Theorem fsumshftm 11324

Description: Negative index shift of a finite sum. (Contributed by NM, 28-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)

Hypotheses

Ref	Expression
fsumrev.1	|- ( ph -> K e. ZZ )
fsumrev.2	|- ( ph -> M e. ZZ )
fsumrev.3	|- ( ph -> N e. ZZ )
fsumrev.4	|- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )
fsumshftm.5	|- ( j = ( k + K ) -> A = B )

Assertion

Ref	Expression
fsumshftm	|- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B )

Distinct variable groups:   A, k   B, j   j, k, K   j, M, k   j, N, k   ph, j, k

Allowed substitution hints:   A( j)   B( k)

Proof of Theorem fsumshftm

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2299	. . 3 |- F/_ m A
2		nfcsb1v 3064	. . 3 |- F/_ j [_ m / j ]_ A
3		csbeq1a 3040	. . 3 |- ( j = m -> A = [_ m / j ]_ A )
4	1, 2, 3	cbvsumi 11241	. 2 |- sum_ j e. ( M ... N ) A = sum_ m e. ( M ... N ) [_ m / j ]_ A
5		fsumrev.1	. . . . 5 |- ( ph -> K e. ZZ )
6	5	znegcld 9271	. . . 4 |- ( ph -> -u K e. ZZ )
7		fsumrev.2	. . . 4 |- ( ph -> M e. ZZ )
8		fsumrev.3	. . . 4 |- ( ph -> N e. ZZ )
9		fsumrev.4	. . . . . 6 |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )
10	9	ralrimiva 2530	. . . . 5 |- ( ph -> A. j e. ( M ... N ) A e. CC )
11	2	nfel1 2310	. . . . . 6 |- F/ j [_ m / j ]_ A e. CC
12	3	eleq1d 2226	. . . . . 6 |- ( j = m -> ( A e. CC <-> [_ m / j ]_ A e. CC ) )
13	11, 12	rspc 2810	. . . . 5 |- ( m e. ( M ... N ) -> ( A. j e. ( M ... N ) A e. CC -> [_ m / j ]_ A e. CC ) )
14	10, 13	mpan9 279	. . . 4 |- ( ( ph /\ m e. ( M ... N ) ) -> [_ m / j ]_ A e. CC )
15		csbeq1 3034	. . . 4 |- ( m = ( k - -u K ) -> [_ m / j ]_ A = [_ ( k - -u K ) / j ]_ A )
16	6, 7, 8, 14, 15	fsumshft 11323	. . 3 |- ( ph -> sum_ m e. ( M ... N ) [_ m / j ]_ A = sum_ k e. ( ( M + -u K ) ... ( N + -u K ) ) [_ ( k - -u K ) / j ]_ A )
17	7	zcnd 9270	. . . . . 6 |- ( ph -> M e. CC )
18	5	zcnd 9270	. . . . . 6 |- ( ph -> K e. CC )
19	17, 18	negsubd 8175	. . . . 5 |- ( ph -> ( M + -u K ) = ( M - K ) )
20	8	zcnd 9270	. . . . . 6 |- ( ph -> N e. CC )
21	20, 18	negsubd 8175	. . . . 5 |- ( ph -> ( N + -u K ) = ( N - K ) )
22	19, 21	oveq12d 5836	. . . 4 |- ( ph -> ( ( M + -u K ) ... ( N + -u K ) ) = ( ( M - K ) ... ( N - K ) ) )
23	22	sumeq1d 11245	. . 3 |- ( ph -> sum_ k e. ( ( M + -u K ) ... ( N + -u K ) ) [_ ( k - -u K ) / j ]_ A = sum_ k e. ( ( M - K ) ... ( N - K ) ) [_ ( k - -u K ) / j ]_ A )
24		elfzelz 9910	. . . . . . . 8 |- ( k e. ( ( M - K ) ... ( N - K ) ) -> k e. ZZ )
25	24	zcnd 9270	. . . . . . 7 |- ( k e. ( ( M - K ) ... ( N - K ) ) -> k e. CC )
26		subneg 8107	. . . . . . 7 |- ( ( k e. CC /\ K e. CC ) -> ( k - -u K ) = ( k + K ) )
27	25, 18, 26	syl2anr 288	. . . . . 6 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> ( k - -u K ) = ( k + K ) )
28	27	csbeq1d 3038	. . . . 5 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> [_ ( k - -u K ) / j ]_ A = [_ ( k + K ) / j ]_ A )
29	24	adantl 275	. . . . . . 7 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> k e. ZZ )
30	5	adantr 274	. . . . . . 7 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> K e. ZZ )
31	29, 30	zaddcld 9273	. . . . . 6 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> ( k + K ) e. ZZ )
32		fsumshftm.5	. . . . . . 7 |- ( j = ( k + K ) -> A = B )
33	32	adantl 275	. . . . . 6 |- ( ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) /\ j = ( k + K ) ) -> A = B )
34	31, 33	csbied 3077	. . . . 5 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> [_ ( k + K ) / j ]_ A = B )
35	28, 34	eqtrd 2190	. . . 4 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> [_ ( k - -u K ) / j ]_ A = B )
36	35	sumeq2dv 11247	. . 3 |- ( ph -> sum_ k e. ( ( M - K ) ... ( N - K ) ) [_ ( k - -u K ) / j ]_ A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B )
37	16, 23, 36	3eqtrd 2194	. 2 |- ( ph -> sum_ m e. ( M ... N ) [_ m / j ]_ A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B )
38	4, 37	syl5eq 2202	1 |- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   A.wral 2435   [_csb 3031  (class class class)co 5818   CCcc 7713   + caddc 7718   - cmin 8029   -ucneg 8030   ZZcz 9150   ...cfz 9894   sum_csu 11232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-mulrcl 7814  ax-addcom 7815  ax-mulcom 7816  ax-addass 7817  ax-mulass 7818  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-1rid 7822  ax-0id 7823  ax-rnegex 7824  ax-precex 7825  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-apti 7830  ax-pre-ltadd 7831  ax-pre-mulgt0 7832  ax-pre-mulext 7833  ax-arch 7834  ax-caucvg 7835

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-po 4255  df-iso 4256  df-iord 4325  df-on 4327  df-ilim 4328  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-isom 5176  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-recs 6246  df-irdg 6311  df-frec 6332  df-1o 6357  df-oadd 6361  df-er 6473  df-en 6679  df-dom 6680  df-fin 6681  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-reap 8433  df-ap 8440  df-div 8529  df-inn 8817  df-2 8875  df-3 8876  df-4 8877  df-n0 9074  df-z 9151  df-uz 9423  df-q 9511  df-rp 9543  df-fz 9895  df-fzo 10024  df-seqfrec 10327  df-exp 10401  df-ihash 10632  df-cj 10724  df-re 10725  df-im 10726  df-rsqrt 10880  df-abs 10881  df-clim 11158  df-sumdc 11233

This theorem is referenced by:  telfsumo  11345  fsumparts  11349  arisum  11377  geo2sum  11393