Theorem fsumshftm 11408

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 2312	. . 3 |- F/_ m A
2		nfcsb1v 3082	. . 3 |- F/_ j [_ m / j ]_ A
3		csbeq1a 3058	. . 3 |- ( j = m -> A = [_ m / j ]_ A )
4	1, 2, 3	cbvsumi 11325	. 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 9336	. . . 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 2543	. . . . 5 |- ( ph -> A. j e. ( M ... N ) A e. CC )
11	2	nfel1 2323	. . . . . 6 |- F/ j [_ m / j ]_ A e. CC
12	3	eleq1d 2239	. . . . . 6 |- ( j = m -> ( A e. CC <-> [_ m / j ]_ A e. CC ) )
13	11, 12	rspc 2828	. . . . 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 3052	. . . 4 |- ( m = ( k - -u K ) -> [_ m / j ]_ A = [_ ( k - -u K ) / j ]_ A )
16	6, 7, 8, 14, 15	fsumshft 11407	. . 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 9335	. . . . . 6 |- ( ph -> M e. CC )
18	5	zcnd 9335	. . . . . 6 |- ( ph -> K e. CC )
19	17, 18	negsubd 8236	. . . . 5 |- ( ph -> ( M + -u K ) = ( M - K ) )
20	8	zcnd 9335	. . . . . 6 |- ( ph -> N e. CC )
21	20, 18	negsubd 8236	. . . . 5 |- ( ph -> ( N + -u K ) = ( N - K ) )
22	19, 21	oveq12d 5871	. . . 4 |- ( ph -> ( ( M + -u K ) ... ( N + -u K ) ) = ( ( M - K ) ... ( N - K ) ) )
23	22	sumeq1d 11329	. . 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 9981	. . . . . . . 8 |- ( k e. ( ( M - K ) ... ( N - K ) ) -> k e. ZZ )
25	24	zcnd 9335	. . . . . . 7 |- ( k e. ( ( M - K ) ... ( N - K ) ) -> k e. CC )
26		subneg 8168	. . . . . . 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 3056	. . . . 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 9338	. . . . . 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 3095	. . . . 5 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> [_ ( k + K ) / j ]_ A = B )
35	28, 34	eqtrd 2203	. . . 4 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> [_ ( k - -u K ) / j ]_ A = B )
36	35	sumeq2dv 11331	. . 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 2207	. 2 |- ( ph -> sum_ m e. ( M ... N ) [_ m / j ]_ A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B )
38	4, 37	eqtrid 2215	1 |- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   A.wral 2448   [_csb 3049  (class class class)co 5853   CCcc 7772   + caddc 7777   - cmin 8090   -ucneg 8091   ZZcz 9212   ...cfz 9965   sum_csu 11316

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317

This theorem is referenced by:  telfsumo  11429  fsumparts  11433  arisum  11461  geo2sum  11477