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Theorem isumshft2 15830
Description: Index shift of an infinite sum.
Assertion
Ref Expression
isumshft2 |- ((F e. A /\ K e. ZZ /\ M e. ZZ) -> sum_k e. (ZZ>=` (M + K))(F` k) = sum_k e. (ZZ>=` M)(F` (K + k)))
Distinct variable groups:   k,F   k,M   k,K
Proof of Theorem isumshft2
StepHypRef Expression
1 fveq1 4680 . . . . 5 |- (F = if(F e. _V, F, (/)) -> (F` k) = (if(F e. _V, F, (/))` k))
21sumeq2sdv 8253 . . . 4 |- (F = if(F e. _V, F, (/)) -> sum_k e. (ZZ>=` (M + K))(F` k) = sum_k e. (ZZ>=` (M + K))(if(F e. _V, F, (/))` k))
3 fveq1 4680 . . . . 5 |- (F = if(F e. _V, F, (/)) -> (F` (K + k)) = (if(F e. _V, F, (/))` (K + k)))
43sumeq2sdv 8253 . . . 4 |- (F = if(F e. _V, F, (/)) -> sum_k e. (ZZ>=` M)(F` (K + k)) = sum_k e. (ZZ>=` M)(if(F e. _V, F, (/))` (K + k)))
52, 4eqeq12d 1899 . . 3 |- (F = if(F e. _V, F, (/)) -> (sum_k e. (ZZ>=` (M + K))(F` k) = sum_k e. (ZZ>=` M)(F` (K + k)) <-> sum_k e. (ZZ>=` (M + K))(if(F e. _V, F, (/))` k) = sum_k e. (ZZ>=` M)(if(F e. _V, F, (/))` (K + k))))
6 opreq2 4890 . . . . . 6 |- (K = if(K e. ZZ, K, 0) -> (M + K) = (M + if(K e. ZZ, K, 0)))
76fveq2d 4685 . . . . 5 |- (K = if(K e. ZZ, K, 0) -> (ZZ>=` (M + K)) = (ZZ>=` (M + if(K e. ZZ, K, 0))))
87sumeq1d 8250 . . . 4 |- (K = if(K e. ZZ, K, 0) -> sum_k e. (ZZ>=` (M + K))(if(F e. _V, F, (/))` k) = sum_k e. (ZZ>=` (M + if(K e. ZZ, K, 0)))(if(F e. _V, F, (/))` k))
9 opreq1 4889 . . . . . 6 |- (K = if(K e. ZZ, K, 0) -> (K + k) = (if(K e. ZZ, K, 0) + k))
109fveq2d 4685 . . . . 5 |- (K = if(K e. ZZ, K, 0) -> (if(F e. _V, F, (/))` (K + k)) = (if(F e. _V, F, (/))` (if(K e. ZZ, K, 0) + k)))
1110sumeq2sdv 8253 . . . 4 |- (K = if(K e. ZZ, K, 0) -> sum_k e. (ZZ>=` M)(if(F e. _V, F, (/))` (K + k)) = sum_k e. (ZZ>=` M)(if(F e. _V, F, (/))` (if(K e. ZZ, K, 0) + k)))
128, 11eqeq12d 1899 . . 3 |- (K = if(K e. ZZ, K, 0) -> (sum_k e. (ZZ>=` (M + K))(if(F e. _V, F, (/))` k) = sum_k e. (ZZ>=` M)(if(F e. _V, F, (/))` (K + k)) <-> sum_k e. (ZZ>=` (M + if(K e. ZZ, K, 0)))(if(F e. _V, F, (/))` k) = sum_k e. (ZZ>=` M)(if(F e. _V, F, (/))` (if(K e. ZZ, K, 0) + k))))
13 opreq1 4889 . . . . . 6 |- (M = if(M e. ZZ, M, 0) -> (M + if(K e. ZZ, K, 0)) = (if(M e. ZZ, M, 0) + if(K e. ZZ, K, 0)))
1413fveq2d 4685 . . . . 5 |- (M = if(M e. ZZ, M, 0) -> (ZZ>=` (M + if(K e. ZZ, K, 0))) = (ZZ>=` (if(M e. ZZ, M, 0) + if(K e. ZZ, K, 0))))
1514sumeq1d 8250 . . . 4 |- (M = if(M e. ZZ, M, 0) -> sum_k e. (ZZ>=` (M + if(K e. ZZ, K, 0)))(if(F e. _V, F, (/))` k) = sum_k e. (ZZ>=` (if(M e. ZZ, M, 0) + if(K e. ZZ, K, 0)))(if(F e. _V, F, (/))` k))
16 fveq2 4681 . . . . 5 |- (M = if(M e. ZZ, M, 0) -> (ZZ>=` M) = (ZZ>=` if(M e. ZZ, M, 0)))
1716sumeq1d 8250 . . . 4 |- (M = if(M e. ZZ, M, 0) -> sum_k e. (ZZ>=` M)(if(F e. _V, F, (/))` (if(K e. ZZ, K, 0) + k)) = sum_k e. (ZZ>=` if(M e. ZZ, M, 0))(if(F e. _V, F, (/))` (if(K e. ZZ, K, 0) + k)))
1815, 17eqeq12d 1899 . . 3 |- (M = if(M e. ZZ, M, 0) -> (sum_k e. (ZZ>=` (M + if(K e. ZZ, K, 0)))(if(F e. _V, F, (/))` k) = sum_k e. (ZZ>=` M)(if(F e. _V, F, (/))` (if(K e. ZZ, K, 0) + k)) <-> sum_k e. (ZZ>=` (if(M e. ZZ, M, 0) + if(K e. ZZ, K, 0)))(if(F e. _V, F, (/))` k) = sum_k e. (ZZ>=` if(M e. ZZ, M, 0))(if(F e. _V, F, (/))` (if(K e. ZZ, K, 0) + k))))
19 0ex 3446 . . . . 5 |- (/) e. _V
2019elimel 3025 . . . 4 |- if(F e. _V, F, (/)) e. _V
21 0z 7355 . . . . 5 |- 0 e. ZZ
2221elimel 3025 . . . 4 |- if(K e. ZZ, K, 0) e. ZZ
2321elimel 3025 . . . 4 |- if(M e. ZZ, M, 0) e. ZZ
24 eqid 1884 . . . 4 |- (if(M e. ZZ, M, 0) + if(K e. ZZ, K, 0)) = (if(M e. ZZ, M, 0) + if(K e. ZZ, K, 0))
2520, 22, 23, 24isumshft2i 8466 . . 3 |- sum_k e. (ZZ>=` (if(M e. ZZ, M, 0) + if(K e. ZZ, K, 0)))(if(F e. _V, F, (/))` k) = sum_k e. (ZZ>=` if(M e. ZZ, M, 0))(if(F e. _V, F, (/))` (if(K e. ZZ, K, 0) + k))
265, 12, 18, 25dedth3h 3016 . 2 |- ((F e. _V /\ K e. ZZ /\ M e. ZZ) -> sum_k e. (ZZ>=` (M + K))(F` k) = sum_k e. (ZZ>=` M)(F` (K + k)))
27 elisset 2299 . 2 |- (F e. A -> F e. _V)
2826, 27syl3an1 1130 1 |- ((F e. A /\ K e. ZZ /\ M e. ZZ) -> sum_k e. (ZZ>=` (M + K))(F` k) = sum_k e. (ZZ>=` M)(F` (K + k)))
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292  (/)c0 2875  ifcif 2982  ` cfv 3998  (class class class)co 4884  0cc0 6386   + caddc 6389  ZZcz 6451  ZZ>=cuz 7586  sum_csu 8239
This theorem is referenced by:  geomcau 15849
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731
This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-sum 8240
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