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Theorem fsump1s 6959
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
StepHypRef Expression
1 class2set 2729 . . . . 5 |- {x e. A | A e. V} e. V
21fsump1slem 6958 . . . 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}))
32adantr 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 2730 . . . . . 6 |- (A e. V -> {x e. A | A e. V} = A)
54r19.20si 1703 . . . . 5 |- (A.k e. (M...(N + 1))A e. V -> A.k e. (M...(N + 1)){x e. A | A e. V} = A)
65sumeq2d 6937 . . . 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)
76adantl 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 6446 . . . . . . . . . 10 |- ((M e. ZZ /\ N e. ZZ) -> (M...N) (_ (M...(N + 1)))
9 eluzel2 6364 . . . . . . . . . 10 |- (N e. (ZZ>` M) -> M e. ZZ)
10 eluzelz 6363 . . . . . . . . . 10 |- (N e. (ZZ>` M) -> N e. ZZ)
118, 9, 10sylanc 471 . . . . . . . . 9 |- (N e. (ZZ>` M) -> (M...N) (_ (M...(N + 1)))
1211sseld 2063 . . . . . . . 8 |- (N e. (ZZ>` M) -> (k e. (M...N) -> k e. (M...(N + 1))))
134a1i 8 . . . . . . . 8 |- (N e. (ZZ>` M) -> (A e. V -> {x e. A | A e. V} = A))
1412, 13imim12d 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)))
1514r19.20dv2 1708 . . . . . 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))
1615imp 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)
1716sumeq2d 6937 . . . 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 1994 . . . . . . 7 |- (((N + 1) e. (M...(N + 1)) /\ A.k e. (M...(N + 1))A e. V) -> [(N + 1) / k]A e. V)
19 peano2uz 6387 . . . . . . . 8 |- (N e. (ZZ>` M) -> (N + 1) e. (ZZ>` M))
20 eluzfz2t 6429 . . . . . . . 8 |- ((N + 1) e. (ZZ>` M) -> (N + 1) e. (M...(N + 1)))
2119, 20syl 10 . . . . . . 7 |- (N e. (ZZ>` M) -> (N + 1) e. (M...(N + 1)))
2218, 21sylan 448 . . . . . 6 |- ((N e. (ZZ>` M) /\ A.k e. (M...(N + 1))A e. V) -> [(N + 1) / k]A e. V)
23 equid 1124 . . . . . . 7 |- x = x
24 oprex 3974 . . . . . . 7 |- (N + 1) e. V
254a1i 8 . . . . . . . 8 |- (x = x -> (A e. V -> {x e. A | A e. V} = A))
2625sbc19.20dv 1981 . . . . . . 7 |- ((x = x /\ (N + 1) e. V) -> ([(N + 1) / k]A e. V -> [(N + 1) / k]{x e. A | A e. V} = A))
2723, 24, 26mp2an 696 . . . . . 6 |- ([(N + 1) / k]A e. V -> [(N + 1) / k]{x e. A | A e. V} = A)
2822, 27syl 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 2008 . . . . . 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))
3024, 29ax-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)
3128, 30sylib 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)
3217, 31opreq12d 3969 . . 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))
333, 7, 323eqtr3d 1512 . 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 1813 . . 3 |- (A e. B -> A e. V)
3534r19.20si 1703 . 2 |- (A.k e. (M...(N + 1))A e. B -> A.k e. (M...(N + 1))A e. V)
3633, 35sylan2 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))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  [wsbc 1168  A.wral 1642  {crab 1645  Vcvv 1807  [_csb 1997   (_ wss 2043  ` cfv 3177  (class class class)co 3954  1c1 5215   + caddc 5217  ZZcz 5278  ZZ>cuz 6357  ...cfz 6407  sum_csu 6925
This theorem is referenced by:  fsumcllem 6960  fsum1ps 6964  fsumsplit 6966  fsumadd 6968  fsumcom 6974  fsumrev 6975  fsummulc1 6979  fsumconst 6984  fsumcmp 6986  fsumabs 6989
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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