_A x. B) = (B x. [_(m - k) / j]_A))
57	56	sumeq2d 8251	. . . . . . . . 9 |- ((((N e. NN0 /\ A.j e. (0...N)A e. CC) /\ m e. (0...N)) /\ A.k e. (0...N)B e. CC) -> sum_k e. (0...m)([_(m - k) / j]_A x. B) = sum_k e. (0...m)(B x. [_(m - k) / j]_A))
58	57	an1rs 547	. . . . . . . 8 |- ((((N e. NN0 /\ A.j e. (0...N)A e. CC) /\ A.k e. (0...N)B e. CC) /\ m e. (0...N)) -> sum_k e. (0...m)([_(m - k) / j]_A x. B) = sum_k e. (0...m)(B x. [_(m - k) / j]_A))
59	58	ex 402	. . . . . . 7 |- (((N e. NN0 /\ A.j e. (0...N)A e. CC) /\ A.k e. (0...N)B e. CC) -> (m e. (0...N) -> sum_k e. (0...m)([_(m - k) / j]_A x. B) = sum_k e. (0...m)(B x. [_(m - k) / j]_A)))
60	59	3impa 1062	. . . . . 6 |- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> (m e. (0...N) -> sum_k e. (0...m)([_(m - k) / j]_A x. B) = sum_k e. (0...m)(B x. [_(m - k) / j]_A)))
61	60	r19.21aiv 2175	. . . . 5 |- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> A.m e. (0...N)sum_k e. (0...m)([_(m - k) / j]_A x. B) = sum_k e. (0...m)(B x. [_(m - k) / j]_A))
62		ax-17 1317	. . . . . . . 8 |- (x e. (0...m) -> A.j x e. (0...m))
63		oprex 4907	. . . . . . . . . 10 |- (m - k) e. _V
64		ax-17 1317	. . . . . . . . . 10 |- (x e. (m - k) -> A.j x e. (m - k))
65	63, 64	hbcsb1 2568	. . . . . . . . 9 |- (x e. [_(m - k) / j]_A -> A.j x e. [_(m - k) / j]_A)
66		ax-17 1317	. . . . . . . . 9 |- (x e. x. -> A.j x e. x. )
67		ax-17 1317	. . . . . . . . 9 |- (x e. B -> A.j x e. B)
68	65, 66, 67	hbopr 4904	. . . . . . . 8 |- (x e. ([_(m - k) / j]_A x. B) -> A.j x e. ([_(m - k) / j]_A x. B))
69	62, 68	hbsum 8244	. . . . . . 7 |- (x e. sum_k e. (0...m)([_(m - k) / j]_A x. B) -> A.j x e. sum_k e. (0...m)([_(m - k) / j]_A x. B))
70	67, 66, 65	hbopr 4904	. . . . . . . 8 |- (x e. (B x. [_(m - k) / j]_A) -> A.j x e. (B x. [_(m - k) / j]_A))
71	62, 70	hbsum 8244	. . . . . . 7 |- (x e. sum_k e. (0...m)(B x. [_(m - k) / j]_A) -> A.j x e. sum_k e. (0...m)(B x. [_(m - k) / j]_A))
72	69, 71	hbeq 1995	. . . . . 6 |- (sum_k e. (0...m)([_(m - k) / j]_A x. B) = sum_k e. (0...m)(B x. [_(m - k) / j]_A) -> A.jsum_k e. (0...m)([_(m - k) / j]_A x. B) = sum_k e. (0...m)(B x. [_(m - k) / j]_A))
73		ax-17 1317	. . . . . 6 |- (sum_k e. (0...j)([_(j - k) / j]_A x. B) = sum_k e. (0...j)(B x. [_(j - k) / j]_A) -> A.msum_k e. (0...j)([_(j - k) / j]_A x. B) = sum_k e. (0...j)(B x. [_(j - k) / j]_A))
74		opreq2 4890	. . . . . . . 8 |- (m = j -> (0...m) = (0...j))
75		opreq1 4889	. . . . . . . . . . 11 |- (m = j -> (m - k) = (j - k))
76	75	csbeq1d 2544	. . . . . . . . . 10 |- (m = j -> [_(m - k) / j]_A = [_(j - k) / j]_A)
77	76	opreq1d 4897	. . . . . . . . 9 |- (m = j -> ([_(m - k) / j]_A x. B) = ([_(j - k) / j]_A x. B))
78	77	adantr 425	. . . . . . . 8 |- ((m = j /\ k e. (0...j)) -> ([_(m - k) / j]_A x. B) = ([_(j - k) / j]_A x. B))
79	74, 78	sumeq12rdv 8256	. . . . . . 7 |- (m = j -> sum_k e. (0...m)([_(m - k) / j]_A x. B) = sum_k e. (0...j)([_(j - k) / j]_A x. B))
80	76	opreq2d 4898	. . . . . . . . 9 |- (m = j -> (B x. [_(m - k) / j]_A) = (B x. [_(j - k) / j]_A))
81	80	adantr 425	. . . . . . . 8 |- ((m = j /\ k e. (0...j)) -> (B x. [_(m - k) / j]_A) = (B x. [_(j - k) / j]_A))
82	74, 81	sumeq12rdv 8256	. . . . . . 7 |- (m = j -> sum_k e. (0...m)(B x. [_(m - k) / j]_A) = sum_k e. (0...j)(B x. [_(j - k) / j]_A))
83	79, 82	eqeq12d 1899	. . . . . 6 |- (m = j -> (sum_k e. (0...m)([_(m - k) / j]_A x. B) = sum_k e. (0...m)(B x. [_(m - k) / j]_A) <-> sum_k e. (0...j)([_(j - k) / j]_A x. B) = sum_k e. (0...j)(B x. [_(j - k) / j]_A)))
84	72, 73, 83	cbvral 2278	. . . . 5 |- (A.m e. (0...N)sum_k e. (0...m)([_(m - k) / j]_A x. B) = sum_k e. (0...m)(B x. [_(m - k) / j]_A) <-> A.j e. (0...N)sum_k e. (0...j)([_(j - k) / j]_A x. B) = sum_k e. (0...j)(B x. [_(j - k) / j]_A))
85	61, 84	sylib 215	. . . 4 |- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> A.j e. (0...N)sum_k e. (0...j)([_(j - k) / j]_A x. B) = sum_k e. (0...j)(B x. [_(j - k) / j]_A))
86	85	sumeq2d 8251	. . 3 |- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> sum_j e. (0...N)sum_k e. (0...j)([_(j - k) / j]_A x. B) = sum_j e. (0...N)sum_k e. (0...j)(B x. [_(j - k) / j]_A))
87		fsum0diag2 8521	. . 3 |- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> sum_j e. (0...N)sum_k e. (0...(N - j))(A x. B) = sum_j e. (0...N)sum_k e. (0...j)([_(j - k) / j]_A x. B))
88		fsum0diag 8520	. . . 4 |- ((N e. NN0 /\ A.k e. (0...N)B e. CC /\ A.j e. (0...N)A e. CC) -> sum_k e. (0...N)sum_j e. (0...(N - k))(B x. A) = sum_j e. (0...N)sum_k e. (0...j)(B x. [_(j - k) / j]_A))
89	88	3com23 1074	. . 3 |- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> sum_k e. (0...N)sum_j e. (0...(N - k))(B x. A) = sum_j e. (0...N)sum_k e. (0...j)(B x. [_(j - k) / j]_A))
90	86, 87, 89	3eqtr4d 1937	. 2 |- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> sum_j e. (0...N)sum_k e. (0...(N - j))(A x. B) = sum_k e. (0...N)sum_j e. (0...(N - k))(B x. A))
91		fsum0diag3 8522	. . 3 |- ((N e. NN0 /\ A.k e. (0...N)B e. CC /\ A.j e. (0...N)A e. CC) -> sum_k e. (0...N)sum_j e. (0...(N - k))(B x. A) = sum_k e. (0...N)(B x. sum_j e. (0...(N - k))A))
92	91	3com23 1074	. 2 |- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> sum_k e. (0...N)sum_j e. (0...(N - k))(B x. A) = sum_k e. (0...N)(B x. sum_j e. (0...(N - k))A))
93	90, 92	eqtrd 1925	1 |- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> sum_j e. (0...N)sum_k e. (0...(N - j))(A x. B) = sum_k e. (0...N)(B x. sum_j e. (0...(N - k))A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  [_csb 2540   C_ wss 2593   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386   x. cmul 6391   - cmin 6445   <_ cle 6448  NN0cn0 6450  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637  sum_csu 8239

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-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-clim 8235  df-sum 8240

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