Theorem fsump1 7006

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

Hypotheses

Ref	Expression
fsump1.1	|- A e. V
fsump1.2	|- B e. V
fsump1.3	|- (k = (N + 1) -> A = B)

Assertion

Ref	Expression
fsump1	|- (N e. (ZZ>` M) -> sum_k e. (M...(N + 1))A = (sum_k e. (M...N)A + B))

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

Proof of Theorem fsump1

Step	Hyp	Ref	Expression
1		peano2uz 6447	. . . 4 |- (N e. (ZZ>` M) -> (N + 1) e. (ZZ>` M))
2		zex 6144	. . . . . 6 |- ZZ e. V
3	2	opabex2 3610	. . . . 5 |- {<.k, y>. | (k e. ZZ /\ y = A)} e. V
4		hbopab1 2813	. . . . 5 |- (x e. {<.k, y>. | (k e. ZZ /\ y = A)} -> A.k x e. {<.k, y>. | (k e. ZZ /\ y = A)})
5	3, 4	fsumserzf 7000	. . . 4 |- ((N + 1) e. (ZZ>` M) -> sum_k e. (M...(N + 1))({<.k, y>. | (k e. ZZ /\ y = A)}` k) = ((<.M, + >. seq {<.k, y>. | (k e. ZZ /\ y = A)})` (N + 1)))
6	1, 5	syl 10	. . 3 |- (N e. (ZZ>` M) -> sum_k e. (M...(N + 1))({<.k, y>. | (k e. ZZ /\ y = A)}` k) = ((<.M, + >. seq {<.k, y>. | (k e. ZZ /\ y = A)})` (N + 1)))
7		sumeq2 6985	. . . 4 |- (A.k e. (M...(N + 1))({<.k, y>. | (k e. ZZ /\ y = A)}` k) = A -> sum_k e. (M...(N + 1))({<.k, y>. | (k e. ZZ /\ y = A)}` k) = sum_k e. (M...(N + 1))A)
8		elfzelz 6482	. . . . 5 |- (k e. (M...(N + 1)) -> k e. ZZ)
9		fsump1.1	. . . . . 6 |- A e. V
10		fvopab2 3791	. . . . . 6 |- ((k e. ZZ /\ A e. V) -> ({<.k, y>. | (k e. ZZ /\ y = A)}` k) = A)
11	9, 10	mpan2 696	. . . . 5 |- (k e. ZZ -> ({<.k, y>. | (k e. ZZ /\ y = A)}` k) = A)
12	8, 11	syl 10	. . . 4 |- (k e. (M...(N + 1)) -> ({<.k, y>. | (k e. ZZ /\ y = A)}` k) = A)
13	7, 12	mprg 1700	. . 3 |- sum_k e. (M...(N + 1))({<.k, y>. | (k e. ZZ /\ y = A)}` k) = sum_k e. (M...(N + 1))A
14	6, 13	syl5eqr 1521	. 2 |- (N e. (ZZ>` M) -> sum_k e. (M...(N + 1))A = ((<.M, + >. seq {<.k, y>. | (k e. ZZ /\ y = A)})` (N + 1)))
15		addex 5317	. . 3 |- + e. V
16	15, 3	seqzp1 6548	. 2 |- (N e. (ZZ>` M) -> ((<.M, + >. seq {<.k, y>. | (k e. ZZ /\ y = A)})` (N + 1)) = (((<.M, + >. seq {<.k, y>. | (k e. ZZ /\ y = A)})` N) + ({<.k, y>. | (k e. ZZ /\ y = A)}` (N + 1))))
17	3, 4	fsumserzf 7000	. . . 4 |- (N e. (ZZ>` M) -> sum_k e. (M...N)({<.k, y>. | (k e. ZZ /\ y = A)}` k) = ((<.M, + >. seq {<.k, y>. | (k e. ZZ /\ y = A)})` N))
18		sumeq2 6985	. . . . 5 |- (A.k e. (M...N)({<.k, y>. | (k e. ZZ /\ y = A)}` k) = A -> sum_k e. (M...N)({<.k, y>. | (k e. ZZ /\ y = A)}` k) = sum_k e. (M...N)A)
19		elfzelz 6482	. . . . . 6 |- (k e. (M...N) -> k e. ZZ)
20	19, 11	syl 10	. . . . 5 |- (k e. (M...N) -> ({<.k, y>. | (k e. ZZ /\ y = A)}` k) = A)
21	18, 20	mprg 1700	. . . 4 |- sum_k e. (M...N)({<.k, y>. | (k e. ZZ /\ y = A)}` k) = sum_k e. (M...N)A
22	17, 21	syl5reqr 1522	. . 3 |- (N e. (ZZ>` M) -> ((<.M, + >. seq {<.k, y>. | (k e. ZZ /\ y = A)})` N) = sum_k e. (M...N)A)
23		eluzelz 6423	. . . 4 |- (N e. (ZZ>` M) -> N e. ZZ)
24		peano2z 6166	. . . 4 |- (N e. ZZ -> (N + 1) e. ZZ)
25		fsump1.3	. . . . 5 |- (k = (N + 1) -> A = B)
26		eqid 1475	. . . . 5 |- {<.k, y>. | (k e. ZZ /\ y = A)} = {<.k, y>. | (k e. ZZ /\ y = A)}
27		fsump1.2	. . . . 5 |- B e. V
28	25, 26, 27	fvopab4 3780	. . . 4 |- ((N + 1) e. ZZ -> ({<.k, y>. | (k e. ZZ /\ y = A)}` (N + 1)) = B)
29	23, 24, 28	3syl 20	. . 3 |- (N e. (ZZ>` M) -> ({<.k, y>. | (k e. ZZ /\ y = A)}` (N + 1)) = B)
30	22, 29	opreq12d 3978	. 2 |- (N e. (ZZ>` M) -> (((<.M, + >. seq {<.k, y>. | (k e. ZZ /\ y = A)})` N) + ({<.k, y>. | (k e. ZZ /\ y = A)}` (N + 1))) = (sum_k e. (M...N)A + B))
31	14, 16, 30	3eqtrd 1511	1 |- (N e. (ZZ>` M) -> sum_k e. (M...(N + 1))A = (sum_k e. (M...N)A + B))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811  <.cop 2411  {copab 2666  ` cfv 3182  (class class class)co 3963  1c1 5235   + caddc 5237  ZZcz 5298  ZZ>cuz 6417  ...cfz 6467   seq cseqz 6531  sum_csu 6979

This theorem is referenced by:  fsump1f 7011  binomlem2 7067  binomlem3 7068  binomlem6 7071  bcxmas 7076  fnsmnt 7226

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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