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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
StepHypRef 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
41, 2, 3sylancl 525 . . . . . . . . . . . . . . . 16 |- ((N e. NN0 /\ m e. (0...N)) -> (0...m) C_ (0...N))
54sseld 2619 . . . . . . . . . . . . . . 15 |- ((N e. NN0 /\ m e. (0...N)) -> (k e. (0...m) -> k e. (0...N)))
65imim1d 33 . . . . . . . . . . . . . 14 |- ((N e. NN0 /\ m e. (0...N)) -> ((k e. (0...N) -> B e. CC) -> (k e. (0...m) -> B e. CC)))
76adantlr 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)))
113, 10mp3an2 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)
1512, 13, 14syl2an 503 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((m e. (0...N) /\ k e. (0...m)) -> (m - k) e. ZZ)
16 nn0z 7363 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- (N e. NN0 -> N e. ZZ)
1711, 15, 16syl2an 503 . . . . . . . . . . . . . . . . . . . . . . . 24 |- (((m e. (0...N) /\ k e. (0...m)) /\ N e. NN0) -> ((m - k) e. (0...N) <-> (0 <_ (m - k) /\ (m - k) <_ N)))
1817ancoms 484 . . . . . . . . . . . . . . . . . . . . . . 23 |- ((N e. NN0 /\ (m e. (0...N) /\ k e. (0...m))) -> ((m - k) e. (0...N) <-> (0 <_ (m - k) /\ (m - k) <_ N)))
1918anassrs 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)
2120adantl 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)
2413, 23syl 12 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- (m e. (0...N) -> m e. RR)
25 zre 7348 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (k e. ZZ -> k e. RR)
2614, 25syl 12 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- (k e. (0...m) -> k e. RR)
2722, 24, 26syl2an 503 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ((m e. (0...N) /\ k e. (0...m)) -> (0 <_ (m - k) <-> k <_ m))
2821, 27mpbird 213 . . . . . . . . . . . . . . . . . . . . . . 23 |- ((m e. (0...N) /\ k e. (0...m)) -> 0 <_ (m - k))
2928adantll 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)
3115, 30syl 12 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ((m e. (0...N) /\ k e. (0...m)) -> (m - k) e. RR)
3231adantll 428 . . . . . . . . . . . . . . . . . . . . . . 23 |- (((N e. NN0 /\ m e. (0...N)) /\ k e. (0...m)) -> (m - k) e. RR)
3324ad2antlr 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)
3534ad2antrr 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)
3736adantl 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))
3938, 24, 26syl2an 503 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((m e. (0...N) /\ k e. (0...m)) -> (0 <_ k <-> (m - k) <_ m))
4037, 39mpbid 212 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ((m e. (0...N) /\ k e. (0...m)) -> (m - k) <_ m)
4140adantll 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)
4342ad2antlr 441 . . . . . . . . . . . . . . . . . . . . . . 23 |- (((N e. NN0 /\ m e. (0...N)) /\ k e. (0...m)) -> m <_ N)
4432, 33, 35, 41, 43letrd 6696 . . . . . . . . . . . . . . . . . . . . . 22 |- (((N e. NN0 /\ m e. (0...N)) /\ k e. (0...m)) -> (m - k) <_ N)
4519, 29, 44mpbir2and 802 . . . . . . . . . . . . . . . . . . . . 21 |- (((N e. NN0 /\ m e. (0...N)) /\ k e. (0...m)) -> (m - k) e. (0...N))
469, 45sylan 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)
4746exp41 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))))
4847com23 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))))
4948com24 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))))
5049imp41 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)
518, 50sylan 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))
5251exp31 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))))
5352a2d 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))))
547, 53syld 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))))
5554ralimdv2 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)))
5655imp 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))
5756sumeq2d 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))
5857an1rs 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))
5958ex 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)))
60593impa 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)))
6160r19.21aiv 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))
6563, 64hbcsb1 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)
6865, 66, 67hbopr 4904 . . . . . . . 8 |- (x e. ([_(m - k) / j]_A x. B) -> A.j x e. ([_(m - k) / j]_A x. B))
6962, 68hbsum 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))
7067, 66, 65hbopr 4904 . . . . . . . 8 |- (x e. (B x. [_(m - k) / j]_A) -> A.j x e. (B x. [_(m - k) / j]_A))
7162, 70hbsum 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))
7269, 71hbeq 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))
7675csbeq1d 2544 . . . . . . . . . 10 |- (m = j -> [_(m - k) / j]_A = [_(j - k) / j]_A)
7776opreq1d 4897 . . . . . . . . 9 |- (m = j -> ([_(m - k) / j]_A x. B) = ([_(j - k) / j]_A x. B))
7877adantr 425 . . . . . . . 8 |- ((m = j /\ k e. (0...j)) -> ([_(m - k) / j]_A x. B) = ([_(j - k) / j]_A x. B))
7974, 78sumeq12rdv 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))
8076opreq2d 4898 . . . . . . . . 9 |- (m = j -> (B x. [_(m - k) / j]_A) = (B x. [_(j - k) / j]_A))
8180adantr 425 . . . . . . . 8 |- ((m = j /\ k e. (0...j)) -> (B x. [_(m - k) / j]_A) = (B x. [_(j - k) / j]_A))
8274, 81sumeq12rdv 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))
8379, 82eqeq12d 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)))
8472, 73, 83cbvral 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))
8561, 84sylib 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))
8685sumeq2d 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))
89883com23 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))
9086, 87, 893eqtr4d 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))
92913com23 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))
9390, 92eqtrd 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))
Colors of variables: wff set class
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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