Theorem seqshft 13825

Description: Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-Feb-2014.)

Hypothesis

Ref	Expression
seqshft.1	|- F e. _V

Assertion

Ref	Expression
seqshft	|- ( ( M e. ZZ /\ N e. ZZ ) -> seq M ( .+ , ( F shift N ) ) = ( seq ( M - N ) ( .+ , F ) shift N ) )

Proof of Theorem seqshft

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqfn 12813	. . 3 |- ( M e. ZZ -> seq M ( .+ , ( F shift N ) ) Fn ( ZZ>= ` M ) )
2	1	adantr 481	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> seq M ( .+ , ( F shift N ) ) Fn ( ZZ>= ` M ) )
3		zsubcl 11419	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
4		seqfn 12813	. . . . 5 |- ( ( M - N ) e. ZZ -> seq ( M - N ) ( .+ , F ) Fn ( ZZ>= ` ( M - N ) ) )
5	3, 4	syl 17	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> seq ( M - N ) ( .+ , F ) Fn ( ZZ>= ` ( M - N ) ) )
6		zcn 11382	. . . . 5 |- ( N e. ZZ -> N e. CC )
7	6	adantl 482	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. CC )
8		seqex 12803	. . . . 5 |- seq ( M - N ) ( .+ , F ) e. _V
9	8	shftfn 13813	. . . 4 |- ( ( seq ( M - N ) ( .+ , F ) Fn ( ZZ>= ` ( M - N ) ) /\ N e. CC ) -> ( seq ( M - N ) ( .+ , F ) shift N ) Fn { x e. CC | ( x - N ) e. ( ZZ>= ` ( M - N ) ) } )
10	5, 7, 9	syl2anc 693	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( seq ( M - N ) ( .+ , F ) shift N ) Fn { x e. CC | ( x - N ) e. ( ZZ>= ` ( M - N ) ) } )
11		simpr 477	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
12		shftuz 13809	. . . . . 6 |- ( ( N e. ZZ /\ ( M - N ) e. ZZ ) -> { x e. CC | ( x - N ) e. ( ZZ>= ` ( M - N ) ) } = ( ZZ>= ` ( ( M - N ) + N ) ) )
13	11, 3, 12	syl2anc 693	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> { x e. CC | ( x - N ) e. ( ZZ>= ` ( M - N ) ) } = ( ZZ>= ` ( ( M - N ) + N ) ) )
14		zcn 11382	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
15		npcan 10290	. . . . . . 7 |- ( ( M e. CC /\ N e. CC ) -> ( ( M - N ) + N ) = M )
16	14, 6, 15	syl2an 494	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M - N ) + N ) = M )
17	16	fveq2d 6195	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ZZ>= ` ( ( M - N ) + N ) ) = ( ZZ>= ` M ) )
18	13, 17	eqtrd 2656	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> { x e. CC | ( x - N ) e. ( ZZ>= ` ( M - N ) ) } = ( ZZ>= ` M ) )
19	18	fneq2d 5982	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( seq ( M - N ) ( .+ , F ) shift N ) Fn { x e. CC | ( x - N ) e. ( ZZ>= ` ( M - N ) ) } <-> ( seq ( M - N ) ( .+ , F ) shift N ) Fn ( ZZ>= ` M ) ) )
20	10, 19	mpbid 222	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( seq ( M - N ) ( .+ , F ) shift N ) Fn ( ZZ>= ` M ) )
21		negsub 10329	. . . . . . 7 |- ( ( M e. CC /\ N e. CC ) -> ( M + -u N ) = ( M - N ) )
22	14, 6, 21	syl2an 494	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M + -u N ) = ( M - N ) )
23	22	adantr 481	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ z e. ( ZZ>= ` M ) ) -> ( M + -u N ) = ( M - N ) )
24	23	seqeq1d 12807	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ z e. ( ZZ>= ` M ) ) -> seq ( M + -u N ) ( .+ , F ) = seq ( M - N ) ( .+ , F ) )
25		eluzelcn 11699	. . . . 5 |- ( z e. ( ZZ>= ` M ) -> z e. CC )
26		negsub 10329	. . . . 5 |- ( ( z e. CC /\ N e. CC ) -> ( z + -u N ) = ( z - N ) )
27	25, 7, 26	syl2anr 495	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ z e. ( ZZ>= ` M ) ) -> ( z + -u N ) = ( z - N ) )
28	24, 27	fveq12d 6197	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ z e. ( ZZ>= ` M ) ) -> ( seq ( M + -u N ) ( .+ , F ) ` ( z + -u N ) ) = ( seq ( M - N ) ( .+ , F ) ` ( z - N ) ) )
29		simpr 477	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ z e. ( ZZ>= ` M ) ) -> z e. ( ZZ>= ` M ) )
30		znegcl 11412	. . . . 5 |- ( N e. ZZ -> -u N e. ZZ )
31	30	ad2antlr 763	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ z e. ( ZZ>= ` M ) ) -> -u N e. ZZ )
32		elfzelz 12342	. . . . . . 7 |- ( y e. ( M ... z ) -> y e. ZZ )
33	32	zcnd 11483	. . . . . 6 |- ( y e. ( M ... z ) -> y e. CC )
34		seqshft.1	. . . . . . . 8 |- F e. _V
35	34	shftval 13814	. . . . . . 7 |- ( ( N e. CC /\ y e. CC ) -> ( ( F shift N ) ` y ) = ( F ` ( y - N ) ) )
36		negsub 10329	. . . . . . . . 9 |- ( ( y e. CC /\ N e. CC ) -> ( y + -u N ) = ( y - N ) )
37	36	ancoms 469	. . . . . . . 8 |- ( ( N e. CC /\ y e. CC ) -> ( y + -u N ) = ( y - N ) )
38	37	fveq2d 6195	. . . . . . 7 |- ( ( N e. CC /\ y e. CC ) -> ( F ` ( y + -u N ) ) = ( F ` ( y - N ) ) )
39	35, 38	eqtr4d 2659	. . . . . 6 |- ( ( N e. CC /\ y e. CC ) -> ( ( F shift N ) ` y ) = ( F ` ( y + -u N ) ) )
40	6, 33, 39	syl2an 494	. . . . 5 |- ( ( N e. ZZ /\ y e. ( M ... z ) ) -> ( ( F shift N ) ` y ) = ( F ` ( y + -u N ) ) )
41	40	ad4ant24 1298	. . . 4 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ z e. ( ZZ>= ` M ) ) /\ y e. ( M ... z ) ) -> ( ( F shift N ) ` y ) = ( F ` ( y + -u N ) ) )
42	29, 31, 41	seqshft2 12827	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ z e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , ( F shift N ) ) ` z ) = ( seq ( M + -u N ) ( .+ , F ) ` ( z + -u N ) ) )
43	8	shftval 13814	. . . 4 |- ( ( N e. CC /\ z e. CC ) -> ( ( seq ( M - N ) ( .+ , F ) shift N ) ` z ) = ( seq ( M - N ) ( .+ , F ) ` ( z - N ) ) )
44	7, 25, 43	syl2an 494	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ z e. ( ZZ>= ` M ) ) -> ( ( seq ( M - N ) ( .+ , F ) shift N ) ` z ) = ( seq ( M - N ) ( .+ , F ) ` ( z - N ) ) )
45	28, 42, 44	3eqtr4d 2666	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ z e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , ( F shift N ) ) ` z ) = ( ( seq ( M - N ) ( .+ , F ) shift N ) ` z ) )
46	2, 20, 45	eqfnfvd 6314	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> seq M ( .+ , ( F shift N ) ) = ( seq ( M - N ) ( .+ , F ) shift N ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   CCcc 9934   + caddc 9939   - cmin 10266   -ucneg 10267   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   shift cshi 13806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-shft 13807

This theorem is referenced by:  isershft  14394  cvgrat  14615  eftlub  14839  dvradcnv2  38546  binomcxplemnotnn0  38555