Theorem fsumshftm 11214

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 2281	. . 3 |- F/_ m A
2		nfcsb1v 3035	. . 3 |- F/_ j [_ m / j ]_ A
3		csbeq1a 3012	. . 3 |- ( j = m -> A = [_ m / j ]_ A )
4	1, 2, 3	cbvsumi 11131	. 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 9175	. . . 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 2505	. . . . 5 |- ( ph -> A. j e. ( M ... N ) A e. CC )
11	2	nfel1 2292	. . . . . 6 |- F/ j [_ m / j ]_ A e. CC
12	3	eleq1d 2208	. . . . . 6 |- ( j = m -> ( A e. CC <-> [_ m / j ]_ A e. CC ) )
13	11, 12	rspc 2783	. . . . 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 3006	. . . 4 |- ( m = ( k - -u K ) -> [_ m / j ]_ A = [_ ( k - -u K ) / j ]_ A )
16	6, 7, 8, 14, 15	fsumshft 11213	. . 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 9174	. . . . . 6 |- ( ph -> M e. CC )
18	5	zcnd 9174	. . . . . 6 |- ( ph -> K e. CC )
19	17, 18	negsubd 8079	. . . . 5 |- ( ph -> ( M + -u K ) = ( M - K ) )
20	8	zcnd 9174	. . . . . 6 |- ( ph -> N e. CC )
21	20, 18	negsubd 8079	. . . . 5 |- ( ph -> ( N + -u K ) = ( N - K ) )
22	19, 21	oveq12d 5792	. . . 4 |- ( ph -> ( ( M + -u K ) ... ( N + -u K ) ) = ( ( M - K ) ... ( N - K ) ) )
23	22	sumeq1d 11135	. . 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 9806	. . . . . . . 8 |- ( k e. ( ( M - K ) ... ( N - K ) ) -> k e. ZZ )
25	24	zcnd 9174	. . . . . . 7 |- ( k e. ( ( M - K ) ... ( N - K ) ) -> k e. CC )
26		subneg 8011	. . . . . . 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 3010	. . . . 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 9177	. . . . . 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 3046	. . . . 5 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> [_ ( k + K ) / j ]_ A = B )
35	28, 34	eqtrd 2172	. . . 4 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> [_ ( k - -u K ) / j ]_ A = B )
36	35	sumeq2dv 11137	. . 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 2176	. 2 |- ( ph -> sum_ m e. ( M ... N ) [_ m / j ]_ A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B )
38	4, 37	syl5eq 2184	1 |- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   A.wral 2416   [_csb 3003  (class class class)co 5774   CCcc 7618   + caddc 7623   - cmin 7933   -ucneg 7934   ZZcz 9054   ...cfz 9790   sum_csu 11122

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-ihash 10522  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123

This theorem is referenced by:  telfsumo  11235  fsumparts  11239  arisum  11267  geo2sum  11283