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 2739	. . . . 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 391	. . 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 2740	. . . . . 6 |- (A e. V -> {x e. A | A e. V} = A)
5	4	r19.20si 1709	. . . . 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 390	. . 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 6507	. . . . . . . . . 10 |- ((M e. ZZ /\ N e. ZZ) -> (M...N) (_ (M...(N + 1)))
9		eluzel2 6425	. . . . . . . . . 10 |- (N e. (ZZ>` M) -> M e. ZZ)
10		eluzelz 6424	. . . . . . . . . 10 |- (N e. (ZZ>` M) -> N e. ZZ)
11	8, 9, 10	sylanc 473	. . . . . . . . 9 |- (N e. (ZZ>` M) -> (M...N) (_ (M...(N + 1)))
12	11	sseld 2070	. . . . . . . 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 1714	. . . . . 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 2001	. . . . . . 7 |- (((N + 1) e. (M...(N + 1)) /\ A.k e. (M...(N + 1))A e. V) -> [(N + 1) / k]A e. V)
19		peano2uz 6448	. . . . . . . 8 |- (N e. (ZZ>` M) -> (N + 1) e. (ZZ>` M))
20		eluzfz2t 6490	. . . . . . . 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 450	. . . . . 6 |- ((N e. (ZZ>` M) /\ A.k e. (M...(N + 1))A e. V) -> [(N + 1) / k]A e. V)
23		equid 1128	. . . . . . 7 |- x = x
24		oprex 3989	. . . . . . 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 1988	. . . . . . 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 699	. . . . . 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 2015	. . . . . 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 3984	. . 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 1518	. 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 1820	. . 3 |- (A e. B -> A e. V)
35	34	r19.20si 1709	. 2 |- (A.k e. (M...(N + 1))A e. B -> A.k e. (M...(N + 1))A e. V)
36	33, 35	sylan2 453	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 958   e. wcel 960  [wsbc 1172  A.wral 1648  {crab 1651  Vcvv 1814  [_csb 2004   (_ wss 2050  ` cfv 3188  (class class class)co 3969  1c1 5247   + caddc 5249  ZZcz 5310  ZZ>cuz 6418  ...cfz 6468  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-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-n 5927  df-n0 6102  df-z 6138  df-seq1 6309  df-shft 6342  df-uz 6419  df-fz 6469  df-seqz 6534  df-sum 6980

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