Theorem fsum0diag 8520

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

Assertion

Ref	Expression
fsum0diag	|- ((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...k)(A x. [_(k - j) / k]_B))

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

Proof of Theorem fsum0diag

Step	Hyp	Ref	Expression
1		fsum0diaglem2 8519	. . 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_k e. (0...n)sum_j e. (0...k)(A x. [_(k - j) / k]_B)) -> ((A.j e. (0...(n + 1))A e. CC /\ A.k e. (0...(n + 1))B e. CC) -> sum_j e. (0...(n + 1))sum_k e. (0...((n + 1) - j))(A x. B) = sum_k e. (0...(n + 1))sum_j e. (0...k)(A x. [_(k - j) / k]_B))))
2		oprex 4907	. . . . . 6 |- ([_0 / j]_A x. [_0 / k]_B) e. _V
3		0z 7355	. . . . . 6 |- 0 e. ZZ
4		ax-17 1317	. . . . . . . . . 10 |- (m e. 0 -> A.j m e. 0)
5	4	hbcsb1g 2567	. . . . . . . . 9 |- (0 e. ZZ -> (m e. [_0 / j]_A -> A.j m e. [_0 / j]_A))
6	3, 5	ax-mp 7	. . . . . . . 8 |- (m e. [_0 / j]_A -> A.j m e. [_0 / j]_A)
7		ax-17 1317	. . . . . . . 8 |- (m e. x. -> A.j m e. x. )
8		ax-17 1317	. . . . . . . 8 |- (m e. [_0 / k]_B -> A.j m e. [_0 / k]_B)
9	6, 7, 8	hbopr 4904	. . . . . . 7 |- (m e. ([_0 / j]_A x. [_0 / k]_B) -> A.j m e. ([_0 / j]_A x. [_0 / k]_B))
10		csbeq1a 2546	. . . . . . . 8 |- (j = 0 -> A = [_0 / j]_A)
11	10	opreq1d 4897	. . . . . . 7 |- (j = 0 -> (A x. [_0 / k]_B) = ([_0 / j]_A x. [_0 / k]_B))
12	9, 11	fsum1fi 8267	. . . . . 6 |- ((([_0 / j]_A x. [_0 / k]_B) e. _V /\ 0 e. ZZ) -> sum_j e. (0...0)(A x. [_0 / k]_B) = ([_0 / j]_A x. [_0 / k]_B))
13	2, 3, 12	mp2an 761	. . . . 5 |- sum_j e. (0...0)(A x. [_0 / k]_B) = ([_0 / j]_A x. [_0 / k]_B)
14		sumex 8241	. . . . . 6 |- sum_j e. (0...0)(A x. [_0 / k]_B) e. _V
15		ax-17 1317	. . . . . . . 8 |- (m e. (0...0) -> A.k m e. (0...0))
16		ax-17 1317	. . . . . . . . 9 |- (m e. A -> A.k m e. A)
17		ax-17 1317	. . . . . . . . 9 |- (m e. x. -> A.k m e. x. )
18		ax-17 1317	. . . . . . . . . . 11 |- (m e. 0 -> A.k m e. 0)
19	18	hbcsb1g 2567	. . . . . . . . . 10 |- (0 e. ZZ -> (m e. [_0 / k]_B -> A.k m e. [_0 / k]_B))
20	3, 19	ax-mp 7	. . . . . . . . 9 |- (m e. [_0 / k]_B -> A.k m e. [_0 / k]_B)
21	16, 17, 20	hbopr 4904	. . . . . . . 8 |- (m e. (A x. [_0 / k]_B) -> A.k m e. (A x. [_0 / k]_B))
22	15, 21	hbsum 8244	. . . . . . 7 |- (m e. sum_j e. (0...0)(A x. [_0 / k]_B) -> A.k m e. sum_j e. (0...0)(A x. [_0 / k]_B))
23		opreq2 4890	. . . . . . . 8 |- (k = 0 -> (0...k) = (0...0))
24		opreq12 4891	. . . . . . . . . . . 12 |- ((k = 0 /\ j = 0) -> (k - j) = (0 - 0))
25		0cn 6481	. . . . . . . . . . . . 13 |- 0 e. CC
26	25	subidi 6551	. . . . . . . . . . . 12 |- (0 - 0) = 0
27	24, 26	syl6eq 1944	. . . . . . . . . . 11 |- ((k = 0 /\ j = 0) -> (k - j) = 0)
28	27	csbeq1d 2544	. . . . . . . . . 10 |- ((k = 0 /\ j = 0) -> [_(k - j) / k]_B = [_0 / k]_B)
29	28	opreq2d 4898	. . . . . . . . 9 |- ((k = 0 /\ j = 0) -> (A x. [_(k - j) / k]_B) = (A x. [_0 / k]_B))
30		elfz1eq 7662	. . . . . . . . 9 |- (j e. (0...0) -> j = 0)
31	29, 30	sylan2 500	. . . . . . . 8 |- ((k = 0 /\ j e. (0...0)) -> (A x. [_(k - j) / k]_B) = (A x. [_0 / k]_B))
32	23, 31	sumeq12rdv 8256	. . . . . . 7 |- (k = 0 -> sum_j e. (0...k)(A x. [_(k - j) / k]_B) = sum_j e. (0...0)(A x. [_0 / k]_B))
33	22, 32	fsum1fi 8267	. . . . . 6 |- ((sum_j e. (0...0)(A x. [_0 / k]_B) e. _V /\ 0 e. ZZ) -> sum_k e. (0...0)sum_j e. (0...k)(A x. [_(k - j) / k]_B) = sum_j e. (0...0)(A x. [_0 / k]_B))
34	14, 3, 33	mp2an 761	. . . . 5 |- sum_k e. (0...0)sum_j e. (0...k)(A x. [_(k - j) / k]_B) = sum_j e. (0...0)(A x. [_0 / k]_B)
35		sumex 8241	. . . . . . 7 |- sum_k e. (0...0)([_0 / j]_A x. B) e. _V
36		ax-17 1317	. . . . . . . . 9 |- (m e. (0...0) -> A.j m e. (0...0))
37		ax-17 1317	. . . . . . . . . 10 |- (m e. B -> A.j m e. B)
38	6, 7, 37	hbopr 4904	. . . . . . . . 9 |- (m e. ([_0 / j]_A x. B) -> A.j m e. ([_0 / j]_A x. B))
39	36, 38	hbsum 8244	. . . . . . . 8 |- (m e. sum_k e. (0...0)([_0 / j]_A x. B) -> A.j m e. sum_k e. (0...0)([_0 / j]_A x. B))
40		opreq2 4890	. . . . . . . . . . 11 |- (j = 0 -> (0 - j) = (0 - 0))
41	40, 26	syl6eq 1944	. . . . . . . . . 10 |- (j = 0 -> (0 - j) = 0)
42	41	opreq2d 4898	. . . . . . . . 9 |- (j = 0 -> (0...(0 - j)) = (0...0))
43	10	opreq1d 4897	. . . . . . . . . 10 |- (j = 0 -> (A x. B) = ([_0 / j]_A x. B))
44	43	adantr 425	. . . . . . . . 9 |- ((j = 0 /\ k e. (0...0)) -> (A x. B) = ([_0 / j]_A x. B))
45	42, 44	sumeq12rdv 8256	. . . . . . . 8 |- (j = 0 -> sum_k e. (0...(0 - j))(A x. B) = sum_k e. (0...0)([_0 / j]_A x. B))
46	39, 45	fsum1fi 8267	. . . . . . 7 |- ((sum_k e. (0...0)([_0 / j]_A x. B) e. _V /\ 0 e. ZZ) -> sum_j e. (0...0)sum_k e. (0...(0 - j))(A x. B) = sum_k e. (0...0)([_0 / j]_A x. B))
47	35, 3, 46	mp2an 761	. . . . . 6 |- sum_j e. (0...0)sum_k e. (0...(0 - j))(A x. B) = sum_k e. (0...0)([_0 / j]_A x. B)
48		ax-17 1317	. . . . . . . . 9 |- (m e. [_0 / j]_A -> A.k m e. [_0 / j]_A)
49	48, 17, 20	hbopr 4904	. . . . . . . 8 |- (m e. ([_0 / j]_A x. [_0 / k]_B) -> A.k m e. ([_0 / j]_A x. [_0 / k]_B))
50		csbeq1a 2546	. . . . . . . . 9 |- (k = 0 -> B = [_0 / k]_B)
51	50	opreq2d 4898	. . . . . . . 8 |- (k = 0 -> ([_0 / j]_A x. B) = ([_0 / j]_A x. [_0 / k]_B))
52	49, 51	fsum1fi 8267	. . . . . . 7 |- ((([_0 / j]_A x. [_0 / k]_B) e. _V /\ 0 e. ZZ) -> sum_k e. (0...0)([_0 / j]_A x. B) = ([_0 / j]_A x. [_0 / k]_B))
53	2, 3, 52	mp2an 761	. . . . . 6 |- sum_k e. (0...0)([_0 / j]_A x. B) = ([_0 / j]_A x. [_0 / k]_B)
54	47, 53	eqtri 1908	. . . . 5 |- sum_j e. (0...0)sum_k e. (0...(0 - j))(A x. B) = ([_0 / j]_A x. [_0 / k]_B)
55	13, 34, 54	3eqtr4ri 1923	. . . 4 |- sum_j e. (0...0)sum_k e. (0...(0 - j))(A x. B) = sum_k e. (0...0)sum_j e. (0...k)(A x. [_(k - j) / k]_B)
56	55	a1i 8	. . 3 |- ((A.j e. (0...0)A e. CC /\ A.k e. (0...0)B e. CC) -> sum_j e. (0...0)sum_k e. (0...(0 - j))(A x. B) = sum_k e. (0...0)sum_j e. (0...k)(A x. [_(k - j) / k]_B))
57		opreq2 4890	. . . . . 6 |- (m = 0 -> (0...m) = (0...0))
58	57	raleqdv 2269	. . . . 5 |- (m = 0 -> (A.j e. (0...m)A e. CC <-> A.j e. (0...0)A e. CC))
59	57	raleqdv 2269	. . . . 5 |- (m = 0 -> (A.k e. (0...m)B e. CC <-> A.k e. (0...0)B e. CC))
60	58, 59	anbi12d 690	. . . 4 |- (m = 0 -> ((A.j e. (0...m)A e. CC /\ A.k e. (0...m)B e. CC) <-> (A.j e. (0...0)A e. CC /\ A.k e. (0...0)B e. CC)))
61		opreq1 4889	. . . . . . . . 9 |- (m = 0 -> (m - j) = (0 - j))
62	61	opreq2d 4898	. . . . . . . 8 |- (m = 0 -> (0...(m - j)) = (0...(0 - j)))
63	62	sumeq1d 8250	. . . . . . 7 |- (m = 0 -> sum_k e. (0...(m - j))(A x. B) = sum_k e. (0...(0 - j))(A x. B))
64	63	adantr 425	. . . . . 6 |- ((m = 0 /\ j e. (0...m)) -> sum_k e. (0...(m - j))(A x. B) = sum_k e. (0...(0 - j))(A x. B))
65	57, 64	sumeq12dv 8255	. . . . 5 |- (m = 0 -> sum_j e. (0...m)sum_k e. (0...(m - j))(A x. B) = sum_j e. (0...0)sum_k e. (0...(0 - j))(A x. B))
66	57	sumeq1d 8250	. . . . 5 |- (m = 0 -> sum_k e. (0...m)sum_j e. (0...k)(A x. [_(k - j) / k]_B) = sum_k e. (0...0)sum_j e. (0...k)(A x. [_(k - j) / k]_B))
67	65, 66	eqeq12d 1899	. . . 4 |- (m = 0 -> (sum_j e. (0...m)sum_k e. (0...(m - j))(A x. B) = sum_k e. (0...m)sum_j e. (0...k)(A x. [_(k - j) / k]_B) <-> sum_j e. (0...0)sum_k e. (0...(0 - j))(A x. B) = sum_k e. (0...0)sum_j e. (0...k)(A x. [_(k - j) / k]_B)))
68	60, 67	imbi12d 688	. . 3 |- (m = 0 -> (((A.j e. (0...m)A e. CC /\ A.k e. (0...m)B e. CC) -> sum_j e. (0...m)sum_k e. (0...(m - j))(A x. B) = sum_k e. (0...m)sum_j e. (0...k)(A x. [_(k - j) / k]_B)) <-> ((A.j e. (0...0)A e. CC /\ A.k e. (0...0)B e. CC) -> sum_j e. (0...0)sum_k e. (0...(0 - j))(A x. B) = sum_k e. (0...0)sum_j e. (0...k)(A x. [_(k - j) / k]_B))))
69		opreq2 4890	. . . . . 6 |- (m = n -> (0...m) = (0...n))
70	69	raleqdv 2269	. . . . 5 |- (m = n -> (A.j e. (0...m)A e. CC <-> A.j e. (0...n)A e. CC))
71	69	raleqdv 2269	. . . . 5 |- (m = n -> (A.k e. (0...m)B e. CC <-> A.k e. (0...n)B e. CC))
72	70, 71	anbi12d 690	. . . 4 |- (m = n -> ((A.j e. (0...m)A e. CC /\ A.k e. (0...m)B e. CC) <-> (A.j e. (0...n)A e. CC /\ A.k e. (0...n)B e. CC)))
73		opreq1 4889	. . . . . . . . 9 |- (m = n -> (m - j) = (n - j))
74	73	opreq2d 4898	. . . . . . . 8 |- (m = n -> (0...(m - j)) = (0...(n - j)))
75	74	sumeq1d 8250	. . . . . . 7 |- (m = n -> sum_k e. (0...(m - j))(A x. B) = sum_k e. (0...(n - j))(A x. B))
76	75	adantr 425	. . . . . 6 |- ((m = n /\ j e. (0...m)) -> sum_k e. (0...(m - j))(A x. B) = sum_k e. (0...(n - j))(A x. B))
77	69, 76	sumeq12dv 8255	. . . . 5 |- (m = n -> sum_j e. (0...m)sum_k e. (0...(m - j))(A x. B) = sum_j e. (0...n)sum_k e. (0...(n - j))(A x. B))
78	69	sumeq1d 8250	. . . . 5 |- (m = n -> sum_k e. (0...m)sum_j e. (0...k)(A x. [_(k - j) / k]_B) = sum_k e. (0...n)sum_j e. (0...k)(A x. [_(k - j) / k]_B))
79	77, 78	eqeq12d 1899	. . . 4 |- (m = n -> (sum_j e. (0...m)sum_k e. (0...(m - j))(A x. B) = sum_k e. (0...m)sum_j e. (0...k)(A x. [_(k - j) / k]_B) <-> sum_j e. (0...n)sum_k e. (0...(n - j))(A x. B) = sum_k e. (0...n)sum_j e. (0...k)(A x. [_(k - j) / k]_B)))
80	72, 79	imbi12d 688	. . 3 |- (m = n -> (((A.j e. (0...m)A e. CC /\ A.k e. (0...m)B e. CC) -> sum_j e. (0...m)sum_k e. (0...(m - j))(A x. B) = sum_k e. (0...m)sum_j e. (0...k)(A x. [_(k - j) / k]_B)) <-> ((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...k)(A x. [_(k - j) / k]_B))))
81		opreq2 4890	. . . . . 6 |- (m = (n + 1) -> (0...m) = (0...(n + 1)))
82	81	raleqdv 2269	. . . . 5 |- (m = (n + 1) -> (A.j e. (0...m)A e. CC <-> A.j e. (0...(n + 1))A e. CC))
83	81	raleqdv 2269	. . . . 5 |- (m = (n + 1) -> (A.k e. (0...m)B e. CC <-> A.k e. (0...(n + 1))B e. CC))
84	82, 83	anbi12d 690	. . . 4 |- (m = (n + 1) -> ((A.j e. (0...m)A e. CC /\ A.k e. (0...m)B e. CC) <-> (A.j e. (0...(n + 1))A e. CC /\ A.k e. (0...(n + 1))B e. CC)))
85		opreq1 4889	. . . . . . . . 9 |- (m = (n + 1) -> (m - j) = ((n + 1) - j))
86	85	opreq2d 4898	. . . . . . . 8 |- (m = (n + 1) -> (0...(m - j)) = (0...((n + 1) - j)))
87	86	sumeq1d 8250	. . . . . . 7 |- (m = (n + 1) -> sum_k e. (0...(m - j))(A x. B) = sum_k e. (0...((n + 1) - j))(A x. B))
88	87	adantr 425	. . . . . 6 |- ((m = (n + 1) /\ j e. (0...m)) -> sum_k e. (0...(m - j))(A x. B) = sum_k e. (0...((n + 1) - j))(A x. B))
89	81, 88	sumeq12dv 8255	. . . . 5 |- (m = (n + 1) -> sum_j e. (0...m)sum_k e. (0...(m - j))(A x. B) = sum_j e. (0...(n + 1))sum_k e. (0...((n + 1) - j))(A x. B))
90	81	sumeq1d 8250	. . . . 5 |- (m = (n + 1) -> sum_k e. (0...m)sum_j e. (0...k)(A x. [_(k - j) / k]_B) = sum_k e. (0...(n + 1))sum_j e. (0...k)(A x. [_(k - j) / k]_B))
91	89, 90	eqeq12d 1899	. . . 4 |- (m = (n + 1) -> (sum_j e. (0...m)sum_k e. (0...(m - j))(A x. B) = sum_k e. (0...m)sum_j e. (0...k)(A x. [_(k - j) / k]_B) <-> sum_j e. (0...(n + 1))sum_k e. (0...((n + 1) - j))(A x. B) = sum_k e. (0...(n + 1))sum_j e. (0...k)(A x. [_(k - j) / k]_B)))
92	84, 91	imbi12d 688	. . 3 |- (m = (n + 1) -> (((A.j e. (0...m)A e. CC /\ A.k e. (0...m)B e. CC) -> sum_j e. (0...m)sum_k e. (0...(m - j))(A x. B) = sum_k e. (0...m)sum_j e. (0...k)(A x. [_(k - j) / k]_B)) <-> ((A.j e. (0...(n + 1))A e. CC /\ A.k e. (0...(n + 1))B e. CC) -> sum_j e. (0...(n + 1))sum_k e. (0...((n + 1) - j))(A x. B) = sum_k e. (0...(n + 1))sum_j e. (0...k)(A x. [_(k - j) / k]_B))))
93		opreq2 4890	. . . . . 6 |- (m = N -> (0...m) = (0...N))
94	93	raleqdv 2269	. . . . 5 |- (m = N -> (A.j e. (0...m)A e. CC <-> A.j e. (0...N)A e. CC))
95	93	raleqdv 2269	. . . . 5 |- (m = N -> (A.k e. (0...m)B e. CC <-> A.k e. (0...N)B e. CC))
96	94, 95	anbi12d 690	. . . 4 |- (m = N -> ((A.j e. (0...m)A e. CC /\ A.k e. (0...m)B e. CC) <-> (A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC)))
97		opreq1 4889	. . . . . . . . 9 |- (m = N -> (m - j) = (N - j))
98	97	opreq2d 4898	. . . . . . . 8 |- (m = N -> (0...(m - j)) = (0...(N - j)))
99	98	sumeq1d 8250	. . . . . . 7 |- (m = N -> sum_k e. (0...(m - j))(A x. B) = sum_k e. (0...(N - j))(A x. B))
100	99	adantr 425	. . . . . 6 |- ((m = N /\ j e. (0...m)) -> sum_k e. (0...(m - j))(A x. B) = sum_k e. (0...(N - j))(A x. B))
101	93, 100	sumeq12dv 8255	. . . . 5 |- (m = N -> sum_j e. (0...m)sum_k e. (0...(m - j))(A x. B) = sum_j e. (0...N)sum_k e. (0...(N - j))(A x. B))
102	93	sumeq1d 8250	. . . . 5 |- (m = N -> sum_k e. (0...m)sum_j e. (0...k)(A x. [_(k - j) / k]_B) = sum_k e. (0...N)sum_j e. (0...k)(A x. [_(k - j) / k]_B))
103	101, 102	eqeq12d 1899	. . . 4 |- (m = N -> (sum_j e. (0...m)sum_k e. (0...(m - j))(A x. B) = sum_k e. (0...m)sum_j e. (0...k)(A x. [_(k - j) / k]_B) <-> sum_j e. (0...N)sum_k e. (0...(N - j))(A x. B) = sum_k e. (0...N)sum_j e. (0...k)(A x. [_(k - j) / k]_B)))
104	96, 103	imbi12d 688	. . 3 |- (m = N -> (((A.j e. (0...m)A e. CC /\ A.k e. (0...m)B e. CC) -> sum_j e. (0...m)sum_k e. (0...(m - j))(A x. B) = sum_k e. (0...m)sum_j e. (0...k)(A x. [_(k - j) / k]_B)) <-> ((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...k)(A x. [_(k - j) / k]_B))))
105	1, 56, 68, 80, 92, 104	nn0indALT 7425	. 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...k)(A x. [_(k - j) / k]_B)))
106	105	3impib 1065	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)sum_j e. (0...k)(A x. [_(k - j) / k]_B))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  [_csb 2540  (class class class)co 4884  CCcc 6384  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445  NN0cn0 6450  ZZcz 6451  ...cfz 7637  sum_csu 8239

This theorem is referenced by:  fsum0diag2 8521  fsum0diag4 8523

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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