Theorem fsump1 6952

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 6387	. . . 4 |- (N e. (ZZ>` M) -> (N + 1) e. (ZZ>` M))
2		zex 6099	. . . . . 6 |- ZZ e. V
3	2	opabex2 3602	. . . . 5 |- {<.k, y>. | (k e. ZZ /\ y = A)} e. V
4		hbopab1 2808	. . . . 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 6946	. . . 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 6931	. . . 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 6422	. . . . 5 |- (k e. (M...(N + 1)) -> k e. ZZ)
9		fsump1.1	. . . . . 6 |- A e. V
10		fvopab2 3782	. . . . . 6 |- ((k e. ZZ /\ A e. V) -> ({<.k, y>. | (k e. ZZ /\ y = A)}` k) = A)
11	9, 10	mpan2 695	. . . . 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 1697	. . 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 1518	. 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 5297	. . 3 |- + e. V
16	15, 3	seqzp1 6488	. 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 6946	. . . 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 6931	. . . . 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 6422	. . . . . 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 1697	. . . 4 |- sum_k e. (M...N)({<.k, y>. | (k e. ZZ /\ y = A)}` k) = sum_k e. (M...N)A
22	17, 21	syl5reqr 1519	. . 3 |- (N e. (ZZ>` M) -> ((<.M, + >. seq {<.k, y>. | (k e. ZZ /\ y = A)})` N) = sum_k e. (M...N)A)
23		eluzelz 6363	. . . 4 |- (N e. (ZZ>` M) -> N e. ZZ)
24		peano2z 6121	. . . 4 |- (N e. ZZ -> (N + 1) e. ZZ)
25		fsump1.3	. . . . 5 |- (k = (N + 1) -> A = B)
26		eqid 1473	. . . . 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 3771	. . . 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 3969	. 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 1508	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 954   e. wcel 956  Vcvv 1807  <.cop 2407  {copab 2661  ` cfv 3177  (class class class)co 3954  1c1 5215   + caddc 5217  ZZcz 5278  ZZ>cuz 6357  ...cfz 6407   seq cseqz 6471  sum_csu 6925

This theorem is referenced by:  fsump1f 6957  binomlem2 7013  binomlem3 7014  binomlem6 7017  bcxmas 7022  fnsmnt 7169

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-n 5881  df-n0 6055  df-z 6091  df-seq1 6253  df-shft 6286  df-uz 6358  df-fz 6408  df-seqz 6473  df-sum 6926

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