Theorem fsumcmp 8300

Description: If all of the terms of finite sums compare, so do the sums.

Assertion

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

Distinct variable groups:   k,M   k,N

Proof of Theorem fsumcmp

Step	Hyp	Ref	Expression
1		ra4sbca 2537	. . . . . . 7 |- ((M e. (M...M) /\ A.k e. (M...M)A <_ B) -> [M / k]A <_ B)
2		elfz3 7661	. . . . . . 7 |- (M e. ZZ -> M e. (M...M))
3		simp3 878	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ A <_ B) -> A <_ B)
4	3	ralimi 2168	. . . . . . 7 |- (A.k e. (M...M)(A e. RR /\ B e. RR /\ A <_ B) -> A.k e. (M...M)A <_ B)
5	1, 2, 4	syl2an 503	. . . . . 6 |- ((M e. ZZ /\ A.k e. (M...M)(A e. RR /\ B e. RR /\ A <_ B)) -> [M / k]A <_ B)
6		sbcbr12g 3392	. . . . . . 7 |- (M e. ZZ -> ([M / k]A <_ B <-> [_M / k]_A <_ [_M / k]_B))
7	6	adantr 425	. . . . . 6 |- ((M e. ZZ /\ A.k e. (M...M)(A e. RR /\ B e. RR /\ A <_ B)) -> ([M / k]A <_ B <-> [_M / k]_A <_ [_M / k]_B))
8	5, 7	mpbid 212	. . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)(A e. RR /\ B e. RR /\ A <_ B)) -> [_M / k]_A <_ [_M / k]_B)
9		fsum1s 8269	. . . . . 6 |- ((M e. ZZ /\ A.k e. (M...M)A e. RR) -> sum_k e. (M...M)A = [_M / k]_A)
10		simp1 876	. . . . . . 7 |- ((A e. RR /\ B e. RR /\ A <_ B) -> A e. RR)
11	10	ralimi 2168	. . . . . 6 |- (A.k e. (M...M)(A e. RR /\ B e. RR /\ A <_ B) -> A.k e. (M...M)A e. RR)
12	9, 11	sylan2 500	. . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)(A e. RR /\ B e. RR /\ A <_ B)) -> sum_k e. (M...M)A = [_M / k]_A)
13		fsum1s 8269	. . . . . 6 |- ((M e. ZZ /\ A.k e. (M...M)B e. RR) -> sum_k e. (M...M)B = [_M / k]_B)
14		simp2 877	. . . . . . 7 |- ((A e. RR /\ B e. RR /\ A <_ B) -> B e. RR)
15	14	ralimi 2168	. . . . . 6 |- (A.k e. (M...M)(A e. RR /\ B e. RR /\ A <_ B) -> A.k e. (M...M)B e. RR)
16	13, 15	sylan2 500	. . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)(A e. RR /\ B e. RR /\ A <_ B)) -> sum_k e. (M...M)B = [_M / k]_B)
17	8, 12, 16	3brtr4d 3367	. . . 4 |- ((M e. ZZ /\ A.k e. (M...M)(A e. RR /\ B e. RR /\ A <_ B)) -> sum_k e. (M...M)A <_ sum_k e. (M...M)B)
18	17	ex 402	. . 3 |- (M e. ZZ -> (A.k e. (M...M)(A e. RR /\ B e. RR /\ A <_ B) -> sum_k e. (M...M)A <_ sum_k e. (M...M)B))
19		eluzel2 7593	. . . . . . . . 9 |- (m e. (ZZ>=` M) -> M e. ZZ)
20		eluzelz 7592	. . . . . . . . 9 |- (m e. (ZZ>=` M) -> m e. ZZ)
21		fzssp1 7679	. . . . . . . . 9 |- ((M e. ZZ /\ m e. ZZ) -> (M...m) C_ (M...(m + 1)))
22	19, 20, 21	syl11anc 524	. . . . . . . 8 |- (m e. (ZZ>=` M) -> (M...m) C_ (M...(m + 1)))
23	22	sseld 2619	. . . . . . 7 |- (m e. (ZZ>=` M) -> (k e. (M...m) -> k e. (M...(m + 1))))
24	23	imim1d 33	. . . . . 6 |- (m e. (ZZ>=` M) -> ((k e. (M...(m + 1)) -> (A e. RR /\ B e. RR /\ A <_ B)) -> (k e. (M...m) -> (A e. RR /\ B e. RR /\ A <_ B))))
25	24	ralimdv2 2173	. . . . 5 |- (m e. (ZZ>=` M) -> (A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B) -> A.k e. (M...m)(A e. RR /\ B e. RR /\ A <_ B)))
26	25	imim1d 33	. . . 4 |- (m e. (ZZ>=` M) -> ((A.k e. (M...m)(A e. RR /\ B e. RR /\ A <_ B) -> sum_k e. (M...m)A <_ sum_k e. (M...m)B) -> (A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B) -> sum_k e. (M...m)A <_ sum_k e. (M...m)B)))
27		fsump1s 8273	. . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. RR) -> sum_k e. (M...(m + 1))A = (sum_k e. (M...m)A + [_(m + 1) / k]_A))
28	10	ralimi 2168	. . . . . . . . . 10 |- (A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B) -> A.k e. (M...(m + 1))A e. RR)
29	27, 28	sylan2 500	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) -> sum_k e. (M...(m + 1))A = (sum_k e. (M...m)A + [_(m + 1) / k]_A))
30	29	adantr 425	. . . . . . . 8 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) /\ sum_k e. (M...m)A <_ sum_k e. (M...m)B) -> sum_k e. (M...(m + 1))A = (sum_k e. (M...m)A + [_(m + 1) / k]_A))
31		simpr 350	. . . . . . . . 9 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) /\ sum_k e. (M...m)A <_ sum_k e. (M...m)B) -> sum_k e. (M...m)A <_ sum_k e. (M...m)B)
32	10	a1i 8	. . . . . . . . . . . . . . 15 |- (m e. (ZZ>=` M) -> ((A e. RR /\ B e. RR /\ A <_ B) -> A e. RR))
33	23, 32	imim12d 69	. . . . . . . . . . . . . 14 |- (m e. (ZZ>=` M) -> ((k e. (M...(m + 1)) -> (A e. RR /\ B e. RR /\ A <_ B)) -> (k e. (M...m) -> A e. RR)))
34	33	ralimdv2 2173	. . . . . . . . . . . . 13 |- (m e. (ZZ>=` M) -> (A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B) -> A.k e. (M...m)A e. RR))
35	34	imp 377	. . . . . . . . . . . 12 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) -> A.k e. (M...m)A e. RR)
36		fsumrecl 8277	. . . . . . . . . . . 12 |- ((m e. (ZZ>=` M) /\ A.k e. (M...m)A e. RR) -> sum_k e. (M...m)A e. RR)
37	35, 36	syldan 516	. . . . . . . . . . 11 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) -> sum_k e. (M...m)A e. RR)
38	37	adantr 425	. . . . . . . . . 10 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) /\ sum_k e. (M...m)A <_ sum_k e. (M...m)B) -> sum_k e. (M...m)A e. RR)
39	14	a1i 8	. . . . . . . . . . . . . . 15 |- (m e. (ZZ>=` M) -> ((A e. RR /\ B e. RR /\ A <_ B) -> B e. RR))
40	23, 39	imim12d 69	. . . . . . . . . . . . . 14 |- (m e. (ZZ>=` M) -> ((k e. (M...(m + 1)) -> (A e. RR /\ B e. RR /\ A <_ B)) -> (k e. (M...m) -> B e. RR)))
41	40	ralimdv2 2173	. . . . . . . . . . . . 13 |- (m e. (ZZ>=` M) -> (A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B) -> A.k e. (M...m)B e. RR))
42	41	imp 377	. . . . . . . . . . . 12 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) -> A.k e. (M...m)B e. RR)
43		fsumrecl 8277	. . . . . . . . . . . 12 |- ((m e. (ZZ>=` M) /\ A.k e. (M...m)B e. RR) -> sum_k e. (M...m)B e. RR)
44	42, 43	syldan 516	. . . . . . . . . . 11 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) -> sum_k e. (M...m)B e. RR)
45	44	adantr 425	. . . . . . . . . 10 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) /\ sum_k e. (M...m)A <_ sum_k e. (M...m)B) -> sum_k e. (M...m)B e. RR)
46		ra4csbela 2587	. . . . . . . . . . . 12 |- (((m + 1) e. (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. RR) -> [_(m + 1) / k]_A e. RR)
47		peano2uz 7616	. . . . . . . . . . . . 13 |- (m e. (ZZ>=` M) -> (m + 1) e. (ZZ>=` M))
48		eluzfz2 7659	. . . . . . . . . . . . 13 |- ((m + 1) e. (ZZ>=` M) -> (m + 1) e. (M...(m + 1)))
49	47, 48	syl 12	. . . . . . . . . . . 12 |- (m e. (ZZ>=` M) -> (m + 1) e. (M...(m + 1)))
50	46, 49, 28	syl2an 503	. . . . . . . . . . 11 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) -> [_(m + 1) / k]_A e. RR)
51	50	adantr 425	. . . . . . . . . 10 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) /\ sum_k e. (M...m)A <_ sum_k e. (M...m)B) -> [_(m + 1) / k]_A e. RR)
52		leadd1 6808	. . . . . . . . . 10 |- ((sum_k e. (M...m)A e. RR /\ sum_k e. (M...m)B e. RR /\ [_(m + 1) / k]_A e. RR) -> (sum_k e. (M...m)A <_ sum_k e. (M...m)B <-> (sum_k e. (M...m)A + [_(m + 1) / k]_A) <_ (sum_k e. (M...m)B + [_(m + 1) / k]_A)))
53	38, 45, 51, 52	syl111anc 1100	. . . . . . . . 9 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) /\ sum_k e. (M...m)A <_ sum_k e. (M...m)B) -> (sum_k e. (M...m)A <_ sum_k e. (M...m)B <-> (sum_k e. (M...m)A + [_(m + 1) / k]_A) <_ (sum_k e. (M...m)B + [_(m + 1) / k]_A)))
54	31, 53	mpbid 212	. . . . . . . 8 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) /\ sum_k e. (M...m)A <_ sum_k e. (M...m)B) -> (sum_k e. (M...m)A + [_(m + 1) / k]_A) <_ (sum_k e. (M...m)B + [_(m + 1) / k]_A))
55	30, 54	eqbrtrd 3357	. . . . . . 7 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) /\ sum_k e. (M...m)A <_ sum_k e. (M...m)B) -> sum_k e. (M...(m + 1))A <_ (sum_k e. (M...m)B + [_(m + 1) / k]_A))
56		ra4sbca 2537	. . . . . . . . . . . 12 |- (((m + 1) e. (M...(m + 1)) /\ A.k e. (M...(m + 1))A <_ B) -> [(m + 1) / k]A <_ B)
57		oprex 4907	. . . . . . . . . . . . 13 |- (m + 1) e. _V
58		sbcbr12g 3392	. . . . . . . . . . . . 13 |- ((m + 1) e. _V -> ([(m + 1) / k]A <_ B <-> [_(m + 1) / k]_A <_ [_(m + 1) / k]_B))
59	57, 58	ax-mp 7	. . . . . . . . . . . 12 |- ([(m + 1) / k]A <_ B <-> [_(m + 1) / k]_A <_ [_(m + 1) / k]_B)
60	56, 59	sylib 215	. . . . . . . . . . 11 |- (((m + 1) e. (M...(m + 1)) /\ A.k e. (M...(m + 1))A <_ B) -> [_(m + 1) / k]_A <_ [_(m + 1) / k]_B)
61	3	ralimi 2168	. . . . . . . . . . 11 |- (A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B) -> A.k e. (M...(m + 1))A <_ B)
62	60, 49, 61	syl2an 503	. . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) -> [_(m + 1) / k]_A <_ [_(m + 1) / k]_B)
63		ra4csbela 2587	. . . . . . . . . . . 12 |- (((m + 1) e. (M...(m + 1)) /\ A.k e. (M...(m + 1))B e. RR) -> [_(m + 1) / k]_B e. RR)
64	14	ralimi 2168	. . . . . . . . . . . 12 |- (A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B) -> A.k e. (M...(m + 1))B e. RR)
65	63, 49, 64	syl2an 503	. . . . . . . . . . 11 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) -> [_(m + 1) / k]_B e. RR)
66		leadd2 6809	. . . . . . . . . . 11 |- (([_(m + 1) / k]_A e. RR /\ [_(m + 1) / k]_B e. RR /\ sum_k e. (M...m)B e. RR) -> ([_(m + 1) / k]_A <_ [_(m + 1) / k]_B <-> (sum_k e. (M...m)B + [_(m + 1) / k]_A) <_ (sum_k e. (M...m)B + [_(m + 1) / k]_B)))
67	50, 65, 44, 66	syl111anc 1100	. . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) -> ([_(m + 1) / k]_A <_ [_(m + 1) / k]_B <-> (sum_k e. (M...m)B + [_(m + 1) / k]_A) <_ (sum_k e. (M...m)B + [_(m + 1) / k]_B)))
68	62, 67	mpbid 212	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) -> (sum_k e. (M...m)B + [_(m + 1) / k]_A) <_ (sum_k e. (M...m)B + [_(m + 1) / k]_B))
69		fsump1s 8273	. . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))B e. RR) -> sum_k e. (M...(m + 1))B = (sum_k e. (M...m)B + [_(m + 1) / k]_B))
70	69, 64	sylan2 500	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) -> sum_k e. (M...(m + 1))B = (sum_k e. (M...m)B + [_(m + 1) / k]_B))
71	68, 70	breqtrrd 3363	. . . . . . . 8 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) -> (sum_k e. (M...m)B + [_(m + 1) / k]_A) <_ sum_k e. (M...(m + 1))B)
72	71	adantr 425	. . . . . . 7 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) /\ sum_k e. (M...m)A <_ sum_k e. (M...m)B) -> (sum_k e. (M...m)B + [_(m + 1) / k]_A) <_ sum_k e. (M...(m + 1))B)
73		fsumrecl 8277	. . . . . . . . . 10 |- (((m + 1) e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. RR) -> sum_k e. (M...(m + 1))A e. RR)
74	73, 47, 28	syl2an 503	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) -> sum_k e. (M...(m + 1))A e. RR)
75		readdcl 6455	. . . . . . . . . 10 |- ((sum_k e. (M...m)B e. RR /\ [_(m + 1) / k]_A e. RR) -> (sum_k e. (M...m)B + [_(m + 1) / k]_A) e. RR)
76	44, 50, 75	syl11anc 524	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) -> (sum_k e. (M...m)B + [_(m + 1) / k]_A) e. RR)
77		fsumrecl 8277	. . . . . . . . . 10 |- (((m + 1) e. (ZZ>=` M) /\ A.k e. (M...(m + 1))B e. RR) -> sum_k e. (M...(m + 1))B e. RR)
78	77, 47, 64	syl2an 503	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) -> sum_k e. (M...(m + 1))B e. RR)
79		letr 6695	. . . . . . . . 9 |- ((sum_k e. (M...(m + 1))A e. RR /\ (sum_k e. (M...m)B + [_(m + 1) / k]_A) e. RR /\ sum_k e. (M...(m + 1))B e. RR) -> ((sum_k e. (M...(m + 1))A <_ (sum_k e. (M...m)B + [_(m + 1) / k]_A) /\ (sum_k e. (M...m)B + [_(m + 1) / k]_A) <_ sum_k e. (M...(m + 1))B) -> sum_k e. (M...(m + 1))A <_ sum_k e. (M...(m + 1))B))
80	74, 76, 78, 79	syl111anc 1100	. . . . . . . 8 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) -> ((sum_k e. (M...(m + 1))A <_ (sum_k e. (M...m)B + [_(m + 1) / k]_A) /\ (sum_k e. (M...m)B + [_(m + 1) / k]_A) <_ sum_k e. (M...(m + 1))B) -> sum_k e. (M...(m + 1))A <_ sum_k e. (M...(m + 1))B))
81	80	adantr 425	. . . . . . 7 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) /\ sum_k e. (M...m)A <_ sum_k e. (M...m)B) -> ((sum_k e. (M...(m + 1))A <_ (sum_k e. (M...m)B + [_(m + 1) / k]_A) /\ (sum_k e. (M...m)B + [_(m + 1) / k]_A) <_ sum_k e. (M...(m + 1))B) -> sum_k e. (M...(m + 1))A <_ sum_k e. (M...(m + 1))B))
82	55, 72, 81	mp2and 767	. . . . . 6 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)) /\ sum_k e. (M...m)A <_ sum_k e. (M...m)B) -> sum_k e. (M...(m + 1))A <_ sum_k e. (M...(m + 1))B)
83	82	exp31 407	. . . . 5 |- (m e. (ZZ>=` M) -> (A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B) -> (sum_k e. (M...m)A <_ sum_k e. (M...m)B -> sum_k e. (M...(m + 1))A <_ sum_k e. (M...(m + 1))B)))
84	83	a2d 16	. . . 4 |- (m e. (ZZ>=` M) -> ((A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B) -> sum_k e. (M...m)A <_ sum_k e. (M...m)B) -> (A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B) -> sum_k e. (M...(m + 1))A <_ sum_k e. (M...(m + 1))B)))
85	26, 84	syld 30	. . 3 |- (m e. (ZZ>=` M) -> ((A.k e. (M...m)(A e. RR /\ B e. RR /\ A <_ B) -> sum_k e. (M...m)A <_ sum_k e. (M...m)B) -> (A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B) -> sum_k e. (M...(m + 1))A <_ sum_k e. (M...(m + 1))B)))
86		opreq2 4890	. . . . 5 |- (j = M -> (M...j) = (M...M))
87	86	raleqdv 2269	. . . 4 |- (j = M -> (A.k e. (M...j)(A e. RR /\ B e. RR /\ A <_ B) <-> A.k e. (M...M)(A e. RR /\ B e. RR /\ A <_ B)))
88	86	sumeq1d 8250	. . . . 5 |- (j = M -> sum_k e. (M...j)A = sum_k e. (M...M)A)
89	86	sumeq1d 8250	. . . . 5 |- (j = M -> sum_k e. (M...j)B = sum_k e. (M...M)B)
90	88, 89	breq12d 3351	. . . 4 |- (j = M -> (sum_k e. (M...j)A <_ sum_k e. (M...j)B <-> sum_k e. (M...M)A <_ sum_k e. (M...M)B))
91	87, 90	imbi12d 688	. . 3 |- (j = M -> ((A.k e. (M...j)(A e. RR /\ B e. RR /\ A <_ B) -> sum_k e. (M...j)A <_ sum_k e. (M...j)B) <-> (A.k e. (M...M)(A e. RR /\ B e. RR /\ A <_ B) -> sum_k e. (M...M)A <_ sum_k e. (M...M)B)))
92		opreq2 4890	. . . . 5 |- (j = m -> (M...j) = (M...m))
93	92	raleqdv 2269	. . . 4 |- (j = m -> (A.k e. (M...j)(A e. RR /\ B e. RR /\ A <_ B) <-> A.k e. (M...m)(A e. RR /\ B e. RR /\ A <_ B)))
94	92	sumeq1d 8250	. . . . 5 |- (j = m -> sum_k e. (M...j)A = sum_k e. (M...m)A)
95	92	sumeq1d 8250	. . . . 5 |- (j = m -> sum_k e. (M...j)B = sum_k e. (M...m)B)
96	94, 95	breq12d 3351	. . . 4 |- (j = m -> (sum_k e. (M...j)A <_ sum_k e. (M...j)B <-> sum_k e. (M...m)A <_ sum_k e. (M...m)B))
97	93, 96	imbi12d 688	. . 3 |- (j = m -> ((A.k e. (M...j)(A e. RR /\ B e. RR /\ A <_ B) -> sum_k e. (M...j)A <_ sum_k e. (M...j)B) <-> (A.k e. (M...m)(A e. RR /\ B e. RR /\ A <_ B) -> sum_k e. (M...m)A <_ sum_k e. (M...m)B)))
98		opreq2 4890	. . . . 5 |- (j = (m + 1) -> (M...j) = (M...(m + 1)))
99	98	raleqdv 2269	. . . 4 |- (j = (m + 1) -> (A.k e. (M...j)(A e. RR /\ B e. RR /\ A <_ B) <-> A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B)))
100	98	sumeq1d 8250	. . . . 5 |- (j = (m + 1) -> sum_k e. (M...j)A = sum_k e. (M...(m + 1))A)
101	98	sumeq1d 8250	. . . . 5 |- (j = (m + 1) -> sum_k e. (M...j)B = sum_k e. (M...(m + 1))B)
102	100, 101	breq12d 3351	. . . 4 |- (j = (m + 1) -> (sum_k e. (M...j)A <_ sum_k e. (M...j)B <-> sum_k e. (M...(m + 1))A <_ sum_k e. (M...(m + 1))B))
103	99, 102	imbi12d 688	. . 3 |- (j = (m + 1) -> ((A.k e. (M...j)(A e. RR /\ B e. RR /\ A <_ B) -> sum_k e. (M...j)A <_ sum_k e. (M...j)B) <-> (A.k e. (M...(m + 1))(A e. RR /\ B e. RR /\ A <_ B) -> sum_k e. (M...(m + 1))A <_ sum_k e. (M...(m + 1))B)))
104		opreq2 4890	. . . . 5 |- (j = N -> (M...j) = (M...N))
105	104	raleqdv 2269	. . . 4 |- (j = N -> (A.k e. (M...j)(A e. RR /\ B e. RR /\ A <_ B) <-> A.k e. (M...N)(A e. RR /\ B e. RR /\ A <_ B)))
106	104	sumeq1d 8250	. . . . 5 |- (j = N -> sum_k e. (M...j)A = sum_k e. (M...N)A)
107	104	sumeq1d 8250	. . . . 5 |- (j = N -> sum_k e. (M...j)B = sum_k e. (M...N)B)
108	106, 107	breq12d 3351	. . . 4 |- (j = N -> (sum_k e. (M...j)A <_ sum_k e. (M...j)B <-> sum_k e. (M...N)A <_ sum_k e. (M...N)B))
109	105, 108	imbi12d 688	. . 3 |- (j = N -> ((A.k e. (M...j)(A e. RR /\ B e. RR /\ A <_ B) -> sum_k e. (M...j)A <_ sum_k e. (M...j)B) <-> (A.k e. (M...N)(A e. RR /\ B e. RR /\ A <_ B) -> sum_k e. (M...N)A <_ sum_k e. (M...N)B)))
110	18, 85, 91, 97, 103, 109	uzind4ALT 7620	. 2 |- (N e. (ZZ>=` M) -> (A.k e. (M...N)(A e. RR /\ B e. RR /\ A <_ B) -> sum_k e. (M...N)A <_ sum_k e. (M...N)B))
111	110	imp 377	1 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ B e. RR /\ A <_ B)) -> sum_k e. (M...N)A <_ sum_k e. (M...N)B)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  _Vcvv 2292  [_csb 2540   C_ wss 2593   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  RRcr 6385  1c1 6387   + caddc 6389   <_ cle 6448  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637  sum_csu 8239

This theorem is referenced by:  fsumcmp0 8301  serzcmp 8314  efaddlem16 8615  efaddlem19 8618  fsumlt1 15831  geomcau 15849

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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