Theorem fsumlt1 15831

Description: A sum of nonnegative numbers is greater than or equal to any one of its terms.

Hypothesis

Ref	Expression
fsumlt1.1	|- (k = K -> A = B)

Assertion

Ref	Expression
fsumlt1	|- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> B <_ sum_k e. (M...N)A)

Distinct variable groups:   k,N   k,M   k,K   B,k

Proof of Theorem fsumlt1

Step	Hyp	Ref	Expression
1		fzm1 15796	. . . . . . . 8 |- (N e. (ZZ>=` M) -> (K e. (M...N) <-> (K e. (M...(N - 1)) \/ K = N)))
2	1	biimpd 170	. . . . . . 7 |- (N e. (ZZ>=` M) -> (K e. (M...N) -> (K e. (M...(N - 1)) \/ K = N)))
3	2	adantr 425	. . . . . 6 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) -> (K e. (M...N) -> (K e. (M...(N - 1)) \/ K = N)))
4	3	imdistani 491	. . . . 5 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...N)) -> ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ (K e. (M...(N - 1)) \/ K = N)))
5		eluzelz 7592	. . . . . . . . 9 |- (N e. (ZZ>=` M) -> N e. ZZ)
6	5	ad2antrr 440	. . . . . . . 8 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...(N - 1))) -> N e. ZZ)
7		simpr 350	. . . . . . . 8 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...(N - 1))) -> K e. (M...(N - 1)))
8		iftrue 2989	. . . . . . . . . . . . . . 15 |- (k = K -> if(k = K, B, 0) = B)
9		fsumlt1.1	. . . . . . . . . . . . . . 15 |- (k = K -> A = B)
10	8, 9	eqtr4d 1928	. . . . . . . . . . . . . 14 |- (k = K -> if(k = K, B, 0) = A)
11	10	eleq1d 1963	. . . . . . . . . . . . 13 |- (k = K -> (if(k = K, B, 0) e. RR <-> A e. RR))
12		simpl 346	. . . . . . . . . . . . 13 |- ((A e. RR /\ 0 <_ A) -> A e. RR)
13	11, 12	syl5cbir 228	. . . . . . . . . . . 12 |- ((A e. RR /\ 0 <_ A) -> (k = K -> if(k = K, B, 0) e. RR))
14		iffalse 2991	. . . . . . . . . . . . . 14 |- (-. k = K -> if(k = K, B, 0) = 0)
15	14	eleq1d 1963	. . . . . . . . . . . . 13 |- (-. k = K -> (if(k = K, B, 0) e. RR <-> 0 e. RR))
16		0re 6603	. . . . . . . . . . . . . 14 |- 0 e. RR
17	16	a1i 8	. . . . . . . . . . . . 13 |- ((A e. RR /\ 0 <_ A) -> 0 e. RR)
18	15, 17	syl5cbir 228	. . . . . . . . . . . 12 |- ((A e. RR /\ 0 <_ A) -> (-. k = K -> if(k = K, B, 0) e. RR))
19	13, 18	pm2.61d 141	. . . . . . . . . . 11 |- ((A e. RR /\ 0 <_ A) -> if(k = K, B, 0) e. RR)
20	19	recnd 6468	. . . . . . . . . 10 |- ((A e. RR /\ 0 <_ A) -> if(k = K, B, 0) e. CC)
21	20	ralimi 2168	. . . . . . . . 9 |- (A.k e. (M...N)(A e. RR /\ 0 <_ A) -> A.k e. (M...N)if(k = K, B, 0) e. CC)
22	21	ad2antlr 441	. . . . . . . 8 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...(N - 1))) -> A.k e. (M...N)if(k = K, B, 0) e. CC)
23		fsumsplit 8280	. . . . . . . 8 |- ((N e. ZZ /\ K e. (M...(N - 1)) /\ A.k e. (M...N)if(k = K, B, 0) e. CC) -> sum_k e. (M...N)if(k = K, B, 0) = (sum_k e. (M...K)if(k = K, B, 0) + sum_k e. ((K + 1)...N)if(k = K, B, 0)))
24	6, 7, 22, 23	syl111anc 1100	. . . . . . 7 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...(N - 1))) -> sum_k e. (M...N)if(k = K, B, 0) = (sum_k e. (M...K)if(k = K, B, 0) + sum_k e. ((K + 1)...N)if(k = K, B, 0)))
25		elfzelz 7652	. . . . . . . . . . . . . . 15 |- (K e. (M...(N - 1)) -> K e. ZZ)
26		zre 7348	. . . . . . . . . . . . . . . . . 18 |- (K e. ZZ -> K e. RR)
27		ltp1 6989	. . . . . . . . . . . . . . . . . 18 |- (K e. RR -> K < (K + 1))
28	26, 27	syl 12	. . . . . . . . . . . . . . . . 17 |- (K e. ZZ -> K < (K + 1))
29		peano2re 6599	. . . . . . . . . . . . . . . . . . 19 |- (K e. RR -> (K + 1) e. RR)
30	26, 29	syl 12	. . . . . . . . . . . . . . . . . 18 |- (K e. ZZ -> (K + 1) e. RR)
31		ltnle 6680	. . . . . . . . . . . . . . . . . 18 |- ((K e. RR /\ (K + 1) e. RR) -> (K < (K + 1) <-> -. (K + 1) <_ K))
32	26, 30, 31	syl11anc 524	. . . . . . . . . . . . . . . . 17 |- (K e. ZZ -> (K < (K + 1) <-> -. (K + 1) <_ K))
33	28, 32	mpbid 212	. . . . . . . . . . . . . . . 16 |- (K e. ZZ -> -. (K + 1) <_ K)
34		eluzle 7594	. . . . . . . . . . . . . . . . . 18 |- (K e. (ZZ>=` (K + 1)) -> (K + 1) <_ K)
35	34	a1i 8	. . . . . . . . . . . . . . . . 17 |- (K e. ZZ -> (K e. (ZZ>=` (K + 1)) -> (K + 1) <_ K))
36		elfzuz 7658	. . . . . . . . . . . . . . . . 17 |- (K e. ((K + 1)...N) -> K e. (ZZ>=` (K + 1)))
37	35, 36	syl5 20	. . . . . . . . . . . . . . . 16 |- (K e. ZZ -> (K e. ((K + 1)...N) -> (K + 1) <_ K))
38	33, 37	mtod 123	. . . . . . . . . . . . . . 15 |- (K e. ZZ -> -. K e. ((K + 1)...N))
39		eleq1 1957	. . . . . . . . . . . . . . . . 17 |- (k = K -> (k e. ((K + 1)...N) <-> K e. ((K + 1)...N)))
40	39	notbid 673	. . . . . . . . . . . . . . . 16 |- (k = K -> (-. k e. ((K + 1)...N) <-> -. K e. ((K + 1)...N)))
41	40	biimprcd 173	. . . . . . . . . . . . . . 15 |- (-. K e. ((K + 1)...N) -> (k = K -> -. k e. ((K + 1)...N)))
42	25, 38, 41	3syl 24	. . . . . . . . . . . . . 14 |- (K e. (M...(N - 1)) -> (k = K -> -. k e. ((K + 1)...N)))
43	42	con2d 107	. . . . . . . . . . . . 13 |- (K e. (M...(N - 1)) -> (k e. ((K + 1)...N) -> -. k = K))
44	43	imp 377	. . . . . . . . . . . 12 |- ((K e. (M...(N - 1)) /\ k e. ((K + 1)...N)) -> -. k = K)
45	44, 14	syl 12	. . . . . . . . . . 11 |- ((K e. (M...(N - 1)) /\ k e. ((K + 1)...N)) -> if(k = K, B, 0) = 0)
46	45	sumeq2dv 8252	. . . . . . . . . 10 |- (K e. (M...(N - 1)) -> sum_k e. ((K + 1)...N)if(k = K, B, 0) = sum_k e. ((K + 1)...N)0)
47	46	adantl 424	. . . . . . . . 9 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...(N - 1))) -> sum_k e. ((K + 1)...N)if(k = K, B, 0) = sum_k e. ((K + 1)...N)0)
48		zcn 7349	. . . . . . . . . . . . . . 15 |- (N e. ZZ -> N e. CC)
49		ax1cn 6422	. . . . . . . . . . . . . . . 16 |- 1 e. CC
50		npcan 6559	. . . . . . . . . . . . . . . 16 |- ((N e. CC /\ 1 e. CC) -> ((N - 1) + 1) = N)
51	49, 50	mpan2 760	. . . . . . . . . . . . . . 15 |- (N e. CC -> ((N - 1) + 1) = N)
52	5, 48, 51	3syl 24	. . . . . . . . . . . . . 14 |- (N e. (ZZ>=` M) -> ((N - 1) + 1) = N)
53	52	adantr 425	. . . . . . . . . . . . 13 |- ((N e. (ZZ>=` M) /\ (N - 1) e. (ZZ>=` K)) -> ((N - 1) + 1) = N)
54		eluzp1p1 7604	. . . . . . . . . . . . . 14 |- ((N - 1) e. (ZZ>=` K) -> ((N - 1) + 1) e. (ZZ>=` (K + 1)))
55	54	adantl 424	. . . . . . . . . . . . 13 |- ((N e. (ZZ>=` M) /\ (N - 1) e. (ZZ>=` K)) -> ((N - 1) + 1) e. (ZZ>=` (K + 1)))
56	53, 55	eqeltrrd 1972	. . . . . . . . . . . 12 |- ((N e. (ZZ>=` M) /\ (N - 1) e. (ZZ>=` K)) -> N e. (ZZ>=` (K + 1)))
57		oprex 4907	. . . . . . . . . . . . 13 |- (N - 1) e. _V
58		elfzuz3 7648	. . . . . . . . . . . . 13 |- (((N - 1) e. _V /\ K e. (M...(N - 1))) -> (N - 1) e. (ZZ>=` K))
59	57, 58	mpan 759	. . . . . . . . . . . 12 |- (K e. (M...(N - 1)) -> (N - 1) e. (ZZ>=` K))
60	56, 59	sylan2 500	. . . . . . . . . . 11 |- ((N e. (ZZ>=` M) /\ K e. (M...(N - 1))) -> N e. (ZZ>=` (K + 1)))
61	60	adantlr 429	. . . . . . . . . 10 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...(N - 1))) -> N e. (ZZ>=` (K + 1)))
62		fsum0 8299	. . . . . . . . . 10 |- (N e. (ZZ>=` (K + 1)) -> sum_k e. ((K + 1)...N)0 = 0)
63	61, 62	syl 12	. . . . . . . . 9 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...(N - 1))) -> sum_k e. ((K + 1)...N)0 = 0)
64	47, 63	eqtrd 1925	. . . . . . . 8 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...(N - 1))) -> sum_k e. ((K + 1)...N)if(k = K, B, 0) = 0)
65	64	opreq2d 4898	. . . . . . 7 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...(N - 1))) -> (sum_k e. (M...K)if(k = K, B, 0) + sum_k e. ((K + 1)...N)if(k = K, B, 0)) = (sum_k e. (M...K)if(k = K, B, 0) + 0))
66		elfzuz 7658	. . . . . . . . . 10 |- (K e. (M...(N - 1)) -> K e. (ZZ>=` M))
67	66	adantl 424	. . . . . . . . 9 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...(N - 1))) -> K e. (ZZ>=` M))
68		elfzuz3 7648	. . . . . . . . . . . . . . . . . 18 |- ((N e. (ZZ>=` M) /\ K e. (M...N)) -> N e. (ZZ>=` K))
69	68	ex 402	. . . . . . . . . . . . . . . . 17 |- (N e. (ZZ>=` M) -> (K e. (M...N) -> N e. (ZZ>=` K)))
70	1, 69	sylbird 222	. . . . . . . . . . . . . . . 16 |- (N e. (ZZ>=` M) -> ((K e. (M...(N - 1)) \/ K = N) -> N e. (ZZ>=` K)))
71		orc 291	. . . . . . . . . . . . . . . 16 |- (K e. (M...(N - 1)) -> (K e. (M...(N - 1)) \/ K = N))
72	70, 71	syl5 20	. . . . . . . . . . . . . . 15 |- (N e. (ZZ>=` M) -> (K e. (M...(N - 1)) -> N e. (ZZ>=` K)))
73	72	imp 377	. . . . . . . . . . . . . 14 |- ((N e. (ZZ>=` M) /\ K e. (M...(N - 1))) -> N e. (ZZ>=` K))
74		elfzel1 7651	. . . . . . . . . . . . . . 15 |- (K e. (M...(N - 1)) -> M e. ZZ)
75	74	adantl 424	. . . . . . . . . . . . . 14 |- ((N e. (ZZ>=` M) /\ K e. (M...(N - 1))) -> M e. ZZ)
76		fzss2 7676	. . . . . . . . . . . . . 14 |- ((N e. (ZZ>=` K) /\ M e. ZZ) -> (M...K) C_ (M...N))
77	73, 75, 76	syl11anc 524	. . . . . . . . . . . . 13 |- ((N e. (ZZ>=` M) /\ K e. (M...(N - 1))) -> (M...K) C_ (M...N))
78		ssralv 2672	. . . . . . . . . . . . 13 |- ((M...K) C_ (M...N) -> (A.k e. (M...N)(A e. RR /\ 0 <_ A) -> A.k e. (M...K)(A e. RR /\ 0 <_ A)))
79	77, 78	syl 12	. . . . . . . . . . . 12 |- ((N e. (ZZ>=` M) /\ K e. (M...(N - 1))) -> (A.k e. (M...N)(A e. RR /\ 0 <_ A) -> A.k e. (M...K)(A e. RR /\ 0 <_ A)))
80	79	imp 377	. . . . . . . . . . 11 |- (((N e. (ZZ>=` M) /\ K e. (M...(N - 1))) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) -> A.k e. (M...K)(A e. RR /\ 0 <_ A))
81	80	an1rs 547	. . . . . . . . . 10 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...(N - 1))) -> A.k e. (M...K)(A e. RR /\ 0 <_ A))
82	20	ralimi 2168	. . . . . . . . . 10 |- (A.k e. (M...K)(A e. RR /\ 0 <_ A) -> A.k e. (M...K)if(k = K, B, 0) e. CC)
83	81, 82	syl 12	. . . . . . . . 9 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...(N - 1))) -> A.k e. (M...K)if(k = K, B, 0) e. CC)
84		fsumcl 8275	. . . . . . . . 9 |- ((K e. (ZZ>=` M) /\ A.k e. (M...K)if(k = K, B, 0) e. CC) -> sum_k e. (M...K)if(k = K, B, 0) e. CC)
85	67, 83, 84	syl11anc 524	. . . . . . . 8 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...(N - 1))) -> sum_k e. (M...K)if(k = K, B, 0) e. CC)
86		addid1 6463	. . . . . . . 8 |- (sum_k e. (M...K)if(k = K, B, 0) e. CC -> (sum_k e. (M...K)if(k = K, B, 0) + 0) = sum_k e. (M...K)if(k = K, B, 0))
87	85, 86	syl 12	. . . . . . 7 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...(N - 1))) -> (sum_k e. (M...K)if(k = K, B, 0) + 0) = sum_k e. (M...K)if(k = K, B, 0))
88	24, 65, 87	3eqtrd 1929	. . . . . 6 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...(N - 1))) -> sum_k e. (M...N)if(k = K, B, 0) = sum_k e. (M...K)if(k = K, B, 0))
89		opreq2 4890	. . . . . . . . 9 |- (K = N -> (M...K) = (M...N))
90	89	eqcomd 1889	. . . . . . . 8 |- (K = N -> (M...N) = (M...K))
91	90	sumeq1d 8250	. . . . . . 7 |- (K = N -> sum_k e. (M...N)if(k = K, B, 0) = sum_k e. (M...K)if(k = K, B, 0))
92	91	adantl 424	. . . . . 6 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K = N) -> sum_k e. (M...N)if(k = K, B, 0) = sum_k e. (M...K)if(k = K, B, 0))
93	88, 92	jaodan 471	. . . . 5 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ (K e. (M...(N - 1)) \/ K = N)) -> sum_k e. (M...N)if(k = K, B, 0) = sum_k e. (M...K)if(k = K, B, 0))
94	4, 93	syl 12	. . . 4 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...N)) -> sum_k e. (M...N)if(k = K, B, 0) = sum_k e. (M...K)if(k = K, B, 0))
95	94	3impa 1062	. . 3 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> sum_k e. (M...N)if(k = K, B, 0) = sum_k e. (M...K)if(k = K, B, 0))
96		elfzuz 7658	. . . . . 6 |- (K e. (M...N) -> K e. (ZZ>=` M))
97		uzm1 15784	. . . . . 6 |- (K e. (ZZ>=` M) -> (K = M \/ (K - 1) e. (ZZ>=` M)))
98	96, 97	syl 12	. . . . 5 |- (K e. (M...N) -> (K = M \/ (K - 1) e. (ZZ>=` M)))
99	98	3ad2ant3 899	. . . 4 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> (K = M \/ (K - 1) e. (ZZ>=` M)))
100		opreq1 4889	. . . . . . . . 9 |- (K = M -> (K...K) = (M...K))
101	100	eqcomd 1889	. . . . . . . 8 |- (K = M -> (M...K) = (K...K))
102	101	sumeq1d 8250	. . . . . . 7 |- (K = M -> sum_k e. (M...K)if(k = K, B, 0) = sum_k e. (K...K)if(k = K, B, 0))
103	102	eqeq1d 1892	. . . . . 6 |- (K = M -> (sum_k e. (M...K)if(k = K, B, 0) = B <-> sum_k e. (K...K)if(k = K, B, 0) = B))
104	9	eleq1d 1963	. . . . . . . . . . 11 |- (k = K -> (A e. RR <-> B e. RR))
105	9	breq2d 3350	. . . . . . . . . . 11 |- (k = K -> (0 <_ A <-> 0 <_ B))
106	104, 105	anbi12d 690	. . . . . . . . . 10 |- (k = K -> ((A e. RR /\ 0 <_ A) <-> (B e. RR /\ 0 <_ B)))
107	106	rcla4cva 2379	. . . . . . . . 9 |- ((A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> (B e. RR /\ 0 <_ B))
108	107	simplld 348	. . . . . . . 8 |- ((A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> B e. RR)
109	108	3adant1 894	. . . . . . 7 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> B e. RR)
110		elfzelz 7652	. . . . . . . 8 |- (K e. (M...N) -> K e. ZZ)
111	110	3ad2ant3 899	. . . . . . 7 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> K e. ZZ)
112	8	fsum1i 8265	. . . . . . 7 |- ((B e. RR /\ K e. ZZ) -> sum_k e. (K...K)if(k = K, B, 0) = B)
113	109, 111, 112	syl11anc 524	. . . . . 6 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> sum_k e. (K...K)if(k = K, B, 0) = B)
114	103, 113	syl5cbir 228	. . . . 5 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> (K = M -> sum_k e. (M...K)if(k = K, B, 0) = B))
115		zcn 7349	. . . . . . . . . . . . . 14 |- (K e. ZZ -> K e. CC)
116		npcan 6559	. . . . . . . . . . . . . . 15 |- ((K e. CC /\ 1 e. CC) -> ((K - 1) + 1) = K)
117	49, 116	mpan2 760	. . . . . . . . . . . . . 14 |- (K e. CC -> ((K - 1) + 1) = K)
118	110, 115, 117	3syl 24	. . . . . . . . . . . . 13 |- (K e. (M...N) -> ((K - 1) + 1) = K)
119	118	eqcomd 1889	. . . . . . . . . . . 12 |- (K e. (M...N) -> K = ((K - 1) + 1))
120	119	3ad2ant3 899	. . . . . . . . . . 11 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> K = ((K - 1) + 1))
121	120	adantr 425	. . . . . . . . . 10 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> K = ((K - 1) + 1))
122	121	opreq2d 4898	. . . . . . . . 9 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> (M...K) = (M...((K - 1) + 1)))
123	122	sumeq1d 8250	. . . . . . . 8 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> sum_k e. (M...K)if(k = K, B, 0) = sum_k e. (M...((K - 1) + 1))if(k = K, B, 0))
124		simpr 350	. . . . . . . . 9 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> (K - 1) e. (ZZ>=` M))
125	118	adantl 424	. . . . . . . . . . . . . . . . 17 |- ((N e. (ZZ>=` M) /\ K e. (M...N)) -> ((K - 1) + 1) = K)
126	125	opreq2d 4898	. . . . . . . . . . . . . . . 16 |- ((N e. (ZZ>=` M) /\ K e. (M...N)) -> (M...((K - 1) + 1)) = (M...K))
127		elfzel1 7651	. . . . . . . . . . . . . . . . . 18 |- (K e. (M...N) -> M e. ZZ)
128	127	adantl 424	. . . . . . . . . . . . . . . . 17 |- ((N e. (ZZ>=` M) /\ K e. (M...N)) -> M e. ZZ)
129	68, 128, 76	syl11anc 524	. . . . . . . . . . . . . . . 16 |- ((N e. (ZZ>=` M) /\ K e. (M...N)) -> (M...K) C_ (M...N))
130	126, 129	eqsstrd 2651	. . . . . . . . . . . . . . 15 |- ((N e. (ZZ>=` M) /\ K e. (M...N)) -> (M...((K - 1) + 1)) C_ (M...N))
131		ssralv 2672	. . . . . . . . . . . . . . 15 |- ((M...((K - 1) + 1)) C_ (M...N) -> (A.k e. (M...N)(A e. RR /\ 0 <_ A) -> A.k e. (M...((K - 1) + 1))(A e. RR /\ 0 <_ A)))
132	130, 131	syl 12	. . . . . . . . . . . . . 14 |- ((N e. (ZZ>=` M) /\ K e. (M...N)) -> (A.k e. (M...N)(A e. RR /\ 0 <_ A) -> A.k e. (M...((K - 1) + 1))(A e. RR /\ 0 <_ A)))
133	19	ralimi 2168	. . . . . . . . . . . . . 14 |- (A.k e. (M...((K - 1) + 1))(A e. RR /\ 0 <_ A) -> A.k e. (M...((K - 1) + 1))if(k = K, B, 0) e. RR)
134	132, 133	syl6 25	. . . . . . . . . . . . 13 |- ((N e. (ZZ>=` M) /\ K e. (M...N)) -> (A.k e. (M...N)(A e. RR /\ 0 <_ A) -> A.k e. (M...((K - 1) + 1))if(k = K, B, 0) e. RR))
135	134	imp 377	. . . . . . . . . . . 12 |- (((N e. (ZZ>=` M) /\ K e. (M...N)) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) -> A.k e. (M...((K - 1) + 1))if(k = K, B, 0) e. RR)
136	135	an1rs 547	. . . . . . . . . . 11 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A)) /\ K e. (M...N)) -> A.k e. (M...((K - 1) + 1))if(k = K, B, 0) e. RR)
137	136	3impa 1062	. . . . . . . . . 10 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> A.k e. (M...((K - 1) + 1))if(k = K, B, 0) e. RR)
138	137	adantr 425	. . . . . . . . 9 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> A.k e. (M...((K - 1) + 1))if(k = K, B, 0) e. RR)
139		fsump1s 8273	. . . . . . . . 9 |- (((K - 1) e. (ZZ>=` M) /\ A.k e. (M...((K - 1) + 1))if(k = K, B, 0) e. RR) -> sum_k e. (M...((K - 1) + 1))if(k = K, B, 0) = (sum_k e. (M...(K - 1))if(k = K, B, 0) + [_((K - 1) + 1) / k]_if(k = K, B, 0)))
140	124, 138, 139	syl11anc 524	. . . . . . . 8 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> sum_k e. (M...((K - 1) + 1))if(k = K, B, 0) = (sum_k e. (M...(K - 1))if(k = K, B, 0) + [_((K - 1) + 1) / k]_if(k = K, B, 0)))
141	123, 140	eqtrd 1925	. . . . . . 7 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> sum_k e. (M...K)if(k = K, B, 0) = (sum_k e. (M...(K - 1))if(k = K, B, 0) + [_((K - 1) + 1) / k]_if(k = K, B, 0)))
142		eleq1 1957	. . . . . . . . . . . . . . . 16 |- (k = K -> (k e. (M...(K - 1)) <-> K e. (M...(K - 1))))
143	142	notbid 673	. . . . . . . . . . . . . . 15 |- (k = K -> (-. k e. (M...(K - 1)) <-> -. K e. (M...(K - 1))))
144		ltm1 6993	. . . . . . . . . . . . . . . . . . 19 |- (K e. RR -> (K - 1) < K)
145		peano2rem 6605	. . . . . . . . . . . . . . . . . . . 20 |- (K e. RR -> (K - 1) e. RR)
146		ltnle 6680	. . . . . . . . . . . . . . . . . . . 20 |- (((K - 1) e. RR /\ K e. RR) -> ((K - 1) < K <-> -. K <_ (K - 1)))
147	145, 146	mpancom 769	. . . . . . . . . . . . . . . . . . 19 |- (K e. RR -> ((K - 1) < K <-> -. K <_ (K - 1)))
148	144, 147	mpbid 212	. . . . . . . . . . . . . . . . . 18 |- (K e. RR -> -. K <_ (K - 1))
149		elfzle2 7654	. . . . . . . . . . . . . . . . . 18 |- (K e. (M...(K - 1)) -> K <_ (K - 1))
150	148, 149	nsyl 131	. . . . . . . . . . . . . . . . 17 |- (K e. RR -> -. K e. (M...(K - 1)))
151	110, 26, 150	3syl 24	. . . . . . . . . . . . . . . 16 |- (K e. (M...N) -> -. K e. (M...(K - 1)))
152	151	adantr 425	. . . . . . . . . . . . . . 15 |- ((K e. (M...N) /\ (K - 1) e. (ZZ>=` M)) -> -. K e. (M...(K - 1)))
153	143, 152	syl5cbir 228	. . . . . . . . . . . . . 14 |- ((K e. (M...N) /\ (K - 1) e. (ZZ>=` M)) -> (k = K -> -. k e. (M...(K - 1))))
154	153	con2d 107	. . . . . . . . . . . . 13 |- ((K e. (M...N) /\ (K - 1) e. (ZZ>=` M)) -> (k e. (M...(K - 1)) -> -. k = K))
155	154	imp 377	. . . . . . . . . . . 12 |- (((K e. (M...N) /\ (K - 1) e. (ZZ>=` M)) /\ k e. (M...(K - 1))) -> -. k = K)
156	155, 14	syl 12	. . . . . . . . . . 11 |- (((K e. (M...N) /\ (K - 1) e. (ZZ>=` M)) /\ k e. (M...(K - 1))) -> if(k = K, B, 0) = 0)
157	156	sumeq2dv 8252	. . . . . . . . . 10 |- ((K e. (M...N) /\ (K - 1) e. (ZZ>=` M)) -> sum_k e. (M...(K - 1))if(k = K, B, 0) = sum_k e. (M...(K - 1))0)
158	157	3ad2antl3 1040	. . . . . . . . 9 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> sum_k e. (M...(K - 1))if(k = K, B, 0) = sum_k e. (M...(K - 1))0)
159		fsum0 8299	. . . . . . . . . 10 |- ((K - 1) e. (ZZ>=` M) -> sum_k e. (M...(K - 1))0 = 0)
160	159	adantl 424	. . . . . . . . 9 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> sum_k e. (M...(K - 1))0 = 0)
161	158, 160	eqtrd 1925	. . . . . . . 8 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> sum_k e. (M...(K - 1))if(k = K, B, 0) = 0)
162	161	opreq1d 4897	. . . . . . 7 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> (sum_k e. (M...(K - 1))if(k = K, B, 0) + [_((K - 1) + 1) / k]_if(k = K, B, 0)) = (0 + [_((K - 1) + 1) / k]_if(k = K, B, 0)))
163	121	eqcomd 1889	. . . . . . . . . . . 12 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> ((K - 1) + 1) = K)
164	163	csbeq1d 2544	. . . . . . . . . . 11 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> [_((K - 1) + 1) / k]_if(k = K, B, 0) = [_K / k]_if(k = K, B, 0))
165		ax-17 1317	. . . . . . . . . . . . . . 15 |- (z e. B -> A.k z e. B)
166	165	a1i 8	. . . . . . . . . . . . . 14 |- (K e. (M...N) -> (z e. B -> A.k z e. B))
167	166, 8	csbiegf 2575	. . . . . . . . . . . . 13 |- (K e. (M...N) -> [_K / k]_if(k = K, B, 0) = B)
168	167	3ad2ant3 899	. . . . . . . . . . . 12 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> [_K / k]_if(k = K, B, 0) = B)
169	168	adantr 425	. . . . . . . . . . 11 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> [_K / k]_if(k = K, B, 0) = B)
170	164, 169	eqtrd 1925	. . . . . . . . . 10 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> [_((K - 1) + 1) / k]_if(k = K, B, 0) = B)
171	108	recnd 6468	. . . . . . . . . . . 12 |- ((A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> B e. CC)
172	171	3adant1 894	. . . . . . . . . . 11 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> B e. CC)
173	172	adantr 425	. . . . . . . . . 10 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> B e. CC)
174	170, 173	eqeltrd 1971	. . . . . . . . 9 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> [_((K - 1) + 1) / k]_if(k = K, B, 0) e. CC)
175		addid2 6482	. . . . . . . . 9 |- ([_((K - 1) + 1) / k]_if(k = K, B, 0) e. CC -> (0 + [_((K - 1) + 1) / k]_if(k = K, B, 0)) = [_((K - 1) + 1) / k]_if(k = K, B, 0))
176	174, 175	syl 12	. . . . . . . 8 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> (0 + [_((K - 1) + 1) / k]_if(k = K, B, 0)) = [_((K - 1) + 1) / k]_if(k = K, B, 0))
177	176, 164, 169	3eqtrd 1929	. . . . . . 7 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> (0 + [_((K - 1) + 1) / k]_if(k = K, B, 0)) = B)
178	141, 162, 177	3eqtrd 1929	. . . . . 6 |- (((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) /\ (K - 1) e. (ZZ>=` M)) -> sum_k e. (M...K)if(k = K, B, 0) = B)
179	178	ex 402	. . . . 5 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> ((K - 1) e. (ZZ>=` M) -> sum_k e. (M...K)if(k = K, B, 0) = B))
180	114, 179	jaod 469	. . . 4 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> ((K = M \/ (K - 1) e. (ZZ>=` M)) -> sum_k e. (M...K)if(k = K, B, 0) = B))
181	99, 180	mpd 29	. . 3 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> sum_k e. (M...K)if(k = K, B, 0) = B)
182	95, 181	eqtrd 1925	. 2 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> sum_k e. (M...N)if(k = K, B, 0) = B)
183		simp1 876	. . 3 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> N e. (ZZ>=` M))
184	10	breq1d 3348	. . . . . . . 8 |- (k = K -> (if(k = K, B, 0) <_ A <-> A <_ A))
185		leid 6701	. . . . . . . . 9 |- (A e. RR -> A <_ A)
186	185	adantr 425	. . . . . . . 8 |- ((A e. RR /\ 0 <_ A) -> A <_ A)
187	184, 186	syl5cbir 228	. . . . . . 7 |- ((A e. RR /\ 0 <_ A) -> (k = K -> if(k = K, B, 0) <_ A))
188	14	breq1d 3348	. . . . . . . 8 |- (-. k = K -> (if(k = K, B, 0) <_ A <-> 0 <_ A))
189		simpr 350	. . . . . . . 8 |- ((A e. RR /\ 0 <_ A) -> 0 <_ A)
190	188, 189	syl5cbir 228	. . . . . . 7 |- ((A e. RR /\ 0 <_ A) -> (-. k = K -> if(k = K, B, 0) <_ A))
191	187, 190	pm2.61d 141	. . . . . 6 |- ((A e. RR /\ 0 <_ A) -> if(k = K, B, 0) <_ A)
192	19, 12, 191	3jca 1050	. . . . 5 |- ((A e. RR /\ 0 <_ A) -> (if(k = K, B, 0) e. RR /\ A e. RR /\ if(k = K, B, 0) <_ A))
193	192	ralimi 2168	. . . 4 |- (A.k e. (M...N)(A e. RR /\ 0 <_ A) -> A.k e. (M...N)(if(k = K, B, 0) e. RR /\ A e. RR /\ if(k = K, B, 0) <_ A))
194	193	3ad2ant2 898	. . 3 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> A.k e. (M...N)(if(k = K, B, 0) e. RR /\ A e. RR /\ if(k = K, B, 0) <_ A))
195		fsumcmp 8300	. . 3 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(if(k = K, B, 0) e. RR /\ A e. RR /\ if(k = K, B, 0) <_ A)) -> sum_k e. (M...N)if(k = K, B, 0) <_ sum_k e. (M...N)A)
196	183, 194, 195	syl11anc 524	. 2 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> sum_k e. (M...N)if(k = K, B, 0) <_ sum_k e. (M...N)A)
197	182, 196	eqbrtrrd 3359	1 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(A e. RR /\ 0 <_ A) /\ K e. (M...N)) -> B <_ sum_k e. (M...N)A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  [_csb 2540   C_ wss 2593  ifcif 2982   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  ZZcz 6451   < clt 6653  ZZ>=cuz 7586  ...cfz 7637  sum_csu 8239

This theorem is referenced by:  rrndstprj1 16017

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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