Theorem fsumshftm 10839

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 2228	. . 3 |- F/_ m A
2		nfcsb1v 2963	. . 3 |- F/_ j [_ m / j ]_ A
3		csbeq1a 2941	. . 3 |- ( j = m -> A = [_ m / j ]_ A )
4	1, 2, 3	cbvsumi 10751	. 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 8870	. . . 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 2446	. . . . 5 |- ( ph -> A. j e. ( M ... N ) A e. CC )
11	2	nfel1 2239	. . . . . 6 |- F/ j [_ m / j ]_ A e. CC
12	3	eleq1d 2156	. . . . . 6 |- ( j = m -> ( A e. CC <-> [_ m / j ]_ A e. CC ) )
13	11, 12	rspc 2716	. . . . 5 |- ( m e. ( M ... N ) -> ( A. j e. ( M ... N ) A e. CC -> [_ m / j ]_ A e. CC ) )
14	10, 13	mpan9 275	. . . 4 |- ( ( ph /\ m e. ( M ... N ) ) -> [_ m / j ]_ A e. CC )
15		csbeq1 2936	. . . 4 |- ( m = ( k - -u K ) -> [_ m / j ]_ A = [_ ( k - -u K ) / j ]_ A )
16	6, 7, 8, 14, 15	fsumshft 10838	. . 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 8869	. . . . . 6 |- ( ph -> M e. CC )
18	5	zcnd 8869	. . . . . 6 |- ( ph -> K e. CC )
19	17, 18	negsubd 7799	. . . . 5 |- ( ph -> ( M + -u K ) = ( M - K ) )
20	8	zcnd 8869	. . . . . 6 |- ( ph -> N e. CC )
21	20, 18	negsubd 7799	. . . . 5 |- ( ph -> ( N + -u K ) = ( N - K ) )
22	19, 21	oveq12d 5670	. . . 4 |- ( ph -> ( ( M + -u K ) ... ( N + -u K ) ) = ( ( M - K ) ... ( N - K ) ) )
23	22	sumeq1d 10755	. . 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 9440	. . . . . . . 8 |- ( k e. ( ( M - K ) ... ( N - K ) ) -> k e. ZZ )
25	24	zcnd 8869	. . . . . . 7 |- ( k e. ( ( M - K ) ... ( N - K ) ) -> k e. CC )
26		subneg 7731	. . . . . . 7 |- ( ( k e. CC /\ K e. CC ) -> ( k - -u K ) = ( k + K ) )
27	25, 18, 26	syl2anr 284	. . . . . 6 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> ( k - -u K ) = ( k + K ) )
28	27	csbeq1d 2939	. . . . 5 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> [_ ( k - -u K ) / j ]_ A = [_ ( k + K ) / j ]_ A )
29	24	adantl 271	. . . . . . 7 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> k e. ZZ )
30	5	adantr 270	. . . . . . 7 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> K e. ZZ )
31	29, 30	zaddcld 8872	. . . . . 6 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> ( k + K ) e. ZZ )
32		fsumshftm.5	. . . . . . 7 |- ( j = ( k + K ) -> A = B )
33	32	adantl 271	. . . . . 6 |- ( ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) /\ j = ( k + K ) ) -> A = B )
34	31, 33	csbied 2974	. . . . 5 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> [_ ( k + K ) / j ]_ A = B )
35	28, 34	eqtrd 2120	. . . 4 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> [_ ( k - -u K ) / j ]_ A = B )
36	35	sumeq2dv 10757	. . 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 2124	. 2 |- ( ph -> sum_ m e. ( M ... N ) [_ m / j ]_ A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B )
38	4, 37	syl5eq 2132	1 |- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1289   e. wcel 1438   A.wral 2359   [_csb 2933  (class class class)co 5652   CCcc 7348   + caddc 7353   - cmin 7653   -ucneg 7654   ZZcz 8750   ...cfz 9424   sum_csu 10742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464  ax-caucvg 7465

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-frec 6156  df-1o 6181  df-oadd 6185  df-er 6292  df-en 6458  df-dom 6459  df-fin 6460  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-n0 8674  df-z 8751  df-uz 9020  df-q 9105  df-rp 9135  df-fz 9425  df-fzo 9554  df-iseq 9853  df-seq3 9854  df-exp 9955  df-ihash 10184  df-cj 10276  df-re 10277  df-im 10278  df-rsqrt 10431  df-abs 10432  df-clim 10667  df-isum 10743

This theorem is referenced by:  telfsumo  10860  fsumparts  10864  arisum  10892  geo2sum  10908