Theorem fsum00 15820

Description: A sum of nonnegative numbers is zero iff all terms are zero.

Assertion

Ref	Expression
fsum00	|- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) -> (sum_k e. (M...N)A = 0 <-> A.k e. (M...N)A = 0))

Distinct variable groups:   k,M   k,N

Proof of Theorem fsum00

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . . 6 |- (m = M -> (M...m) = (M...M))
2	1	raleqdv 2269	. . . . 5 |- (m = M -> (A.k e. (M...m)(A e. RR /\ 0 <_ A) <-> A.k e. (M...M)(A e. RR /\ 0 <_ A)))
3	1	sumeq1d 8250	. . . . . . 7 |- (m = M -> sum_k e. (M...m)A = sum_k e. (M...M)A)
4	3	eqeq1d 1892	. . . . . 6 |- (m = M -> (sum_k e. (M...m)A = 0 <-> sum_k e. (M...M)A = 0))
5	1	raleqdv 2269	. . . . . 6 |- (m = M -> (A.k e. (M...m)A = 0 <-> A.k e. (M...M)A = 0))
6	4, 5	imbi12d 688	. . . . 5 |- (m = M -> ((sum_k e. (M...m)A = 0 -> A.k e. (M...m)A = 0) <-> (sum_k e. (M...M)A = 0 -> A.k e. (M...M)A = 0)))
7	2, 6	imbi12d 688	. . . 4 |- (m = M -> ((A.k e. (M...m)(A e. RR /\ 0 <_ A) -> (sum_k e. (M...m)A = 0 -> A.k e. (M...m)A = 0)) <-> (A.k e. (M...M)(A e. RR /\ 0 <_ A) -> (sum_k e. (M...M)A = 0 -> A.k e. (M...M)A = 0))))
8		opreq2 4890	. . . . . 6 |- (m = n -> (M...m) = (M...n))
9	8	raleqdv 2269	. . . . 5 |- (m = n -> (A.k e. (M...m)(A e. RR /\ 0 <_ A) <-> A.k e. (M...n)(A e. RR /\ 0 <_ A)))
10	8	sumeq1d 8250	. . . . . . 7 |- (m = n -> sum_k e. (M...m)A = sum_k e. (M...n)A)
11	10	eqeq1d 1892	. . . . . 6 |- (m = n -> (sum_k e. (M...m)A = 0 <-> sum_k e. (M...n)A = 0))
12	8	raleqdv 2269	. . . . . 6 |- (m = n -> (A.k e. (M...m)A = 0 <-> A.k e. (M...n)A = 0))
13	11, 12	imbi12d 688	. . . . 5 |- (m = n -> ((sum_k e. (M...m)A = 0 -> A.k e. (M...m)A = 0) <-> (sum_k e. (M...n)A = 0 -> A.k e. (M...n)A = 0)))
14	9, 13	imbi12d 688	. . . 4 |- (m = n -> ((A.k e. (M...m)(A e. RR /\ 0 <_ A) -> (sum_k e. (M...m)A = 0 -> A.k e. (M...m)A = 0)) <-> (A.k e. (M...n)(A e. RR /\ 0 <_ A) -> (sum_k e. (M...n)A = 0 -> A.k e. (M...n)A = 0))))
15		opreq2 4890	. . . . . 6 |- (m = (n + 1) -> (M...m) = (M...(n + 1)))
16	15	raleqdv 2269	. . . . 5 |- (m = (n + 1) -> (A.k e. (M...m)(A e. RR /\ 0 <_ A) <-> A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)))
17	15	sumeq1d 8250	. . . . . . 7 |- (m = (n + 1) -> sum_k e. (M...m)A = sum_k e. (M...(n + 1))A)
18	17	eqeq1d 1892	. . . . . 6 |- (m = (n + 1) -> (sum_k e. (M...m)A = 0 <-> sum_k e. (M...(n + 1))A = 0))
19	15	raleqdv 2269	. . . . . 6 |- (m = (n + 1) -> (A.k e. (M...m)A = 0 <-> A.k e. (M...(n + 1))A = 0))
20	18, 19	imbi12d 688	. . . . 5 |- (m = (n + 1) -> ((sum_k e. (M...m)A = 0 -> A.k e. (M...m)A = 0) <-> (sum_k e. (M...(n + 1))A = 0 -> A.k e. (M...(n + 1))A = 0)))
21	16, 20	imbi12d 688	. . . 4 |- (m = (n + 1) -> ((A.k e. (M...m)(A e. RR /\ 0 <_ A) -> (sum_k e. (M...m)A = 0 -> A.k e. (M...m)A = 0)) <-> (A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A) -> (sum_k e. (M...(n + 1))A = 0 -> A.k e. (M...(n + 1))A = 0))))
22		opreq2 4890	. . . . . 6 |- (m = N -> (M...m) = (M...N))
23	22	raleqdv 2269	. . . . 5 |- (m = N -> (A.k e. (M...m)(A e. RR /\ 0 <_ A) <-> A.k e. (M...N)(A e. RR /\ 0 <_ A)))
24	22	sumeq1d 8250	. . . . . . 7 |- (m = N -> sum_k e. (M...m)A = sum_k e. (M...N)A)
25	24	eqeq1d 1892	. . . . . 6 |- (m = N -> (sum_k e. (M...m)A = 0 <-> sum_k e. (M...N)A = 0))
26	22	raleqdv 2269	. . . . . 6 |- (m = N -> (A.k e. (M...m)A = 0 <-> A.k e. (M...N)A = 0))
27	25, 26	imbi12d 688	. . . . 5 |- (m = N -> ((sum_k e. (M...m)A = 0 -> A.k e. (M...m)A = 0) <-> (sum_k e. (M...N)A = 0 -> A.k e. (M...N)A = 0)))
28	23, 27	imbi12d 688	. . . 4 |- (m = N -> ((A.k e. (M...m)(A e. RR /\ 0 <_ A) -> (sum_k e. (M...m)A = 0 -> A.k e. (M...m)A = 0)) <-> (A.k e. (M...N)(A e. RR /\ 0 <_ A) -> (sum_k e. (M...N)A = 0 -> A.k e. (M...N)A = 0))))
29		fsum1s 8269	. . . . . . . 8 |- ((M e. ZZ /\ A.k e. (M...M)A e. RR) -> sum_k e. (M...M)A = [_M / k]_A)
30	29	eqeq1d 1892	. . . . . . 7 |- ((M e. ZZ /\ A.k e. (M...M)A e. RR) -> (sum_k e. (M...M)A = 0 <-> [_M / k]_A = 0))
31		sbceq1dig 2557	. . . . . . . . . 10 |- (M e. ZZ -> ([M / k]A = 0 <-> [_M / k]_A = 0))
32	31	biimprd 171	. . . . . . . . 9 |- (M e. ZZ -> ([_M / k]_A = 0 -> [M / k]A = 0))
33		fz1sbc 7696	. . . . . . . . 9 |- (M e. ZZ -> (A.k e. (M...M)A = 0 <-> [M / k]A = 0))
34	32, 33	sylibrd 221	. . . . . . . 8 |- (M e. ZZ -> ([_M / k]_A = 0 -> A.k e. (M...M)A = 0))
35	34	adantr 425	. . . . . . 7 |- ((M e. ZZ /\ A.k e. (M...M)A e. RR) -> ([_M / k]_A = 0 -> A.k e. (M...M)A = 0))
36	30, 35	sylbid 220	. . . . . 6 |- ((M e. ZZ /\ A.k e. (M...M)A e. RR) -> (sum_k e. (M...M)A = 0 -> A.k e. (M...M)A = 0))
37		simpl 346	. . . . . . 7 |- ((A e. RR /\ 0 <_ A) -> A e. RR)
38	37	ralimi 2168	. . . . . 6 |- (A.k e. (M...M)(A e. RR /\ 0 <_ A) -> A.k e. (M...M)A e. RR)
39	36, 38	sylan2 500	. . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)(A e. RR /\ 0 <_ A)) -> (sum_k e. (M...M)A = 0 -> A.k e. (M...M)A = 0))
40	39	ex 402	. . . 4 |- (M e. ZZ -> (A.k e. (M...M)(A e. RR /\ 0 <_ A) -> (sum_k e. (M...M)A = 0 -> A.k e. (M...M)A = 0)))
41		eluzel2 7593	. . . . . . . 8 |- (n e. (ZZ>=` M) -> M e. ZZ)
42		eluzelz 7592	. . . . . . . 8 |- (n e. (ZZ>=` M) -> n e. ZZ)
43		fzssp1 7679	. . . . . . . 8 |- ((M e. ZZ /\ n e. ZZ) -> (M...n) C_ (M...(n + 1)))
44	41, 42, 43	syl11anc 524	. . . . . . 7 |- (n e. (ZZ>=` M) -> (M...n) C_ (M...(n + 1)))
45		ssralv 2672	. . . . . . 7 |- ((M...n) C_ (M...(n + 1)) -> (A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A) -> A.k e. (M...n)(A e. RR /\ 0 <_ A)))
46	44, 45	syl 12	. . . . . 6 |- (n e. (ZZ>=` M) -> (A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A) -> A.k e. (M...n)(A e. RR /\ 0 <_ A)))
47	46	imim1d 33	. . . . 5 |- (n e. (ZZ>=` M) -> ((A.k e. (M...n)(A e. RR /\ 0 <_ A) -> (sum_k e. (M...n)A = 0 -> A.k e. (M...n)A = 0)) -> (A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A) -> (sum_k e. (M...n)A = 0 -> A.k e. (M...n)A = 0))))
48		pm3.45 621	. . . . . . . . . 10 |- ((sum_k e. (M...n)A = 0 -> A.k e. (M...n)A = 0) -> ((sum_k e. (M...n)A = 0 /\ [_(n + 1) / k]_A = 0) -> (A.k e. (M...n)A = 0 /\ [_(n + 1) / k]_A = 0)))
49		peano2uz 7616	. . . . . . . . . . . . . . . . 17 |- (n e. (ZZ>=` M) -> (n + 1) e. (ZZ>=` M))
50		eluzle 7594	. . . . . . . . . . . . . . . . 17 |- ((n + 1) e. (ZZ>=` M) -> M <_ (n + 1))
51	49, 50	syl 12	. . . . . . . . . . . . . . . 16 |- (n e. (ZZ>=` M) -> M <_ (n + 1))
52		fzsuc 7678	. . . . . . . . . . . . . . . 16 |- ((M e. ZZ /\ n e. ZZ /\ M <_ (n + 1)) -> (M...(n + 1)) = ((M...n) u. {(n + 1)}))
53	41, 42, 51, 52	syl111anc 1100	. . . . . . . . . . . . . . 15 |- (n e. (ZZ>=` M) -> (M...(n + 1)) = ((M...n) u. {(n + 1)}))
54		peano2z 7375	. . . . . . . . . . . . . . . . . 18 |- (n e. ZZ -> (n + 1) e. ZZ)
55		fzsn 7684	. . . . . . . . . . . . . . . . . 18 |- ((n + 1) e. ZZ -> ((n + 1)...(n + 1)) = {(n + 1)})
56	42, 54, 55	3syl 24	. . . . . . . . . . . . . . . . 17 |- (n e. (ZZ>=` M) -> ((n + 1)...(n + 1)) = {(n + 1)})
57	56	eqcomd 1889	. . . . . . . . . . . . . . . 16 |- (n e. (ZZ>=` M) -> {(n + 1)} = ((n + 1)...(n + 1)))
58	57	uneq2d 2755	. . . . . . . . . . . . . . 15 |- (n e. (ZZ>=` M) -> ((M...n) u. {(n + 1)}) = ((M...n) u. ((n + 1)...(n + 1))))
59	53, 58	eqtrd 1925	. . . . . . . . . . . . . 14 |- (n e. (ZZ>=` M) -> (M...(n + 1)) = ((M...n) u. ((n + 1)...(n + 1))))
60	59	raleqdv 2269	. . . . . . . . . . . . 13 |- (n e. (ZZ>=` M) -> (A.k e. (M...(n + 1))A = 0 <-> A.k e. ((M...n) u. ((n + 1)...(n + 1)))A = 0))
61		ralun 2785	. . . . . . . . . . . . 13 |- ((A.k e. (M...n)A = 0 /\ A.k e. ((n + 1)...(n + 1))A = 0) -> A.k e. ((M...n) u. ((n + 1)...(n + 1)))A = 0)
62	60, 61	syl5bir 227	. . . . . . . . . . . 12 |- (n e. (ZZ>=` M) -> ((A.k e. (M...n)A = 0 /\ A.k e. ((n + 1)...(n + 1))A = 0) -> A.k e. (M...(n + 1))A = 0))
63	62	adantr 425	. . . . . . . . . . 11 |- ((n e. (ZZ>=` M) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> ((A.k e. (M...n)A = 0 /\ A.k e. ((n + 1)...(n + 1))A = 0) -> A.k e. (M...(n + 1))A = 0))
64		fz1sbc 7696	. . . . . . . . . . . . . . 15 |- ((n + 1) e. ZZ -> (A.k e. ((n + 1)...(n + 1))A = 0 <-> [(n + 1) / k]A = 0))
65	42, 54, 64	3syl 24	. . . . . . . . . . . . . 14 |- (n e. (ZZ>=` M) -> (A.k e. ((n + 1)...(n + 1))A = 0 <-> [(n + 1) / k]A = 0))
66		oprex 4907	. . . . . . . . . . . . . . 15 |- (n + 1) e. _V
67		sbceq1dig 2557	. . . . . . . . . . . . . . 15 |- ((n + 1) e. _V -> ([(n + 1) / k]A = 0 <-> [_(n + 1) / k]_A = 0))
68	66, 67	ax-mp 7	. . . . . . . . . . . . . 14 |- ([(n + 1) / k]A = 0 <-> [_(n + 1) / k]_A = 0)
69	65, 68	syl6bb 595	. . . . . . . . . . . . 13 |- (n e. (ZZ>=` M) -> (A.k e. ((n + 1)...(n + 1))A = 0 <-> [_(n + 1) / k]_A = 0))
70	69	biimprd 171	. . . . . . . . . . . 12 |- (n e. (ZZ>=` M) -> ([_(n + 1) / k]_A = 0 -> A.k e. ((n + 1)...(n + 1))A = 0))
71	70	adantr 425	. . . . . . . . . . 11 |- ((n e. (ZZ>=` M) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> ([_(n + 1) / k]_A = 0 -> A.k e. ((n + 1)...(n + 1))A = 0))
72	63, 71	sylan2d 507	. . . . . . . . . 10 |- ((n e. (ZZ>=` M) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> ((A.k e. (M...n)A = 0 /\ [_(n + 1) / k]_A = 0) -> A.k e. (M...(n + 1))A = 0))
73	48, 72	syl9r 72	. . . . . . . . 9 |- ((n e. (ZZ>=` M) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> ((sum_k e. (M...n)A = 0 -> A.k e. (M...n)A = 0) -> ((sum_k e. (M...n)A = 0 /\ [_(n + 1) / k]_A = 0) -> A.k e. (M...(n + 1))A = 0)))
74	37	ralimi 2168	. . . . . . . . . . . . . 14 |- (A.k e. (M...n)(A e. RR /\ 0 <_ A) -> A.k e. (M...n)A e. RR)
75	46, 74	syl6 25	. . . . . . . . . . . . 13 |- (n e. (ZZ>=` M) -> (A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A) -> A.k e. (M...n)A e. RR))
76	75	imdistani 491	. . . . . . . . . . . 12 |- ((n e. (ZZ>=` M) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> (n e. (ZZ>=` M) /\ A.k e. (M...n)A e. RR))
77		fsumrecl 8277	. . . . . . . . . . . 12 |- ((n e. (ZZ>=` M) /\ A.k e. (M...n)A e. RR) -> sum_k e. (M...n)A e. RR)
78	76, 77	syl 12	. . . . . . . . . . 11 |- ((n e. (ZZ>=` M) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> sum_k e. (M...n)A e. RR)
79	46	imdistani 491	. . . . . . . . . . . 12 |- ((n e. (ZZ>=` M) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> (n e. (ZZ>=` M) /\ A.k e. (M...n)(A e. RR /\ 0 <_ A)))
80		fsumcmp0 8301	. . . . . . . . . . . 12 |- ((n e. (ZZ>=` M) /\ A.k e. (M...n)(A e. RR /\ 0 <_ A)) -> 0 <_ sum_k e. (M...n)A)
81	79, 80	syl 12	. . . . . . . . . . 11 |- ((n e. (ZZ>=` M) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> 0 <_ sum_k e. (M...n)A)
82		ra4sbca 2537	. . . . . . . . . . . . 13 |- (((n + 1) e. (M...(n + 1)) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> [(n + 1) / k](A e. RR /\ 0 <_ A))
83		sbcang 2497	. . . . . . . . . . . . . . 15 |- ((n + 1) e. _V -> ([(n + 1) / k](A e. RR /\ 0 <_ A) <-> ([(n + 1) / k]A e. RR /\ [(n + 1) / k]0 <_ A)))
84	66, 83	ax-mp 7	. . . . . . . . . . . . . 14 |- ([(n + 1) / k](A e. RR /\ 0 <_ A) <-> ([(n + 1) / k]A e. RR /\ [(n + 1) / k]0 <_ A))
85		sbcel1g 2556	. . . . . . . . . . . . . . . 16 |- ((n + 1) e. _V -> ([(n + 1) / k]A e. RR <-> [_(n + 1) / k]_A e. RR))
86	66, 85	ax-mp 7	. . . . . . . . . . . . . . 15 |- ([(n + 1) / k]A e. RR <-> [_(n + 1) / k]_A e. RR)
87		sbcbr2g 3394	. . . . . . . . . . . . . . . 16 |- ((n + 1) e. _V -> ([(n + 1) / k]0 <_ A <-> 0 <_ [_(n + 1) / k]_A))
88	66, 87	ax-mp 7	. . . . . . . . . . . . . . 15 |- ([(n + 1) / k]0 <_ A <-> 0 <_ [_(n + 1) / k]_A)
89	86, 88	anbi12i 540	. . . . . . . . . . . . . 14 |- (([(n + 1) / k]A e. RR /\ [(n + 1) / k]0 <_ A) <-> ([_(n + 1) / k]_A e. RR /\ 0 <_ [_(n + 1) / k]_A))
90	84, 89	bitri 190	. . . . . . . . . . . . 13 |- ([(n + 1) / k](A e. RR /\ 0 <_ A) <-> ([_(n + 1) / k]_A e. RR /\ 0 <_ [_(n + 1) / k]_A))
91	82, 90	sylib 215	. . . . . . . . . . . 12 |- (((n + 1) e. (M...(n + 1)) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> ([_(n + 1) / k]_A e. RR /\ 0 <_ [_(n + 1) / k]_A))
92		eluzfz2 7659	. . . . . . . . . . . . 13 |- ((n + 1) e. (ZZ>=` M) -> (n + 1) e. (M...(n + 1)))
93	49, 92	syl 12	. . . . . . . . . . . 12 |- (n e. (ZZ>=` M) -> (n + 1) e. (M...(n + 1)))
94	91, 93	sylan 497	. . . . . . . . . . 11 |- ((n e. (ZZ>=` M) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> ([_(n + 1) / k]_A e. RR /\ 0 <_ [_(n + 1) / k]_A))
95		add20 15777	. . . . . . . . . . 11 |- (((sum_k e. (M...n)A e. RR /\ 0 <_ sum_k e. (M...n)A) /\ ([_(n + 1) / k]_A e. RR /\ 0 <_ [_(n + 1) / k]_A)) -> ((sum_k e. (M...n)A + [_(n + 1) / k]_A) = 0 <-> (sum_k e. (M...n)A = 0 /\ [_(n + 1) / k]_A = 0)))
96	78, 81, 94, 95	syl21anc 1099	. . . . . . . . . 10 |- ((n e. (ZZ>=` M) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> ((sum_k e. (M...n)A + [_(n + 1) / k]_A) = 0 <-> (sum_k e. (M...n)A = 0 /\ [_(n + 1) / k]_A = 0)))
97	96	biimpd 170	. . . . . . . . 9 |- ((n e. (ZZ>=` M) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> ((sum_k e. (M...n)A + [_(n + 1) / k]_A) = 0 -> (sum_k e. (M...n)A = 0 /\ [_(n + 1) / k]_A = 0)))
98	73, 97	syl5d 66	. . . . . . . 8 |- ((n e. (ZZ>=` M) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> ((sum_k e. (M...n)A = 0 -> A.k e. (M...n)A = 0) -> ((sum_k e. (M...n)A + [_(n + 1) / k]_A) = 0 -> A.k e. (M...(n + 1))A = 0)))
99		fsump1s 8273	. . . . . . . . . . 11 |- ((n e. (ZZ>=` M) /\ A.k e. (M...(n + 1))A e. RR) -> sum_k e. (M...(n + 1))A = (sum_k e. (M...n)A + [_(n + 1) / k]_A))
100	37	ralimi 2168	. . . . . . . . . . 11 |- (A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A) -> A.k e. (M...(n + 1))A e. RR)
101	99, 100	sylan2 500	. . . . . . . . . 10 |- ((n e. (ZZ>=` M) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> sum_k e. (M...(n + 1))A = (sum_k e. (M...n)A + [_(n + 1) / k]_A))
102	101	eqeq1d 1892	. . . . . . . . 9 |- ((n e. (ZZ>=` M) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> (sum_k e. (M...(n + 1))A = 0 <-> (sum_k e. (M...n)A + [_(n + 1) / k]_A) = 0))
103	102	biimpd 170	. . . . . . . 8 |- ((n e. (ZZ>=` M) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> (sum_k e. (M...(n + 1))A = 0 -> (sum_k e. (M...n)A + [_(n + 1) / k]_A) = 0))
104	98, 103	syl5d 66	. . . . . . 7 |- ((n e. (ZZ>=` M) /\ A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A)) -> ((sum_k e. (M...n)A = 0 -> A.k e. (M...n)A = 0) -> (sum_k e. (M...(n + 1))A = 0 -> A.k e. (M...(n + 1))A = 0)))
105	104	ex 402	. . . . . 6 |- (n e. (ZZ>=` M) -> (A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A) -> ((sum_k e. (M...n)A = 0 -> A.k e. (M...n)A = 0) -> (sum_k e. (M...(n + 1))A = 0 -> A.k e. (M...(n + 1))A = 0))))
106	105	a2d 16	. . . . 5 |- (n e. (ZZ>=` M) -> ((A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A) -> (sum_k e. (M...n)A = 0 -> A.k e. (M...n)A = 0)) -> (A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A) -> (sum_k e. (M...(n + 1))A = 0 -> A.k e. (M...(n + 1))A = 0))))
107	47, 106	syld 30	. . . 4 |- (n e. (ZZ>=` M) -> ((A.k e. (M...n)(A e. RR /\ 0 <_ A) -> (sum_k e. (M...n)A = 0 -> A.k e. (M...n)A = 0)) -> (A.k e. (M...(n + 1))(A e. RR /\ 0 <_ A) -> (sum_k e. (M...(n + 1))A = 0 -> A.k e. (M...(n + 1))A = 0))))
108	7, 14, 21, 28, 40, 107	uzind4 7619	. . 3 |- (N e. (ZZ>=` M) -> (A.k e. (M...N)(A e. RR /\ 0 <_ A) -> (sum_k e. (M...N)A = 0 -> A.k e. (M...N)A = 0)))
109	108	imp 377	. 2 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) -> (sum_k e. (M...N)A = 0 -> A.k e. (M...N)A = 0))
110		sumeq2 8245	. . . . 5 |- (A.k e. (M...N)A = 0 -> sum_k e. (M...N)A = sum_k e. (M...N)0)
111	110	eqeq1d 1892	. . . 4 |- (A.k e. (M...N)A = 0 -> (sum_k e. (M...N)A = 0 <-> sum_k e. (M...N)0 = 0))
112		fsum0 8299	. . . 4 |- (N e. (ZZ>=` M) -> sum_k e. (M...N)0 = 0)
113	111, 112	syl5cbir 228	. . 3 |- (N e. (ZZ>=` M) -> (A.k e. (M...N)A = 0 -> sum_k e. (M...N)A = 0))
114	113	adantr 425	. 2 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) -> (A.k e. (M...N)A = 0 -> sum_k e. (M...N)A = 0))
115	109, 114	impbid 574	1 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) -> (sum_k e. (M...N)A = 0 <-> A.k e. (M...N)A = 0))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  _Vcvv 2292  [_csb 2540   u. cun 2591   C_ wss 2593  {csn 3044   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   <_ cle 6448  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637  sum_csu 8239

This theorem is referenced by:  rrnmet 16016

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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