Theorem fsumconst 8298

Description: The sum of constant terms (k is not free in A).

Assertion

Ref	Expression
fsumconst	|- ((N e. (ZZ>=` M) /\ A e. CC) -> sum_k e. (M...N)A = (((N - M) + 1) x. A))

Distinct variable groups:   A,k   k,M   k,N

Proof of Theorem fsumconst

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . . 6 |- (j = M -> (M...j) = (M...M))
2	1	sumeq1d 8250	. . . . 5 |- (j = M -> sum_k e. (M...j)A = sum_k e. (M...M)A)
3		opreq1 4889	. . . . . . 7 |- (j = M -> (j - M) = (M - M))
4	3	opreq1d 4897	. . . . . 6 |- (j = M -> ((j - M) + 1) = ((M - M) + 1))
5	4	opreq1d 4897	. . . . 5 |- (j = M -> (((j - M) + 1) x. A) = (((M - M) + 1) x. A))
6	2, 5	eqeq12d 1899	. . . 4 |- (j = M -> (sum_k e. (M...j)A = (((j - M) + 1) x. A) <-> sum_k e. (M...M)A = (((M - M) + 1) x. A)))
7	6	imbi2d 674	. . 3 |- (j = M -> ((A e. CC -> sum_k e. (M...j)A = (((j - M) + 1) x. A)) <-> (A e. CC -> sum_k e. (M...M)A = (((M - M) + 1) x. A))))
8		opreq2 4890	. . . . . 6 |- (j = m -> (M...j) = (M...m))
9	8	sumeq1d 8250	. . . . 5 |- (j = m -> sum_k e. (M...j)A = sum_k e. (M...m)A)
10		opreq1 4889	. . . . . . 7 |- (j = m -> (j - M) = (m - M))
11	10	opreq1d 4897	. . . . . 6 |- (j = m -> ((j - M) + 1) = ((m - M) + 1))
12	11	opreq1d 4897	. . . . 5 |- (j = m -> (((j - M) + 1) x. A) = (((m - M) + 1) x. A))
13	9, 12	eqeq12d 1899	. . . 4 |- (j = m -> (sum_k e. (M...j)A = (((j - M) + 1) x. A) <-> sum_k e. (M...m)A = (((m - M) + 1) x. A)))
14	13	imbi2d 674	. . 3 |- (j = m -> ((A e. CC -> sum_k e. (M...j)A = (((j - M) + 1) x. A)) <-> (A e. CC -> sum_k e. (M...m)A = (((m - M) + 1) x. A))))
15		opreq2 4890	. . . . . 6 |- (j = (m + 1) -> (M...j) = (M...(m + 1)))
16	15	sumeq1d 8250	. . . . 5 |- (j = (m + 1) -> sum_k e. (M...j)A = sum_k e. (M...(m + 1))A)
17		opreq1 4889	. . . . . . 7 |- (j = (m + 1) -> (j - M) = ((m + 1) - M))
18	17	opreq1d 4897	. . . . . 6 |- (j = (m + 1) -> ((j - M) + 1) = (((m + 1) - M) + 1))
19	18	opreq1d 4897	. . . . 5 |- (j = (m + 1) -> (((j - M) + 1) x. A) = ((((m + 1) - M) + 1) x. A))
20	16, 19	eqeq12d 1899	. . . 4 |- (j = (m + 1) -> (sum_k e. (M...j)A = (((j - M) + 1) x. A) <-> sum_k e. (M...(m + 1))A = ((((m + 1) - M) + 1) x. A)))
21	20	imbi2d 674	. . 3 |- (j = (m + 1) -> ((A e. CC -> sum_k e. (M...j)A = (((j - M) + 1) x. A)) <-> (A e. CC -> sum_k e. (M...(m + 1))A = ((((m + 1) - M) + 1) x. A))))
22		opreq2 4890	. . . . . 6 |- (j = N -> (M...j) = (M...N))
23	22	sumeq1d 8250	. . . . 5 |- (j = N -> sum_k e. (M...j)A = sum_k e. (M...N)A)
24		opreq1 4889	. . . . . . 7 |- (j = N -> (j - M) = (N - M))
25	24	opreq1d 4897	. . . . . 6 |- (j = N -> ((j - M) + 1) = ((N - M) + 1))
26	25	opreq1d 4897	. . . . 5 |- (j = N -> (((j - M) + 1) x. A) = (((N - M) + 1) x. A))
27	23, 26	eqeq12d 1899	. . . 4 |- (j = N -> (sum_k e. (M...j)A = (((j - M) + 1) x. A) <-> sum_k e. (M...N)A = (((N - M) + 1) x. A)))
28	27	imbi2d 674	. . 3 |- (j = N -> ((A e. CC -> sum_k e. (M...j)A = (((j - M) + 1) x. A)) <-> (A e. CC -> sum_k e. (M...N)A = (((N - M) + 1) x. A))))
29		eqidd 1885	. . . . . . 7 |- (k = M -> A = A)
30	29	fsum1i 8265	. . . . . 6 |- ((A e. CC /\ M e. ZZ) -> sum_k e. (M...M)A = A)
31	30	ancoms 484	. . . . 5 |- ((M e. ZZ /\ A e. CC) -> sum_k e. (M...M)A = A)
32		subid 6555	. . . . . . . . . 10 |- (M e. CC -> (M - M) = 0)
33	32	opreq1d 4897	. . . . . . . . 9 |- (M e. CC -> ((M - M) + 1) = (0 + 1))
34		ax1cn 6422	. . . . . . . . . 10 |- 1 e. CC
35	34	addid2i 6484	. . . . . . . . 9 |- (0 + 1) = 1
36	33, 35	syl6eq 1944	. . . . . . . 8 |- (M e. CC -> ((M - M) + 1) = 1)
37	36	opreq1d 4897	. . . . . . 7 |- (M e. CC -> (((M - M) + 1) x. A) = (1 x. A))
38		mulid2 6578	. . . . . . 7 |- (A e. CC -> (1 x. A) = A)
39	37, 38	sylan9eq 1948	. . . . . 6 |- ((M e. CC /\ A e. CC) -> (((M - M) + 1) x. A) = A)
40		zcn 7349	. . . . . 6 |- (M e. ZZ -> M e. CC)
41	39, 40	sylan 497	. . . . 5 |- ((M e. ZZ /\ A e. CC) -> (((M - M) + 1) x. A) = A)
42	31, 41	eqtr4d 1928	. . . 4 |- ((M e. ZZ /\ A e. CC) -> sum_k e. (M...M)A = (((M - M) + 1) x. A))
43	42	ex 402	. . 3 |- (M e. ZZ -> (A e. CC -> sum_k e. (M...M)A = (((M - M) + 1) x. A)))
44		fsump1s 8273	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> sum_k e. (M...(m + 1))A = (sum_k e. (M...m)A + [_(m + 1) / k]_A))
45		ax-1 4	. . . . . . . . . 10 |- (A e. CC -> (k e. (M...(m + 1)) -> A e. CC))
46	45	r19.21aiv 2175	. . . . . . . . 9 |- (A e. CC -> A.k e. (M...(m + 1))A e. CC)
47	44, 46	sylan2 500	. . . . . . . 8 |- ((m e. (ZZ>=` M) /\ A e. CC) -> sum_k e. (M...(m + 1))A = (sum_k e. (M...m)A + [_(m + 1) / k]_A))
48		oprex 4907	. . . . . . . . . 10 |- (m + 1) e. _V
49		ax-17 1317	. . . . . . . . . . 11 |- (j e. A -> A.k j e. A)
50	49	csbconstgf 2551	. . . . . . . . . 10 |- ((m + 1) e. _V -> [_(m + 1) / k]_A = A)
51	48, 50	ax-mp 7	. . . . . . . . 9 |- [_(m + 1) / k]_A = A
52	51	opreq2i 4893	. . . . . . . 8 |- (sum_k e. (M...m)A + [_(m + 1) / k]_A) = (sum_k e. (M...m)A + A)
53	47, 52	syl6eq 1944	. . . . . . 7 |- ((m e. (ZZ>=` M) /\ A e. CC) -> sum_k e. (M...(m + 1))A = (sum_k e. (M...m)A + A))
54		opreq1 4889	. . . . . . 7 |- (sum_k e. (M...m)A = (((m - M) + 1) x. A) -> (sum_k e. (M...m)A + A) = ((((m - M) + 1) x. A) + A))
55	53, 54	sylan9eq 1948	. . . . . 6 |- (((m e. (ZZ>=` M) /\ A e. CC) /\ sum_k e. (M...m)A = (((m - M) + 1) x. A)) -> sum_k e. (M...(m + 1))A = ((((m - M) + 1) x. A) + A))
56		addsub 6542	. . . . . . . . . . . . 13 |- ((m e. CC /\ 1 e. CC /\ M e. CC) -> ((m + 1) - M) = ((m - M) + 1))
57	34, 56	mp3an2 1179	. . . . . . . . . . . 12 |- ((m e. CC /\ M e. CC) -> ((m + 1) - M) = ((m - M) + 1))
58	57	adantr 425	. . . . . . . . . . 11 |- (((m e. CC /\ M e. CC) /\ A e. CC) -> ((m + 1) - M) = ((m - M) + 1))
59	58	opreq1d 4897	. . . . . . . . . 10 |- (((m e. CC /\ M e. CC) /\ A e. CC) -> (((m + 1) - M) + 1) = (((m - M) + 1) + 1))
60	59	opreq1d 4897	. . . . . . . . 9 |- (((m e. CC /\ M e. CC) /\ A e. CC) -> ((((m + 1) - M) + 1) x. A) = ((((m - M) + 1) + 1) x. A))
61		adddir 6472	. . . . . . . . . . 11 |- ((((m - M) + 1) e. CC /\ 1 e. CC /\ A e. CC) -> ((((m - M) + 1) + 1) x. A) = ((((m - M) + 1) x. A) + (1 x. A)))
62	34, 61	mp3an2 1179	. . . . . . . . . 10 |- ((((m - M) + 1) e. CC /\ A e. CC) -> ((((m - M) + 1) + 1) x. A) = ((((m - M) + 1) x. A) + (1 x. A)))
63		subcl 6524	. . . . . . . . . . 11 |- ((m e. CC /\ M e. CC) -> (m - M) e. CC)
64		peano2cn 6498	. . . . . . . . . . 11 |- ((m - M) e. CC -> ((m - M) + 1) e. CC)
65	63, 64	syl 12	. . . . . . . . . 10 |- ((m e. CC /\ M e. CC) -> ((m - M) + 1) e. CC)
66	62, 65	sylan 497	. . . . . . . . 9 |- (((m e. CC /\ M e. CC) /\ A e. CC) -> ((((m - M) + 1) + 1) x. A) = ((((m - M) + 1) x. A) + (1 x. A)))
67	38	opreq2d 4898	. . . . . . . . . 10 |- (A e. CC -> ((((m - M) + 1) x. A) + (1 x. A)) = ((((m - M) + 1) x. A) + A))
68	67	adantl 424	. . . . . . . . 9 |- (((m e. CC /\ M e. CC) /\ A e. CC) -> ((((m - M) + 1) x. A) + (1 x. A)) = ((((m - M) + 1) x. A) + A))
69	60, 66, 68	3eqtrd 1929	. . . . . . . 8 |- (((m e. CC /\ M e. CC) /\ A e. CC) -> ((((m + 1) - M) + 1) x. A) = ((((m - M) + 1) x. A) + A))
70		eluzelz 7592	. . . . . . . . . 10 |- (m e. (ZZ>=` M) -> m e. ZZ)
71		zcn 7349	. . . . . . . . . 10 |- (m e. ZZ -> m e. CC)
72	70, 71	syl 12	. . . . . . . . 9 |- (m e. (ZZ>=` M) -> m e. CC)
73		eluzel2 7593	. . . . . . . . . 10 |- (m e. (ZZ>=` M) -> M e. ZZ)
74	73, 40	syl 12	. . . . . . . . 9 |- (m e. (ZZ>=` M) -> M e. CC)
75	72, 74	jca 310	. . . . . . . 8 |- (m e. (ZZ>=` M) -> (m e. CC /\ M e. CC))
76	69, 75	sylan 497	. . . . . . 7 |- ((m e. (ZZ>=` M) /\ A e. CC) -> ((((m + 1) - M) + 1) x. A) = ((((m - M) + 1) x. A) + A))
77	76	adantr 425	. . . . . 6 |- (((m e. (ZZ>=` M) /\ A e. CC) /\ sum_k e. (M...m)A = (((m - M) + 1) x. A)) -> ((((m + 1) - M) + 1) x. A) = ((((m - M) + 1) x. A) + A))
78	55, 77	eqtr4d 1928	. . . . 5 |- (((m e. (ZZ>=` M) /\ A e. CC) /\ sum_k e. (M...m)A = (((m - M) + 1) x. A)) -> sum_k e. (M...(m + 1))A = ((((m + 1) - M) + 1) x. A))
79	78	exp31 407	. . . 4 |- (m e. (ZZ>=` M) -> (A e. CC -> (sum_k e. (M...m)A = (((m - M) + 1) x. A) -> sum_k e. (M...(m + 1))A = ((((m + 1) - M) + 1) x. A))))
80	79	a2d 16	. . 3 |- (m e. (ZZ>=` M) -> ((A e. CC -> sum_k e. (M...m)A = (((m - M) + 1) x. A)) -> (A e. CC -> sum_k e. (M...(m + 1))A = ((((m + 1) - M) + 1) x. A))))
81	7, 14, 21, 28, 43, 80	uzind4 7619	. 2 |- (N e. (ZZ>=` M) -> (A e. CC -> sum_k e. (M...N)A = (((N - M) + 1) x. A)))
82	81	imp 377	1 |- ((N e. (ZZ>=` M) /\ A e. CC) -> sum_k e. (M...N)A = (((N - M) + 1) x. A))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  [_csb 2540  ` cfv 3998  (class class class)co 4884  CCcc 6384  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637  sum_csu 8239

This theorem is referenced by:  fsum0 8299  arisumi 8487  efaddlem16 8615  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

Copyright terms: Public domain