Theorem seqid2 12847

Description: The last few terms of a sequence that ends with all zeroes (or whatever the identity Z is for operation .+) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Hypotheses

Ref	Expression
seqid2.1	|- ( ( ph /\ x e. S ) -> ( x .+ Z ) = x )
seqid2.2	|- ( ph -> K e. ( ZZ>= ` M ) )
seqid2.3	|- ( ph -> N e. ( ZZ>= ` K ) )
seqid2.4	|- ( ph -> ( seq M ( .+ , F ) ` K ) e. S )
seqid2.5	|- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> ( F ` x ) = Z )

Assertion

Ref	Expression
seqid2	|- ( ph -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` N ) )

Distinct variable groups:   x, F   x, K   x, M   x, N   ph, x   x, S   x, .+   x, Z

Proof of Theorem seqid2

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqid2.3	. . 3 |- ( ph -> N e. ( ZZ>= ` K ) )
2		eluzfz2 12349	. . 3 |- ( N e. ( ZZ>= ` K ) -> N e. ( K ... N ) )
3	1, 2	syl 17	. 2 |- ( ph -> N e. ( K ... N ) )
4		eleq1 2689	. . . . . 6 |- ( x = K -> ( x e. ( K ... N ) <-> K e. ( K ... N ) ) )
5		fveq2 6191	. . . . . . 7 |- ( x = K -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` K ) )
6	5	eqeq2d 2632	. . . . . 6 |- ( x = K -> ( ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) <-> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` K ) ) )
7	4, 6	imbi12d 334	. . . . 5 |- ( x = K -> ( ( x e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) ) <-> ( K e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` K ) ) ) )
8	7	imbi2d 330	. . . 4 |- ( x = K -> ( ( ph -> ( x e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) ) ) <-> ( ph -> ( K e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` K ) ) ) ) )
9		eleq1 2689	. . . . . 6 |- ( x = n -> ( x e. ( K ... N ) <-> n e. ( K ... N ) ) )
10		fveq2 6191	. . . . . . 7 |- ( x = n -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` n ) )
11	10	eqeq2d 2632	. . . . . 6 |- ( x = n -> ( ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) <-> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` n ) ) )
12	9, 11	imbi12d 334	. . . . 5 |- ( x = n -> ( ( x e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) ) <-> ( n e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` n ) ) ) )
13	12	imbi2d 330	. . . 4 |- ( x = n -> ( ( ph -> ( x e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) ) ) <-> ( ph -> ( n e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` n ) ) ) ) )
14		eleq1 2689	. . . . . 6 |- ( x = ( n + 1 ) -> ( x e. ( K ... N ) <-> ( n + 1 ) e. ( K ... N ) ) )
15		fveq2 6191	. . . . . . 7 |- ( x = ( n + 1 ) -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` ( n + 1 ) ) )
16	15	eqeq2d 2632	. . . . . 6 |- ( x = ( n + 1 ) -> ( ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) <-> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` ( n + 1 ) ) ) )
17	14, 16	imbi12d 334	. . . . 5 |- ( x = ( n + 1 ) -> ( ( x e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) ) <-> ( ( n + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` ( n + 1 ) ) ) ) )
18	17	imbi2d 330	. . . 4 |- ( x = ( n + 1 ) -> ( ( ph -> ( x e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) ) ) <-> ( ph -> ( ( n + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` ( n + 1 ) ) ) ) ) )
19		eleq1 2689	. . . . . 6 |- ( x = N -> ( x e. ( K ... N ) <-> N e. ( K ... N ) ) )
20		fveq2 6191	. . . . . . 7 |- ( x = N -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` N ) )
21	20	eqeq2d 2632	. . . . . 6 |- ( x = N -> ( ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) <-> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` N ) ) )
22	19, 21	imbi12d 334	. . . . 5 |- ( x = N -> ( ( x e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) ) <-> ( N e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` N ) ) ) )
23	22	imbi2d 330	. . . 4 |- ( x = N -> ( ( ph -> ( x e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) ) ) <-> ( ph -> ( N e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` N ) ) ) ) )
24		eqidd 2623	. . . . 5 |- ( K e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` K ) )
25	24	2a1i 12	. . . 4 |- ( K e. ZZ -> ( ph -> ( K e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` K ) ) ) )
26		peano2fzr 12354	. . . . . . . . . 10 |- ( ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) -> n e. ( K ... N ) )
27	26	adantl 482	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> n e. ( K ... N ) )
28	27	expr 643	. . . . . . . 8 |- ( ( ph /\ n e. ( ZZ>= ` K ) ) -> ( ( n + 1 ) e. ( K ... N ) -> n e. ( K ... N ) ) )
29	28	imim1d 82	. . . . . . 7 |- ( ( ph /\ n e. ( ZZ>= ` K ) ) -> ( ( n e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` n ) ) -> ( ( n + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` n ) ) ) )
30		oveq1 6657	. . . . . . . . . 10 |- ( ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` n ) -> ( ( seq M ( .+ , F ) ` K ) .+ ( F ` ( n + 1 ) ) ) = ( ( seq M ( .+ , F ) ` n ) .+ ( F ` ( n + 1 ) ) ) )
31		eluzp1p1 11713	. . . . . . . . . . . . . . . 16 |- ( n e. ( ZZ>= ` K ) -> ( n + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
32	31	ad2antrl 764	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( n + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
33		elfzuz3 12339	. . . . . . . . . . . . . . . 16 |- ( ( n + 1 ) e. ( K ... N ) -> N e. ( ZZ>= ` ( n + 1 ) ) )
34	33	ad2antll 765	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> N e. ( ZZ>= ` ( n + 1 ) ) )
35		elfzuzb 12336	. . . . . . . . . . . . . . 15 |- ( ( n + 1 ) e. ( ( K + 1 ) ... N ) <-> ( ( n + 1 ) e. ( ZZ>= ` ( K + 1 ) ) /\ N e. ( ZZ>= ` ( n + 1 ) ) ) )
36	32, 34, 35	sylanbrc 698	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( n + 1 ) e. ( ( K + 1 ) ... N ) )
37		seqid2.5	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> ( F ` x ) = Z )
38	37	ralrimiva 2966	. . . . . . . . . . . . . . 15 |- ( ph -> A. x e. ( ( K + 1 ) ... N ) ( F ` x ) = Z )
39	38	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> A. x e. ( ( K + 1 ) ... N ) ( F ` x ) = Z )
40		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( x = ( n + 1 ) -> ( F ` x ) = ( F ` ( n + 1 ) ) )
41	40	eqeq1d 2624	. . . . . . . . . . . . . . 15 |- ( x = ( n + 1 ) -> ( ( F ` x ) = Z <-> ( F ` ( n + 1 ) ) = Z ) )
42	41	rspcv 3305	. . . . . . . . . . . . . 14 |- ( ( n + 1 ) e. ( ( K + 1 ) ... N ) -> ( A. x e. ( ( K + 1 ) ... N ) ( F ` x ) = Z -> ( F ` ( n + 1 ) ) = Z ) )
43	36, 39, 42	sylc 65	. . . . . . . . . . . . 13 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( F ` ( n + 1 ) ) = Z )
44	43	oveq2d 6666	. . . . . . . . . . . 12 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( ( seq M ( .+ , F ) ` K ) .+ ( F ` ( n + 1 ) ) ) = ( ( seq M ( .+ , F ) ` K ) .+ Z ) )
45		seqid2.4	. . . . . . . . . . . . . 14 |- ( ph -> ( seq M ( .+ , F ) ` K ) e. S )
46		seqid2.1	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. S ) -> ( x .+ Z ) = x )
47	46	ralrimiva 2966	. . . . . . . . . . . . . 14 |- ( ph -> A. x e. S ( x .+ Z ) = x )
48		oveq1 6657	. . . . . . . . . . . . . . . 16 |- ( x = ( seq M ( .+ , F ) ` K ) -> ( x .+ Z ) = ( ( seq M ( .+ , F ) ` K ) .+ Z ) )
49		id 22	. . . . . . . . . . . . . . . 16 |- ( x = ( seq M ( .+ , F ) ` K ) -> x = ( seq M ( .+ , F ) ` K ) )
50	48, 49	eqeq12d 2637	. . . . . . . . . . . . . . 15 |- ( x = ( seq M ( .+ , F ) ` K ) -> ( ( x .+ Z ) = x <-> ( ( seq M ( .+ , F ) ` K ) .+ Z ) = ( seq M ( .+ , F ) ` K ) ) )
51	50	rspcv 3305	. . . . . . . . . . . . . 14 |- ( ( seq M ( .+ , F ) ` K ) e. S -> ( A. x e. S ( x .+ Z ) = x -> ( ( seq M ( .+ , F ) ` K ) .+ Z ) = ( seq M ( .+ , F ) ` K ) ) )
52	45, 47, 51	sylc 65	. . . . . . . . . . . . 13 |- ( ph -> ( ( seq M ( .+ , F ) ` K ) .+ Z ) = ( seq M ( .+ , F ) ` K ) )
53	52	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( ( seq M ( .+ , F ) ` K ) .+ Z ) = ( seq M ( .+ , F ) ` K ) )
54	44, 53	eqtr2d 2657	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( seq M ( .+ , F ) ` K ) = ( ( seq M ( .+ , F ) ` K ) .+ ( F ` ( n + 1 ) ) ) )
55		simprl 794	. . . . . . . . . . . . 13 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> n e. ( ZZ>= ` K ) )
56		seqid2.2	. . . . . . . . . . . . . 14 |- ( ph -> K e. ( ZZ>= ` M ) )
57	56	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> K e. ( ZZ>= ` M ) )
58		uztrn 11704	. . . . . . . . . . . . 13 |- ( ( n e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> n e. ( ZZ>= ` M ) )
59	55, 57, 58	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> n e. ( ZZ>= ` M ) )
60		seqp1 12816	. . . . . . . . . . . 12 |- ( n e. ( ZZ>= ` M ) -> ( seq M ( .+ , F ) ` ( n + 1 ) ) = ( ( seq M ( .+ , F ) ` n ) .+ ( F ` ( n + 1 ) ) ) )
61	59, 60	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( seq M ( .+ , F ) ` ( n + 1 ) ) = ( ( seq M ( .+ , F ) ` n ) .+ ( F ` ( n + 1 ) ) ) )
62	54, 61	eqeq12d 2637	. . . . . . . . . 10 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` ( n + 1 ) ) <-> ( ( seq M ( .+ , F ) ` K ) .+ ( F ` ( n + 1 ) ) ) = ( ( seq M ( .+ , F ) ` n ) .+ ( F ` ( n + 1 ) ) ) ) )
63	30, 62	syl5ibr 236	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` n ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` ( n + 1 ) ) ) )
64	63	expr 643	. . . . . . . 8 |- ( ( ph /\ n e. ( ZZ>= ` K ) ) -> ( ( n + 1 ) e. ( K ... N ) -> ( ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` n ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` ( n + 1 ) ) ) ) )
65	64	a2d 29	. . . . . . 7 |- ( ( ph /\ n e. ( ZZ>= ` K ) ) -> ( ( ( n + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` n ) ) -> ( ( n + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` ( n + 1 ) ) ) ) )
66	29, 65	syld 47	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` K ) ) -> ( ( n e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` n ) ) -> ( ( n + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` ( n + 1 ) ) ) ) )
67	66	expcom 451	. . . . 5 |- ( n e. ( ZZ>= ` K ) -> ( ph -> ( ( n e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` n ) ) -> ( ( n + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` ( n + 1 ) ) ) ) ) )
68	67	a2d 29	. . . 4 |- ( n e. ( ZZ>= ` K ) -> ( ( ph -> ( n e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` n ) ) ) -> ( ph -> ( ( n + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` ( n + 1 ) ) ) ) ) )
69	8, 13, 18, 23, 25, 68	uzind4 11746	. . 3 |- ( N e. ( ZZ>= ` K ) -> ( ph -> ( N e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` N ) ) ) )
70	1, 69	mpcom 38	. 2 |- ( ph -> ( N e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` N ) ) )
71	3, 70	mpd 15	1 |- ( ph -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

This theorem is referenced by:  seqcoll  13248  seqcoll2  13249  fsumcvg  14443  fprodcvg  14660  ovolicc1  23284  lgsdilem2  25058