Theorem bcxmas 8336

Description: Parallel summation (Christmas Stocking) theorem for Pascal's Triangle. (Contributed by Paul Chapman, 18-May-2007.)

Assertion

Ref	Expression
bcxmas	|- ((N e. NN0 /\ M e. NN0) -> (((N + 1) + M) _C M) = sum_j e. (0...M)((N + j) _C j))

Distinct variable group:   j,N

Proof of Theorem bcxmas

Step	Hyp	Ref	Expression
1		bcxmaslem1 8334	. . . . 5 |- (m = 0 -> (((N + 1) + m) _C m) = (((N + 1) + 0) _C 0))
2		opreq2 4890	. . . . . 6 |- (m = 0 -> (0...m) = (0...0))
3	2	sumeq1d 8250	. . . . 5 |- (m = 0 -> sum_j e. (0...m)((N + j) _C j) = sum_j e. (0...0)((N + j) _C j))
4	1, 3	eqeq12d 1899	. . . 4 |- (m = 0 -> ((((N + 1) + m) _C m) = sum_j e. (0...m)((N + j) _C j) <-> (((N + 1) + 0) _C 0) = sum_j e. (0...0)((N + j) _C j)))
5	4	imbi2d 674	. . 3 |- (m = 0 -> ((N e. NN0 -> (((N + 1) + m) _C m) = sum_j e. (0...m)((N + j) _C j)) <-> (N e. NN0 -> (((N + 1) + 0) _C 0) = sum_j e. (0...0)((N + j) _C j))))
6		bcxmaslem1 8334	. . . . 5 |- (m = k -> (((N + 1) + m) _C m) = (((N + 1) + k) _C k))
7		opreq2 4890	. . . . . 6 |- (m = k -> (0...m) = (0...k))
8	7	sumeq1d 8250	. . . . 5 |- (m = k -> sum_j e. (0...m)((N + j) _C j) = sum_j e. (0...k)((N + j) _C j))
9	6, 8	eqeq12d 1899	. . . 4 |- (m = k -> ((((N + 1) + m) _C m) = sum_j e. (0...m)((N + j) _C j) <-> (((N + 1) + k) _C k) = sum_j e. (0...k)((N + j) _C j)))
10	9	imbi2d 674	. . 3 |- (m = k -> ((N e. NN0 -> (((N + 1) + m) _C m) = sum_j e. (0...m)((N + j) _C j)) <-> (N e. NN0 -> (((N + 1) + k) _C k) = sum_j e. (0...k)((N + j) _C j))))
11		bcxmaslem1 8334	. . . . 5 |- (m = (k + 1) -> (((N + 1) + m) _C m) = (((N + 1) + (k + 1)) _C (k + 1)))
12		opreq2 4890	. . . . . 6 |- (m = (k + 1) -> (0...m) = (0...(k + 1)))
13	12	sumeq1d 8250	. . . . 5 |- (m = (k + 1) -> sum_j e. (0...m)((N + j) _C j) = sum_j e. (0...(k + 1))((N + j) _C j))
14	11, 13	eqeq12d 1899	. . . 4 |- (m = (k + 1) -> ((((N + 1) + m) _C m) = sum_j e. (0...m)((N + j) _C j) <-> (((N + 1) + (k + 1)) _C (k + 1)) = sum_j e. (0...(k + 1))((N + j) _C j)))
15	14	imbi2d 674	. . 3 |- (m = (k + 1) -> ((N e. NN0 -> (((N + 1) + m) _C m) = sum_j e. (0...m)((N + j) _C j)) <-> (N e. NN0 -> (((N + 1) + (k + 1)) _C (k + 1)) = sum_j e. (0...(k + 1))((N + j) _C j))))
16		bcxmaslem1 8334	. . . . 5 |- (m = M -> (((N + 1) + m) _C m) = (((N + 1) + M) _C M))
17		opreq2 4890	. . . . . 6 |- (m = M -> (0...m) = (0...M))
18	17	sumeq1d 8250	. . . . 5 |- (m = M -> sum_j e. (0...m)((N + j) _C j) = sum_j e. (0...M)((N + j) _C j))
19	16, 18	eqeq12d 1899	. . . 4 |- (m = M -> ((((N + 1) + m) _C m) = sum_j e. (0...m)((N + j) _C j) <-> (((N + 1) + M) _C M) = sum_j e. (0...M)((N + j) _C j)))
20	19	imbi2d 674	. . 3 |- (m = M -> ((N e. NN0 -> (((N + 1) + m) _C m) = sum_j e. (0...m)((N + j) _C j)) <-> (N e. NN0 -> (((N + 1) + M) _C M) = sum_j e. (0...M)((N + j) _C j))))
21		0nn0 7322	. . . . 5 |- 0 e. NN0
22		nn0addcl 7329	. . . . . 6 |- ((N e. NN0 /\ 0 e. NN0) -> (N + 0) e. NN0)
23		bcn0 8215	. . . . . 6 |- ((N + 0) e. NN0 -> ((N + 0) _C 0) = 1)
24	22, 23	syl 12	. . . . 5 |- ((N e. NN0 /\ 0 e. NN0) -> ((N + 0) _C 0) = 1)
25	21, 24	mpan2 760	. . . 4 |- (N e. NN0 -> ((N + 0) _C 0) = 1)
26		bcxmaslem1 8334	. . . . . 6 |- (j = 0 -> ((N + j) _C j) = ((N + 0) _C 0))
27	26	fsum1i 8265	. . . . 5 |- ((((N + 0) _C 0) e. NN0 /\ 0 e. ZZ) -> sum_j e. (0...0)((N + j) _C j) = ((N + 0) _C 0))
28		1nn0 7323	. . . . . 6 |- 1 e. NN0
29	25, 28	syl6eqel 1979	. . . . 5 |- (N e. NN0 -> ((N + 0) _C 0) e. NN0)
30		0z 7355	. . . . 5 |- 0 e. ZZ
31	27, 29, 30	sylancl 525	. . . 4 |- (N e. NN0 -> sum_j e. (0...0)((N + j) _C j) = ((N + 0) _C 0))
32		peano2nn0 7333	. . . . . 6 |- (N e. NN0 -> (N + 1) e. NN0)
33	32, 21	jctir 317	. . . . 5 |- (N e. NN0 -> ((N + 1) e. NN0 /\ 0 e. NN0))
34		nn0addcl 7329	. . . . 5 |- (((N + 1) e. NN0 /\ 0 e. NN0) -> ((N + 1) + 0) e. NN0)
35		bcn0 8215	. . . . 5 |- (((N + 1) + 0) e. NN0 -> (((N + 1) + 0) _C 0) = 1)
36	33, 34, 35	3syl 24	. . . 4 |- (N e. NN0 -> (((N + 1) + 0) _C 0) = 1)
37	25, 31, 36	3eqtr4rd 1939	. . 3 |- (N e. NN0 -> (((N + 1) + 0) _C 0) = sum_j e. (0...0)((N + j) _C j))
38		simpr 350	. . . . . . . . . . 11 |- ((N e. NN0 /\ k e. NN0) -> k e. NN0)
39		elnn0uz 7610	. . . . . . . . . . 11 |- (k e. NN0 <-> k e. (ZZ>=` 0))
40	38, 39	sylib 215	. . . . . . . . . 10 |- ((N e. NN0 /\ k e. NN0) -> k e. (ZZ>=` 0))
41		oprex 4907	. . . . . . . . . . 11 |- ((N + j) _C j) e. _V
42		oprex 4907	. . . . . . . . . . 11 |- ((N + (k + 1)) _C (k + 1)) e. _V
43		bcxmaslem1 8334	. . . . . . . . . . 11 |- (j = (k + 1) -> ((N + j) _C j) = ((N + (k + 1)) _C (k + 1)))
44	41, 42, 43	fsump1i 8266	. . . . . . . . . 10 |- (k e. (ZZ>=` 0) -> sum_j e. (0...(k + 1))((N + j) _C j) = (sum_j e. (0...k)((N + j) _C j) + ((N + (k + 1)) _C (k + 1))))
45	40, 44	syl 12	. . . . . . . . 9 |- ((N e. NN0 /\ k e. NN0) -> sum_j e. (0...(k + 1))((N + j) _C j) = (sum_j e. (0...k)((N + j) _C j) + ((N + (k + 1)) _C (k + 1))))
46		nn0cn 7318	. . . . . . . . . . . . 13 |- (N e. NN0 -> N e. CC)
47	46	adantr 425	. . . . . . . . . . . 12 |- ((N e. NN0 /\ k e. NN0) -> N e. CC)
48		nn0cn 7318	. . . . . . . . . . . . 13 |- (k e. NN0 -> k e. CC)
49	48	adantl 424	. . . . . . . . . . . 12 |- ((N e. NN0 /\ k e. NN0) -> k e. CC)
50		ax1cn 6422	. . . . . . . . . . . . 13 |- 1 e. CC
51	50	a1i 8	. . . . . . . . . . . 12 |- ((N e. NN0 /\ k e. NN0) -> 1 e. CC)
52		bcxmaslem2 8335	. . . . . . . . . . . 12 |- ((N e. CC /\ k e. CC /\ 1 e. CC) -> (N + (k + 1)) = ((N + 1) + k))
53	47, 49, 51, 52	syl111anc 1100	. . . . . . . . . . 11 |- ((N e. NN0 /\ k e. NN0) -> (N + (k + 1)) = ((N + 1) + k))
54	53	opreq1d 4897	. . . . . . . . . 10 |- ((N e. NN0 /\ k e. NN0) -> ((N + (k + 1)) _C (k + 1)) = (((N + 1) + k) _C (k + 1)))
55	54	opreq2d 4898	. . . . . . . . 9 |- ((N e. NN0 /\ k e. NN0) -> (sum_j e. (0...k)((N + j) _C j) + ((N + (k + 1)) _C (k + 1))) = (sum_j e. (0...k)((N + j) _C j) + (((N + 1) + k) _C (k + 1))))
56	45, 55	eqtrd 1925	. . . . . . . 8 |- ((N e. NN0 /\ k e. NN0) -> sum_j e. (0...(k + 1))((N + j) _C j) = (sum_j e. (0...k)((N + j) _C j) + (((N + 1) + k) _C (k + 1))))
57	56	adantr 425	. . . . . . 7 |- (((N e. NN0 /\ k e. NN0) /\ (((N + 1) + k) _C k) = sum_j e. (0...k)((N + j) _C j)) -> sum_j e. (0...(k + 1))((N + j) _C j) = (sum_j e. (0...k)((N + j) _C j) + (((N + 1) + k) _C (k + 1))))
58		opreq1 4889	. . . . . . . 8 |- ((((N + 1) + k) _C k) = sum_j e. (0...k)((N + j) _C j) -> ((((N + 1) + k) _C k) + (((N + 1) + k) _C (k + 1))) = (sum_j e. (0...k)((N + j) _C j) + (((N + 1) + k) _C (k + 1))))
59	58	adantl 424	. . . . . . 7 |- (((N e. NN0 /\ k e. NN0) /\ (((N + 1) + k) _C k) = sum_j e. (0...k)((N + j) _C j)) -> ((((N + 1) + k) _C k) + (((N + 1) + k) _C (k + 1))) = (sum_j e. (0...k)((N + j) _C j) + (((N + 1) + k) _C (k + 1))))
60		pncan 6557	. . . . . . . . . . . . 13 |- ((k e. CC /\ 1 e. CC) -> ((k + 1) - 1) = k)
61	60, 49, 50	sylancl 525	. . . . . . . . . . . 12 |- ((N e. NN0 /\ k e. NN0) -> ((k + 1) - 1) = k)
62	61	opreq2d 4898	. . . . . . . . . . 11 |- ((N e. NN0 /\ k e. NN0) -> (((N + 1) + k) _C ((k + 1) - 1)) = (((N + 1) + k) _C k))
63	62	opreq2d 4898	. . . . . . . . . 10 |- ((N e. NN0 /\ k e. NN0) -> ((((N + 1) + k) _C (k + 1)) + (((N + 1) + k) _C ((k + 1) - 1))) = ((((N + 1) + k) _C (k + 1)) + (((N + 1) + k) _C k)))
64		nnnn0addcl 7334	. . . . . . . . . . . 12 |- (((N + 1) e. NN /\ k e. NN0) -> ((N + 1) + k) e. NN)
65		nn0p1nn 7384	. . . . . . . . . . . 12 |- (N e. NN0 -> (N + 1) e. NN)
66	64, 65	sylan 497	. . . . . . . . . . 11 |- ((N e. NN0 /\ k e. NN0) -> ((N + 1) + k) e. NN)
67		nn0p1nn 7384	. . . . . . . . . . . 12 |- (k e. NN0 -> (k + 1) e. NN)
68	67	adantl 424	. . . . . . . . . . 11 |- ((N e. NN0 /\ k e. NN0) -> (k + 1) e. NN)
69		nnre 7112	. . . . . . . . . . . . . 14 |- ((k + 1) e. NN -> (k + 1) e. RR)
70	68, 69	syl 12	. . . . . . . . . . . . 13 |- ((N e. NN0 /\ k e. NN0) -> (k + 1) e. RR)
71		simpl 346	. . . . . . . . . . . . 13 |- ((N e. NN0 /\ k e. NN0) -> N e. NN0)
72		nn0addge2 7340	. . . . . . . . . . . . 13 |- (((k + 1) e. RR /\ N e. NN0) -> (k + 1) <_ (N + (k + 1)))
73	70, 71, 72	syl11anc 524	. . . . . . . . . . . 12 |- ((N e. NN0 /\ k e. NN0) -> (k + 1) <_ (N + (k + 1)))
74	73, 53	breqtrd 3361	. . . . . . . . . . 11 |- ((N e. NN0 /\ k e. NN0) -> (k + 1) <_ ((N + 1) + k))
75		bcpasc2 8220	. . . . . . . . . . 11 |- ((((N + 1) + k) e. NN /\ (k + 1) e. NN /\ (k + 1) <_ ((N + 1) + k)) -> ((((N + 1) + k) _C (k + 1)) + (((N + 1) + k) _C ((k + 1) - 1))) = ((((N + 1) + k) + 1) _C (k + 1)))
76	66, 68, 74, 75	syl111anc 1100	. . . . . . . . . 10 |- ((N e. NN0 /\ k e. NN0) -> ((((N + 1) + k) _C (k + 1)) + (((N + 1) + k) _C ((k + 1) - 1))) = ((((N + 1) + k) + 1) _C (k + 1)))
77	63, 76	eqtr3d 1927	. . . . . . . . 9 |- ((N e. NN0 /\ k e. NN0) -> ((((N + 1) + k) _C (k + 1)) + (((N + 1) + k) _C k)) = ((((N + 1) + k) + 1) _C (k + 1)))
78		nnnn0 7315	. . . . . . . . . . . . 13 |- (((N + 1) + k) e. NN -> ((N + 1) + k) e. NN0)
79	66, 78	syl 12	. . . . . . . . . . . 12 |- ((N e. NN0 /\ k e. NN0) -> ((N + 1) + k) e. NN0)
80		nnz 7362	. . . . . . . . . . . . 13 |- ((k + 1) e. NN -> (k + 1) e. ZZ)
81	68, 80	syl 12	. . . . . . . . . . . 12 |- ((N e. NN0 /\ k e. NN0) -> (k + 1) e. ZZ)
82		bccl 8224	. . . . . . . . . . . 12 |- ((((N + 1) + k) e. NN0 /\ (k + 1) e. ZZ) -> (((N + 1) + k) _C (k + 1)) e. NN0)
83	79, 81, 82	syl11anc 524	. . . . . . . . . . 11 |- ((N e. NN0 /\ k e. NN0) -> (((N + 1) + k) _C (k + 1)) e. NN0)
84		nn0cn 7318	. . . . . . . . . . 11 |- ((((N + 1) + k) _C (k + 1)) e. NN0 -> (((N + 1) + k) _C (k + 1)) e. CC)
85	83, 84	syl 12	. . . . . . . . . 10 |- ((N e. NN0 /\ k e. NN0) -> (((N + 1) + k) _C (k + 1)) e. CC)
86		nn0addcl 7329	. . . . . . . . . . . . 13 |- (((N + 1) e. NN0 /\ k e. NN0) -> ((N + 1) + k) e. NN0)
87		nn0z 7363	. . . . . . . . . . . . . 14 |- (k e. NN0 -> k e. ZZ)
88	87	adantl 424	. . . . . . . . . . . . 13 |- (((N + 1) e. NN0 /\ k e. NN0) -> k e. ZZ)
89		bccl 8224	. . . . . . . . . . . . 13 |- ((((N + 1) + k) e. NN0 /\ k e. ZZ) -> (((N + 1) + k) _C k) e. NN0)
90	86, 88, 89	syl11anc 524	. . . . . . . . . . . 12 |- (((N + 1) e. NN0 /\ k e. NN0) -> (((N + 1) + k) _C k) e. NN0)
91	90, 32	sylan 497	. . . . . . . . . . 11 |- ((N e. NN0 /\ k e. NN0) -> (((N + 1) + k) _C k) e. NN0)
92		nn0cn 7318	. . . . . . . . . . 11 |- ((((N + 1) + k) _C k) e. NN0 -> (((N + 1) + k) _C k) e. CC)
93	91, 92	syl 12	. . . . . . . . . 10 |- ((N e. NN0 /\ k e. NN0) -> (((N + 1) + k) _C k) e. CC)
94		addcom 6458	. . . . . . . . . 10 |- (((((N + 1) + k) _C (k + 1)) e. CC /\ (((N + 1) + k) _C k) e. CC) -> ((((N + 1) + k) _C (k + 1)) + (((N + 1) + k) _C k)) = ((((N + 1) + k) _C k) + (((N + 1) + k) _C (k + 1))))
95	85, 93, 94	syl11anc 524	. . . . . . . . 9 |- ((N e. NN0 /\ k e. NN0) -> ((((N + 1) + k) _C (k + 1)) + (((N + 1) + k) _C k)) = ((((N + 1) + k) _C k) + (((N + 1) + k) _C (k + 1))))
96		peano2cn 6498	. . . . . . . . . . . . 13 |- (N e. CC -> (N + 1) e. CC)
97	46, 96	syl 12	. . . . . . . . . . . 12 |- (N e. NN0 -> (N + 1) e. CC)
98	97	adantr 425	. . . . . . . . . . 11 |- ((N e. NN0 /\ k e. NN0) -> (N + 1) e. CC)
99		addass 6460	. . . . . . . . . . 11 |- (((N + 1) e. CC /\ k e. CC /\ 1 e. CC) -> (((N + 1) + k) + 1) = ((N + 1) + (k + 1)))
100	98, 49, 51, 99	syl111anc 1100	. . . . . . . . . 10 |- ((N e. NN0 /\ k e. NN0) -> (((N + 1) + k) + 1) = ((N + 1) + (k + 1)))
101	100	opreq1d 4897	. . . . . . . . 9 |- ((N e. NN0 /\ k e. NN0) -> ((((N + 1) + k) + 1) _C (k + 1)) = (((N + 1) + (k + 1)) _C (k + 1)))
102	77, 95, 101	3eqtr3d 1934	. . . . . . . 8 |- ((N e. NN0 /\ k e. NN0) -> ((((N + 1) + k) _C k) + (((N + 1) + k) _C (k + 1))) = (((N + 1) + (k + 1)) _C (k + 1)))
103	102	adantr 425	. . . . . . 7 |- (((N e. NN0 /\ k e. NN0) /\ (((N + 1) + k) _C k) = sum_j e. (0...k)((N + j) _C j)) -> ((((N + 1) + k) _C k) + (((N + 1) + k) _C (k + 1))) = (((N + 1) + (k + 1)) _C (k + 1)))
104	57, 59, 103	3eqtr2rd 1933	. . . . . 6 |- (((N e. NN0 /\ k e. NN0) /\ (((N + 1) + k) _C k) = sum_j e. (0...k)((N + j) _C j)) -> (((N + 1) + (k + 1)) _C (k + 1)) = sum_j e. (0...(k + 1))((N + j) _C j))
105	104	ex 402	. . . . 5 |- ((N e. NN0 /\ k e. NN0) -> ((((N + 1) + k) _C k) = sum_j e. (0...k)((N + j) _C j) -> (((N + 1) + (k + 1)) _C (k + 1)) = sum_j e. (0...(k + 1))((N + j) _C j)))
106	105	expcom 403	. . . 4 |- (k e. NN0 -> (N e. NN0 -> ((((N + 1) + k) _C k) = sum_j e. (0...k)((N + j) _C j) -> (((N + 1) + (k + 1)) _C (k + 1)) = sum_j e. (0...(k + 1))((N + j) _C j))))
107	106	a2d 16	. . 3 |- (k e. NN0 -> ((N e. NN0 -> (((N + 1) + k) _C k) = sum_j e. (0...k)((N + j) _C j)) -> (N e. NN0 -> (((N + 1) + (k + 1)) _C (k + 1)) = sum_j e. (0...(k + 1))((N + j) _C j))))
108	5, 10, 15, 20, 37, 107	nn0ind 7424	. 2 |- (M e. NN0 -> (N e. NN0 -> (((N + 1) + M) _C M) = sum_j e. (0...M)((N + j) _C j)))
109	108	impcom 378	1 |- ((N e. NN0 /\ M e. NN0) -> (((N + 1) + M) _C M) = sum_j e. (0...M)((N + j) _C j))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   - cmin 6445   <_ cle 6448  NNcn 6449  NN0cn0 6450  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637   _C cbc 8208  sum_csu 8239

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-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-div 6892  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-fac 8184  df-bc 8209  df-sum 8240

Copyright terms: Public domain