Theorem fsump1s 7013

Description: The addition of the next term in a finite sum of A(k) is the previous term plus A(N + 1).

Assertion

Ref	Expression
fsump1s	|- ((N e. (ZZ>` M) /\ A.k e. (M...(N + 1))A e. B) -> sum_k e. (M...(N + 1))A = (sum_k e. (M...N)A + [_(N + 1) / k]_A))

Distinct variable groups:   k,M   k,N

Proof of Theorem fsump1s

Step	Hyp	Ref	Expression
1		class2set 2734	. . . . 5 |- {x e. A | A e. V} e. V
2	1	fsump1slem 7012	. . . 4 |- (N e. (ZZ>` M) -> sum_k e. (M...(N + 1)){x e. A | A e. V} = (sum_k e. (M...N){x e. A | A e. V} + [_(N + 1) / k]_{x e. A | A e. V}))
3	2	adantr 389	. . 3 |- ((N e. (ZZ>` M) /\ A.k e. (M...(N + 1))A e. V) -> sum_k e. (M...(N + 1)){x e. A | A e. V} = (sum_k e. (M...N){x e. A | A e. V} + [_(N + 1) / k]_{x e. A | A e. V}))
4		class2seteq 2735	. . . . . 6 |- (A e. V -> {x e. A | A e. V} = A)
5	4	r19.20si 1706	. . . . 5 |- (A.k e. (M...(N + 1))A e. V -> A.k e. (M...(N + 1)){x e. A | A e. V} = A)
6	5	sumeq2d 6991	. . . 4 |- (A.k e. (M...(N + 1))A e. V -> sum_k e. (M...(N + 1)){x e. A | A e. V} = sum_k e. (M...(N + 1))A)
7	6	adantl 388	. . 3 |- ((N e. (ZZ>` M) /\ A.k e. (M...(N + 1))A e. V) -> sum_k e. (M...(N + 1)){x e. A | A e. V} = sum_k e. (M...(N + 1))A)
8		fzssp1t 6506	. . . . . . . . . 10 |- ((M e. ZZ /\ N e. ZZ) -> (M...N) (_ (M...(N + 1)))
9		eluzel2 6424	. . . . . . . . . 10 |- (N e. (ZZ>` M) -> M e. ZZ)
10		eluzelz 6423	. . . . . . . . . 10 |- (N e. (ZZ>` M) -> N e. ZZ)
11	8, 9, 10	sylanc 471	. . . . . . . . 9 |- (N e. (ZZ>` M) -> (M...N) (_ (M...(N + 1)))
12	11	sseld 2067	. . . . . . . 8 |- (N e. (ZZ>` M) -> (k e. (M...N) -> k e. (M...(N + 1))))
13	4	a1i 8	. . . . . . . 8 |- (N e. (ZZ>` M) -> (A e. V -> {x e. A | A e. V} = A))
14	12, 13	imim12d 29	. . . . . . 7 |- (N e. (ZZ>` M) -> ((k e. (M...(N + 1)) -> A e. V) -> (k e. (M...N) -> {x e. A | A e. V} = A)))
15	14	r19.20dv2 1711	. . . . . 6 |- (N e. (ZZ>` M) -> (A.k e. (M...(N + 1))A e. V -> A.k e. (M...N){x e. A | A e. V} = A))
16	15	imp 350	. . . . 5 |- ((N e. (ZZ>` M) /\ A.k e. (M...(N + 1))A e. V) -> A.k e. (M...N){x e. A | A e. V} = A)
17	16	sumeq2d 6991	. . . 4 |- ((N e. (ZZ>` M) /\ A.k e. (M...(N + 1))A e. V) -> sum_k e. (M...N){x e. A | A e. V} = sum_k e. (M...N)A)
18		ra4sbca 1998	. . . . . . 7 |- (((N + 1) e. (M...(N + 1)) /\ A.k e. (M...(N + 1))A e. V) -> [(N + 1) / k]A e. V)
19		peano2uz 6447	. . . . . . . 8 |- (N e. (ZZ>` M) -> (N + 1) e. (ZZ>` M))
20		eluzfz2t 6489	. . . . . . . 8 |- ((N + 1) e. (ZZ>` M) -> (N + 1) e. (M...(N + 1)))
21	19, 20	syl 10	. . . . . . 7 |- (N e. (ZZ>` M) -> (N + 1) e. (M...(N + 1)))
22	18, 21	sylan 448	. . . . . 6 |- ((N e. (ZZ>` M) /\ A.k e. (M...(N + 1))A e. V) -> [(N + 1) / k]A e. V)
23		equid 1126	. . . . . . 7 |- x = x
24		oprex 3983	. . . . . . 7 |- (N + 1) e. V
25	4	a1i 8	. . . . . . . 8 |- (x = x -> (A e. V -> {x e. A | A e. V} = A))
26	25	sbc19.20dv 1985	. . . . . . 7 |- ((x = x /\ (N + 1) e. V) -> ([(N + 1) / k]A e. V -> [(N + 1) / k]{x e. A | A e. V} = A))
27	23, 24, 26	mp2an 697	. . . . . 6 |- ([(N + 1) / k]A e. V -> [(N + 1) / k]{x e. A | A e. V} = A)
28	22, 27	syl 10	. . . . 5 |- ((N e. (ZZ>` M) /\ A.k e. (M...(N + 1))A e. V) -> [(N + 1) / k]{x e. A | A e. V} = A)
29		sbceqdig 2012	. . . . . 6 |- ((N + 1) e. V -> ([(N + 1) / k]{x e. A | A e. V} = A <-> [_(N + 1) / k]_{x e. A | A e. V} = [_(N + 1) / k]_A))
30	24, 29	ax-mp 7	. . . . 5 |- ([(N + 1) / k]{x e. A | A e. V} = A <-> [_(N + 1) / k]_{x e. A | A e. V} = [_(N + 1) / k]_A)
31	28, 30	sylib 198	. . . 4 |- ((N e. (ZZ>` M) /\ A.k e. (M...(N + 1))A e. V) -> [_(N + 1) / k]_{x e. A | A e. V} = [_(N + 1) / k]_A)
32	17, 31	opreq12d 3978	. . 3 |- ((N e. (ZZ>` M) /\ A.k e. (M...(N + 1))A e. V) -> (sum_k e. (M...N){x e. A | A e. V} + [_(N + 1) / k]_{x e. A | A e. V}) = (sum_k e. (M...N)A + [_(N + 1) / k]_A))
33	3, 7, 32	3eqtr3d 1515	. 2 |- ((N e. (ZZ>` M) /\ A.k e. (M...(N + 1))A e. V) -> sum_k e. (M...(N + 1))A = (sum_k e. (M...N)A + [_(N + 1) / k]_A))
34		elisset 1817	. . 3 |- (A e. B -> A e. V)
35	34	r19.20si 1706	. 2 |- (A.k e. (M...(N + 1))A e. B -> A.k e. (M...(N + 1))A e. V)
36	33, 35	sylan2 451	1 |- ((N e. (ZZ>` M) /\ A.k e. (M...(N + 1))A e. B) -> sum_k e. (M...(N + 1))A = (sum_k e. (M...N)A + [_(N + 1) / k]_A))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  [wsbc 1170  A.wral 1645  {crab 1648  Vcvv 1811  [_csb 2001   (_ wss 2047  ` cfv 3182  (class class class)co 3963  1c1 5235   + caddc 5237  ZZcz 5298  ZZ>cuz 6417  ...cfz 6467  sum_csu 6979

This theorem is referenced by:  fsumcllem 7014  fsum1ps 7018  fsumsplit 7020  fsumadd 7022  fsumcom 7028  fsumrev 7029  fsummulc1 7033  fsumconst 7038  fsumcmp 7040  fsumabs 7043

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-n 5925  df-n0 6100  df-z 6136  df-seq1 6308  df-shft 6341  df-uz 6418  df-fz 6468  df-seqz 6533  df-sum 6980

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