Theorem fsumlt 15821

Description: If every term in one finite sum is less than the corresponding term in another, then the first sum is less than the second.

Assertion

Ref	Expression
fsumlt	|- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(B e. RR /\ C e. RR /\ B < C)) -> sum_k e. (M...N)B < sum_k e. (M...N)C)

Distinct variable groups:   k,M   k,N

Proof of Theorem fsumlt

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . 5 |- (n = M -> (M...n) = (M...M))
2	1	raleqdv 2269	. . . 4 |- (n = M -> (A.k e. (M...n)(B e. RR /\ C e. RR /\ B < C) <-> A.k e. (M...M)(B e. RR /\ C e. RR /\ B < C)))
3	1	sumeq1d 8250	. . . . 5 |- (n = M -> sum_k e. (M...n)B = sum_k e. (M...M)B)
4	1	sumeq1d 8250	. . . . 5 |- (n = M -> sum_k e. (M...n)C = sum_k e. (M...M)C)
5	3, 4	breq12d 3351	. . . 4 |- (n = M -> (sum_k e. (M...n)B < sum_k e. (M...n)C <-> sum_k e. (M...M)B < sum_k e. (M...M)C))
6	2, 5	imbi12d 688	. . 3 |- (n = M -> ((A.k e. (M...n)(B e. RR /\ C e. RR /\ B < C) -> sum_k e. (M...n)B < sum_k e. (M...n)C) <-> (A.k e. (M...M)(B e. RR /\ C e. RR /\ B < C) -> sum_k e. (M...M)B < sum_k e. (M...M)C)))
7		opreq2 4890	. . . . 5 |- (n = j -> (M...n) = (M...j))
8	7	raleqdv 2269	. . . 4 |- (n = j -> (A.k e. (M...n)(B e. RR /\ C e. RR /\ B < C) <-> A.k e. (M...j)(B e. RR /\ C e. RR /\ B < C)))
9	7	sumeq1d 8250	. . . . 5 |- (n = j -> sum_k e. (M...n)B = sum_k e. (M...j)B)
10	7	sumeq1d 8250	. . . . 5 |- (n = j -> sum_k e. (M...n)C = sum_k e. (M...j)C)
11	9, 10	breq12d 3351	. . . 4 |- (n = j -> (sum_k e. (M...n)B < sum_k e. (M...n)C <-> sum_k e. (M...j)B < sum_k e. (M...j)C))
12	8, 11	imbi12d 688	. . 3 |- (n = j -> ((A.k e. (M...n)(B e. RR /\ C e. RR /\ B < C) -> sum_k e. (M...n)B < sum_k e. (M...n)C) <-> (A.k e. (M...j)(B e. RR /\ C e. RR /\ B < C) -> sum_k e. (M...j)B < sum_k e. (M...j)C)))
13		opreq2 4890	. . . . 5 |- (n = (j + 1) -> (M...n) = (M...(j + 1)))
14	13	raleqdv 2269	. . . 4 |- (n = (j + 1) -> (A.k e. (M...n)(B e. RR /\ C e. RR /\ B < C) <-> A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C)))
15	13	sumeq1d 8250	. . . . 5 |- (n = (j + 1) -> sum_k e. (M...n)B = sum_k e. (M...(j + 1))B)
16	13	sumeq1d 8250	. . . . 5 |- (n = (j + 1) -> sum_k e. (M...n)C = sum_k e. (M...(j + 1))C)
17	15, 16	breq12d 3351	. . . 4 |- (n = (j + 1) -> (sum_k e. (M...n)B < sum_k e. (M...n)C <-> sum_k e. (M...(j + 1))B < sum_k e. (M...(j + 1))C))
18	14, 17	imbi12d 688	. . 3 |- (n = (j + 1) -> ((A.k e. (M...n)(B e. RR /\ C e. RR /\ B < C) -> sum_k e. (M...n)B < sum_k e. (M...n)C) <-> (A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C) -> sum_k e. (M...(j + 1))B < sum_k e. (M...(j + 1))C)))
19		opreq2 4890	. . . . 5 |- (n = N -> (M...n) = (M...N))
20	19	raleqdv 2269	. . . 4 |- (n = N -> (A.k e. (M...n)(B e. RR /\ C e. RR /\ B < C) <-> A.k e. (M...N)(B e. RR /\ C e. RR /\ B < C)))
21	19	sumeq1d 8250	. . . . 5 |- (n = N -> sum_k e. (M...n)B = sum_k e. (M...N)B)
22	19	sumeq1d 8250	. . . . 5 |- (n = N -> sum_k e. (M...n)C = sum_k e. (M...N)C)
23	21, 22	breq12d 3351	. . . 4 |- (n = N -> (sum_k e. (M...n)B < sum_k e. (M...n)C <-> sum_k e. (M...N)B < sum_k e. (M...N)C))
24	20, 23	imbi12d 688	. . 3 |- (n = N -> ((A.k e. (M...n)(B e. RR /\ C e. RR /\ B < C) -> sum_k e. (M...n)B < sum_k e. (M...n)C) <-> (A.k e. (M...N)(B e. RR /\ C e. RR /\ B < C) -> sum_k e. (M...N)B < sum_k e. (M...N)C)))
25		ra4sbca 2537	. . . . . . 7 |- ((M e. (M...M) /\ A.k e. (M...M)B < C) -> [M / k]B < C)
26		sbcbr12g 3392	. . . . . . . 8 |- (M e. (M...M) -> ([M / k]B < C <-> [_M / k]_B < [_M / k]_C))
27	26	adantr 425	. . . . . . 7 |- ((M e. (M...M) /\ A.k e. (M...M)B < C) -> ([M / k]B < C <-> [_M / k]_B < [_M / k]_C))
28	25, 27	mpbid 212	. . . . . 6 |- ((M e. (M...M) /\ A.k e. (M...M)B < C) -> [_M / k]_B < [_M / k]_C)
29		elfz3 7661	. . . . . 6 |- (M e. ZZ -> M e. (M...M))
30		simp3 878	. . . . . . 7 |- ((B e. RR /\ C e. RR /\ B < C) -> B < C)
31	30	ralimi 2168	. . . . . 6 |- (A.k e. (M...M)(B e. RR /\ C e. RR /\ B < C) -> A.k e. (M...M)B < C)
32	28, 29, 31	syl2an 503	. . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)(B e. RR /\ C e. RR /\ B < C)) -> [_M / k]_B < [_M / k]_C)
33		fsum1s 8269	. . . . . 6 |- ((M e. ZZ /\ A.k e. (M...M)B e. RR) -> sum_k e. (M...M)B = [_M / k]_B)
34		simp1 876	. . . . . . 7 |- ((B e. RR /\ C e. RR /\ B < C) -> B e. RR)
35	34	ralimi 2168	. . . . . 6 |- (A.k e. (M...M)(B e. RR /\ C e. RR /\ B < C) -> A.k e. (M...M)B e. RR)
36	33, 35	sylan2 500	. . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)(B e. RR /\ C e. RR /\ B < C)) -> sum_k e. (M...M)B = [_M / k]_B)
37		fsum1s 8269	. . . . . 6 |- ((M e. ZZ /\ A.k e. (M...M)C e. RR) -> sum_k e. (M...M)C = [_M / k]_C)
38		simp2 877	. . . . . . 7 |- ((B e. RR /\ C e. RR /\ B < C) -> C e. RR)
39	38	ralimi 2168	. . . . . 6 |- (A.k e. (M...M)(B e. RR /\ C e. RR /\ B < C) -> A.k e. (M...M)C e. RR)
40	37, 39	sylan2 500	. . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)(B e. RR /\ C e. RR /\ B < C)) -> sum_k e. (M...M)C = [_M / k]_C)
41	32, 36, 40	3brtr4d 3367	. . . 4 |- ((M e. ZZ /\ A.k e. (M...M)(B e. RR /\ C e. RR /\ B < C)) -> sum_k e. (M...M)B < sum_k e. (M...M)C)
42	41	ex 402	. . 3 |- (M e. ZZ -> (A.k e. (M...M)(B e. RR /\ C e. RR /\ B < C) -> sum_k e. (M...M)B < sum_k e. (M...M)C))
43		eluzel2 7593	. . . . . . 7 |- (j e. (ZZ>=` M) -> M e. ZZ)
44		eluzelz 7592	. . . . . . 7 |- (j e. (ZZ>=` M) -> j e. ZZ)
45		fzssp1 7679	. . . . . . 7 |- ((M e. ZZ /\ j e. ZZ) -> (M...j) C_ (M...(j + 1)))
46	43, 44, 45	syl11anc 524	. . . . . 6 |- (j e. (ZZ>=` M) -> (M...j) C_ (M...(j + 1)))
47		ssralv 2672	. . . . . 6 |- ((M...j) C_ (M...(j + 1)) -> (A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C) -> A.k e. (M...j)(B e. RR /\ C e. RR /\ B < C)))
48	46, 47	syl 12	. . . . 5 |- (j e. (ZZ>=` M) -> (A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C) -> A.k e. (M...j)(B e. RR /\ C e. RR /\ B < C)))
49	48	imim1d 33	. . . 4 |- (j e. (ZZ>=` M) -> ((A.k e. (M...j)(B e. RR /\ C e. RR /\ B < C) -> sum_k e. (M...j)B < sum_k e. (M...j)C) -> (A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C) -> sum_k e. (M...j)B < sum_k e. (M...j)C)))
50		ra4sbca 2537	. . . . . . . . . . 11 |- (((j + 1) e. (M...(j + 1)) /\ A.k e. (M...(j + 1))B < C) -> [(j + 1) / k]B < C)
51		sbcbr12g 3392	. . . . . . . . . . . 12 |- ((j + 1) e. (M...(j + 1)) -> ([(j + 1) / k]B < C <-> [_(j + 1) / k]_B < [_(j + 1) / k]_C))
52	51	adantr 425	. . . . . . . . . . 11 |- (((j + 1) e. (M...(j + 1)) /\ A.k e. (M...(j + 1))B < C) -> ([(j + 1) / k]B < C <-> [_(j + 1) / k]_B < [_(j + 1) / k]_C))
53	50, 52	mpbid 212	. . . . . . . . . 10 |- (((j + 1) e. (M...(j + 1)) /\ A.k e. (M...(j + 1))B < C) -> [_(j + 1) / k]_B < [_(j + 1) / k]_C)
54		peano2uz 7616	. . . . . . . . . . 11 |- (j e. (ZZ>=` M) -> (j + 1) e. (ZZ>=` M))
55		eluzfz2 7659	. . . . . . . . . . 11 |- ((j + 1) e. (ZZ>=` M) -> (j + 1) e. (M...(j + 1)))
56	54, 55	syl 12	. . . . . . . . . 10 |- (j e. (ZZ>=` M) -> (j + 1) e. (M...(j + 1)))
57	30	ralimi 2168	. . . . . . . . . 10 |- (A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C) -> A.k e. (M...(j + 1))B < C)
58	53, 56, 57	syl2an 503	. . . . . . . . 9 |- ((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C)) -> [_(j + 1) / k]_B < [_(j + 1) / k]_C)
59	58	adantr 425	. . . . . . . 8 |- (((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C)) /\ sum_k e. (M...j)B < sum_k e. (M...j)C) -> [_(j + 1) / k]_B < [_(j + 1) / k]_C)
60	34	ralimi 2168	. . . . . . . . . . . . 13 |- (A.k e. (M...j)(B e. RR /\ C e. RR /\ B < C) -> A.k e. (M...j)B e. RR)
61	48, 60	syl6 25	. . . . . . . . . . . 12 |- (j e. (ZZ>=` M) -> (A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C) -> A.k e. (M...j)B e. RR))
62	61	imdistani 491	. . . . . . . . . . 11 |- ((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C)) -> (j e. (ZZ>=` M) /\ A.k e. (M...j)B e. RR))
63		fsumrecl 8277	. . . . . . . . . . 11 |- ((j e. (ZZ>=` M) /\ A.k e. (M...j)B e. RR) -> sum_k e. (M...j)B e. RR)
64	62, 63	syl 12	. . . . . . . . . 10 |- ((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C)) -> sum_k e. (M...j)B e. RR)
65		ra4csbela 2587	. . . . . . . . . . 11 |- (((j + 1) e. (M...(j + 1)) /\ A.k e. (M...(j + 1))B e. RR) -> [_(j + 1) / k]_B e. RR)
66	34	ralimi 2168	. . . . . . . . . . 11 |- (A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C) -> A.k e. (M...(j + 1))B e. RR)
67	65, 56, 66	syl2an 503	. . . . . . . . . 10 |- ((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C)) -> [_(j + 1) / k]_B e. RR)
68	38	ralimi 2168	. . . . . . . . . . . . 13 |- (A.k e. (M...j)(B e. RR /\ C e. RR /\ B < C) -> A.k e. (M...j)C e. RR)
69	48, 68	syl6 25	. . . . . . . . . . . 12 |- (j e. (ZZ>=` M) -> (A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C) -> A.k e. (M...j)C e. RR))
70	69	imdistani 491	. . . . . . . . . . 11 |- ((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C)) -> (j e. (ZZ>=` M) /\ A.k e. (M...j)C e. RR))
71		fsumrecl 8277	. . . . . . . . . . 11 |- ((j e. (ZZ>=` M) /\ A.k e. (M...j)C e. RR) -> sum_k e. (M...j)C e. RR)
72	70, 71	syl 12	. . . . . . . . . 10 |- ((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C)) -> sum_k e. (M...j)C e. RR)
73		ra4csbela 2587	. . . . . . . . . . 11 |- (((j + 1) e. (M...(j + 1)) /\ A.k e. (M...(j + 1))C e. RR) -> [_(j + 1) / k]_C e. RR)
74	38	ralimi 2168	. . . . . . . . . . 11 |- (A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C) -> A.k e. (M...(j + 1))C e. RR)
75	73, 56, 74	syl2an 503	. . . . . . . . . 10 |- ((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C)) -> [_(j + 1) / k]_C e. RR)
76		lt2add 6827	. . . . . . . . . 10 |- (((sum_k e. (M...j)B e. RR /\ [_(j + 1) / k]_B e. RR) /\ (sum_k e. (M...j)C e. RR /\ [_(j + 1) / k]_C e. RR)) -> ((sum_k e. (M...j)B < sum_k e. (M...j)C /\ [_(j + 1) / k]_B < [_(j + 1) / k]_C) -> (sum_k e. (M...j)B + [_(j + 1) / k]_B) < (sum_k e. (M...j)C + [_(j + 1) / k]_C)))
77	64, 67, 72, 75, 76	syl22anc 1101	. . . . . . . . 9 |- ((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C)) -> ((sum_k e. (M...j)B < sum_k e. (M...j)C /\ [_(j + 1) / k]_B < [_(j + 1) / k]_C) -> (sum_k e. (M...j)B + [_(j + 1) / k]_B) < (sum_k e. (M...j)C + [_(j + 1) / k]_C)))
78	77	expdimp 406	. . . . . . . 8 |- (((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C)) /\ sum_k e. (M...j)B < sum_k e. (M...j)C) -> ([_(j + 1) / k]_B < [_(j + 1) / k]_C -> (sum_k e. (M...j)B + [_(j + 1) / k]_B) < (sum_k e. (M...j)C + [_(j + 1) / k]_C)))
79	59, 78	mpd 29	. . . . . . 7 |- (((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C)) /\ sum_k e. (M...j)B < sum_k e. (M...j)C) -> (sum_k e. (M...j)B + [_(j + 1) / k]_B) < (sum_k e. (M...j)C + [_(j + 1) / k]_C))
80		fsump1s 8273	. . . . . . . . 9 |- ((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))B e. RR) -> sum_k e. (M...(j + 1))B = (sum_k e. (M...j)B + [_(j + 1) / k]_B))
81	80, 66	sylan2 500	. . . . . . . 8 |- ((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C)) -> sum_k e. (M...(j + 1))B = (sum_k e. (M...j)B + [_(j + 1) / k]_B))
82	81	adantr 425	. . . . . . 7 |- (((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C)) /\ sum_k e. (M...j)B < sum_k e. (M...j)C) -> sum_k e. (M...(j + 1))B = (sum_k e. (M...j)B + [_(j + 1) / k]_B))
83		fsump1s 8273	. . . . . . . . 9 |- ((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))C e. RR) -> sum_k e. (M...(j + 1))C = (sum_k e. (M...j)C + [_(j + 1) / k]_C))
84	83, 74	sylan2 500	. . . . . . . 8 |- ((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C)) -> sum_k e. (M...(j + 1))C = (sum_k e. (M...j)C + [_(j + 1) / k]_C))
85	84	adantr 425	. . . . . . 7 |- (((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C)) /\ sum_k e. (M...j)B < sum_k e. (M...j)C) -> sum_k e. (M...(j + 1))C = (sum_k e. (M...j)C + [_(j + 1) / k]_C))
86	79, 82, 85	3brtr4d 3367	. . . . . 6 |- (((j e. (ZZ>=` M) /\ A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C)) /\ sum_k e. (M...j)B < sum_k e. (M...j)C) -> sum_k e. (M...(j + 1))B < sum_k e. (M...(j + 1))C)
87	86	exp31 407	. . . . 5 |- (j e. (ZZ>=` M) -> (A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C) -> (sum_k e. (M...j)B < sum_k e. (M...j)C -> sum_k e. (M...(j + 1))B < sum_k e. (M...(j + 1))C)))
88	87	a2d 16	. . . 4 |- (j e. (ZZ>=` M) -> ((A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C) -> sum_k e. (M...j)B < sum_k e. (M...j)C) -> (A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C) -> sum_k e. (M...(j + 1))B < sum_k e. (M...(j + 1))C)))
89	49, 88	syld 30	. . 3 |- (j e. (ZZ>=` M) -> ((A.k e. (M...j)(B e. RR /\ C e. RR /\ B < C) -> sum_k e. (M...j)B < sum_k e. (M...j)C) -> (A.k e. (M...(j + 1))(B e. RR /\ C e. RR /\ B < C) -> sum_k e. (M...(j + 1))B < sum_k e. (M...(j + 1))C)))
90	6, 12, 18, 24, 42, 89	uzind4 7619	. 2 |- (N e. (ZZ>=` M) -> (A.k e. (M...N)(B e. RR /\ C e. RR /\ B < C) -> sum_k e. (M...N)B < sum_k e. (M...N)C))
91	90	imp 377	1 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(B e. RR /\ C e. RR /\ B < C)) -> sum_k e. (M...N)B < sum_k e. (M...N)C)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  [_csb 2540   C_ wss 2593   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  RRcr 6385  1c1 6387   + caddc 6389  ZZcz 6451   < clt 6653  ZZ>=cuz 7586  ...cfz 7637  sum_csu 8239

This theorem is referenced by:  rrndstprj2 16018

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-n 7108  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-sum 8240

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