Theorem seq3id2 10791

Description: The last few partial sums of a sequence that ends with all zeroes (or any element which is a right-identity for .+) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 12-Nov-2022.)

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 )
seq3id2.f	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
seq3id2.cl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )

Assertion

Ref	Expression
seq3id2	|- ( ph -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` N ) )

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

Allowed substitution hint:   Z( y)

Proof of Theorem seq3id2

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqid2.3	. . 3 |- ( ph -> N e. ( ZZ>= ` K ) )
2		eluzfz2 10269	. . 3 |- ( N e. ( ZZ>= ` K ) -> N e. ( K ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( K ... N ) )
4		eleq1 2293	. . . . . 6 |- ( x = K -> ( x e. ( K ... N ) <-> K e. ( K ... N ) ) )
5		fveq2 5639	. . . . . . 7 |- ( x = K -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` K ) )
6	5	eqeq2d 2242	. . . . . 6 |- ( x = K -> ( ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) <-> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` K ) ) )
7	4, 6	imbi12d 234	. . . . 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 230	. . . 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 2293	. . . . . 6 |- ( x = n -> ( x e. ( K ... N ) <-> n e. ( K ... N ) ) )
10		fveq2 5639	. . . . . . 7 |- ( x = n -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` n ) )
11	10	eqeq2d 2242	. . . . . 6 |- ( x = n -> ( ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) <-> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` n ) ) )
12	9, 11	imbi12d 234	. . . . 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 230	. . . 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 2293	. . . . . 6 |- ( x = ( n + 1 ) -> ( x e. ( K ... N ) <-> ( n + 1 ) e. ( K ... N ) ) )
15		fveq2 5639	. . . . . . 7 |- ( x = ( n + 1 ) -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` ( n + 1 ) ) )
16	15	eqeq2d 2242	. . . . . 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 234	. . . . 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 230	. . . 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 2293	. . . . . 6 |- ( x = N -> ( x e. ( K ... N ) <-> N e. ( K ... N ) ) )
20		fveq2 5639	. . . . . . 7 |- ( x = N -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` N ) )
21	20	eqeq2d 2242	. . . . . 6 |- ( x = N -> ( ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) <-> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` N ) ) )
22	19, 21	imbi12d 234	. . . . 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 230	. . . 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 2231	. . . . 5 |- ( K e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` K ) )
25	24	2a1i 27	. . . 4 |- ( K e. ZZ -> ( ph -> ( K e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` K ) ) ) )
26		peano2fzr 10274	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) -> n e. ( K ... N ) )
27	26	adantl 277	. . . . . . 7 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> n e. ( K ... N ) )
28	27	expr 375	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` K ) ) -> ( ( n + 1 ) e. ( K ... N ) -> n e. ( K ... N ) ) )
29	28	imim1d 75	. . . . 5 |- ( ( 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 6027	. . . . . 6 |- ( ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` n ) -> ( ( seq M ( .+ , F ) ` K ) .+ ( F ` ( n + 1 ) ) ) = ( ( seq M ( .+ , F ) ` n ) .+ ( F ` ( n + 1 ) ) ) )
31		fveqeq2 5648	. . . . . . . . . 10 |- ( x = ( n + 1 ) -> ( ( F ` x ) = Z <-> ( F ` ( n + 1 ) ) = Z ) )
32		seqid2.5	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> ( F ` x ) = Z )
33	32	ralrimiva 2604	. . . . . . . . . . 11 |- ( ph -> A. x e. ( ( K + 1 ) ... N ) ( F ` x ) = Z )
34	33	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> A. x e. ( ( K + 1 ) ... N ) ( F ` x ) = Z )
35		eluzp1p1 9784	. . . . . . . . . . . 12 |- ( n e. ( ZZ>= ` K ) -> ( n + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
36	35	ad2antrl 490	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( n + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
37		elfzuz3 10259	. . . . . . . . . . . 12 |- ( ( n + 1 ) e. ( K ... N ) -> N e. ( ZZ>= ` ( n + 1 ) ) )
38	37	ad2antll 491	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> N e. ( ZZ>= ` ( n + 1 ) ) )
39		elfzuzb 10256	. . . . . . . . . . 11 |- ( ( n + 1 ) e. ( ( K + 1 ) ... N ) <-> ( ( n + 1 ) e. ( ZZ>= ` ( K + 1 ) ) /\ N e. ( ZZ>= ` ( n + 1 ) ) ) )
40	36, 38, 39	sylanbrc 417	. . . . . . . . . 10 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( n + 1 ) e. ( ( K + 1 ) ... N ) )
41	31, 34, 40	rspcdva 2914	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( F ` ( n + 1 ) ) = Z )
42	41	oveq2d 6036	. . . . . . . 8 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( ( seq M ( .+ , F ) ` K ) .+ ( F ` ( n + 1 ) ) ) = ( ( seq M ( .+ , F ) ` K ) .+ Z ) )
43		oveq1 6027	. . . . . . . . . . 11 |- ( x = ( seq M ( .+ , F ) ` K ) -> ( x .+ Z ) = ( ( seq M ( .+ , F ) ` K ) .+ Z ) )
44		id 19	. . . . . . . . . . 11 |- ( x = ( seq M ( .+ , F ) ` K ) -> x = ( seq M ( .+ , F ) ` K ) )
45	43, 44	eqeq12d 2245	. . . . . . . . . 10 |- ( x = ( seq M ( .+ , F ) ` K ) -> ( ( x .+ Z ) = x <-> ( ( seq M ( .+ , F ) ` K ) .+ Z ) = ( seq M ( .+ , F ) ` K ) ) )
46		seqid2.1	. . . . . . . . . . 11 |- ( ( ph /\ x e. S ) -> ( x .+ Z ) = x )
47	46	ralrimiva 2604	. . . . . . . . . 10 |- ( ph -> A. x e. S ( x .+ Z ) = x )
48		seqid2.4	. . . . . . . . . 10 |- ( ph -> ( seq M ( .+ , F ) ` K ) e. S )
49	45, 47, 48	rspcdva 2914	. . . . . . . . 9 |- ( ph -> ( ( seq M ( .+ , F ) ` K ) .+ Z ) = ( seq M ( .+ , F ) ` K ) )
50	49	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( ( seq M ( .+ , F ) ` K ) .+ Z ) = ( seq M ( .+ , F ) ` K ) )
51	42, 50	eqtr2d 2264	. . . . . . 7 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( seq M ( .+ , F ) ` K ) = ( ( seq M ( .+ , F ) ` K ) .+ ( F ` ( n + 1 ) ) ) )
52		simprl 531	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> n e. ( ZZ>= ` K ) )
53		seqid2.2	. . . . . . . . . 10 |- ( ph -> K e. ( ZZ>= ` M ) )
54	53	adantr 276	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> K e. ( ZZ>= ` M ) )
55		uztrn 9775	. . . . . . . . 9 |- ( ( n e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> n e. ( ZZ>= ` M ) )
56	52, 54, 55	syl2anc 411	. . . . . . . 8 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> n e. ( ZZ>= ` M ) )
57		seq3id2.f	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
58	57	adantlr 477	. . . . . . . 8 |- ( ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
59		seq3id2.cl	. . . . . . . . 9 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
60	59	adantlr 477	. . . . . . . 8 |- ( ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
61	56, 58, 60	seq3p1 10730	. . . . . . 7 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( seq M ( .+ , F ) ` ( n + 1 ) ) = ( ( seq M ( .+ , F ) ` n ) .+ ( F ` ( n + 1 ) ) ) )
62	51, 61	eqeq12d 2245	. . . . . 6 |- ( ( 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	imbitrrid 156	. . . . 5 |- ( ( 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	29, 63	animpimp2impd 561	. . . 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 ) ) ) ) ) )
65	8, 13, 18, 23, 25, 64	uzind4 9824	. . 3 |- ( N e. ( ZZ>= ` K ) -> ( ph -> ( N e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` N ) ) ) )
66	1, 65	mpcom 36	. 2 |- ( ph -> ( N e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` N ) ) )
67	3, 66	mpd 13	1 |- ( ph -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2201   A.wral 2509   ` cfv 5325  (class class class)co 6020   1c1 8035   + caddc 8037   ZZcz 9481   ZZ>=cuz 9757   ...cfz 10245   seqcseq 10712

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4203  ax-sep 4206  ax-nul 4214  ax-pow 4263  ax-pr 4298  ax-un 4529  ax-setind 4634  ax-iinf 4685  ax-cnex 8125  ax-resscn 8126  ax-1cn 8127  ax-1re 8128  ax-icn 8129  ax-addcl 8130  ax-addrcl 8131  ax-mulcl 8132  ax-addcom 8134  ax-addass 8136  ax-distr 8138  ax-i2m1 8139  ax-0lt1 8140  ax-0id 8142  ax-rnegex 8143  ax-cnre 8145  ax-pre-ltirr 8146  ax-pre-ltwlin 8147  ax-pre-lttrn 8148  ax-pre-ltadd 8150

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-int 3928  df-iun 3971  df-br 4088  df-opab 4150  df-mpt 4151  df-tr 4187  df-id 4389  df-iord 4462  df-on 4464  df-ilim 4465  df-suc 4467  df-iom 4688  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737  df-iota 5285  df-fun 5327  df-fn 5328  df-f 5329  df-f1 5330  df-fo 5331  df-f1o 5332  df-fv 5333  df-riota 5973  df-ov 6023  df-oprab 6024  df-mpo 6025  df-1st 6305  df-2nd 6306  df-recs 6473  df-frec 6559  df-pnf 8218  df-mnf 8219  df-xr 8220  df-ltxr 8221  df-le 8222  df-sub 8354  df-neg 8355  df-inn 9146  df-n0 9405  df-z 9482  df-uz 9758  df-fz 10246  df-seqfrec 10713

This theorem is referenced by:  seq3coll  11109  fsum3cvg  11959  fproddccvg  12153  lgsdilem2  15791