Theorem fsumshftm 11956

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 2372	. . 3 |- F/_ m A
2		nfcsb1v 3157	. . 3 |- F/_ j [_ m / j ]_ A
3		csbeq1a 3133	. . 3 |- ( j = m -> A = [_ m / j ]_ A )
4	1, 2, 3	cbvsumi 11873	. 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 9571	. . . 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 2603	. . . . 5 |- ( ph -> A. j e. ( M ... N ) A e. CC )
11	2	nfel1 2383	. . . . . 6 |- F/ j [_ m / j ]_ A e. CC
12	3	eleq1d 2298	. . . . . 6 |- ( j = m -> ( A e. CC <-> [_ m / j ]_ A e. CC ) )
13	11, 12	rspc 2901	. . . . 5 |- ( m e. ( M ... N ) -> ( A. j e. ( M ... N ) A e. CC -> [_ m / j ]_ A e. CC ) )
14	10, 13	mpan9 281	. . . 4 |- ( ( ph /\ m e. ( M ... N ) ) -> [_ m / j ]_ A e. CC )
15		csbeq1 3127	. . . 4 |- ( m = ( k - -u K ) -> [_ m / j ]_ A = [_ ( k - -u K ) / j ]_ A )
16	6, 7, 8, 14, 15	fsumshft 11955	. . 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 9570	. . . . . 6 |- ( ph -> M e. CC )
18	5	zcnd 9570	. . . . . 6 |- ( ph -> K e. CC )
19	17, 18	negsubd 8463	. . . . 5 |- ( ph -> ( M + -u K ) = ( M - K ) )
20	8	zcnd 9570	. . . . . 6 |- ( ph -> N e. CC )
21	20, 18	negsubd 8463	. . . . 5 |- ( ph -> ( N + -u K ) = ( N - K ) )
22	19, 21	oveq12d 6019	. . . 4 |- ( ph -> ( ( M + -u K ) ... ( N + -u K ) ) = ( ( M - K ) ... ( N - K ) ) )
23	22	sumeq1d 11877	. . 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 10221	. . . . . . . 8 |- ( k e. ( ( M - K ) ... ( N - K ) ) -> k e. ZZ )
25	24	zcnd 9570	. . . . . . 7 |- ( k e. ( ( M - K ) ... ( N - K ) ) -> k e. CC )
26		subneg 8395	. . . . . . 7 |- ( ( k e. CC /\ K e. CC ) -> ( k - -u K ) = ( k + K ) )
27	25, 18, 26	syl2anr 290	. . . . . 6 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> ( k - -u K ) = ( k + K ) )
28	27	csbeq1d 3131	. . . . 5 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> [_ ( k - -u K ) / j ]_ A = [_ ( k + K ) / j ]_ A )
29	24	adantl 277	. . . . . . 7 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> k e. ZZ )
30	5	adantr 276	. . . . . . 7 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> K e. ZZ )
31	29, 30	zaddcld 9573	. . . . . 6 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> ( k + K ) e. ZZ )
32		fsumshftm.5	. . . . . . 7 |- ( j = ( k + K ) -> A = B )
33	32	adantl 277	. . . . . 6 |- ( ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) /\ j = ( k + K ) ) -> A = B )
34	31, 33	csbied 3171	. . . . 5 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> [_ ( k + K ) / j ]_ A = B )
35	28, 34	eqtrd 2262	. . . 4 |- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> [_ ( k - -u K ) / j ]_ A = B )
36	35	sumeq2dv 11879	. . 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 2266	. 2 |- ( ph -> sum_ m e. ( M ... N ) [_ m / j ]_ A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B )
38	4, 37	eqtrid 2274	1 |- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   [_csb 3124  (class class class)co 6001   CCcc 7997   + caddc 8002   - cmin 8317   -ucneg 8318   ZZcz 9446   ...cfz 10204   sum_csu 11864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-oadd 6566  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-seqfrec 10670  df-exp 10761  df-ihash 10998  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-clim 11790  df-sumdc 11865

This theorem is referenced by:  telfsumo  11977  fsumparts  11981  arisum  12009  geo2sum  12025  dvply1  15439