Theorem seq3id2 10444

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 9967	. . 3 |- ( N e. ( ZZ>= ` K ) -> N e. ( K ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( K ... N ) )
4		eleq1 2229	. . . . . 6 |- ( x = K -> ( x e. ( K ... N ) <-> K e. ( K ... N ) ) )
5		fveq2 5486	. . . . . . 7 |- ( x = K -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` K ) )
6	5	eqeq2d 2177	. . . . . 6 |- ( x = K -> ( ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) <-> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` K ) ) )
7	4, 6	imbi12d 233	. . . . 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 229	. . . 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 2229	. . . . . 6 |- ( x = n -> ( x e. ( K ... N ) <-> n e. ( K ... N ) ) )
10		fveq2 5486	. . . . . . 7 |- ( x = n -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` n ) )
11	10	eqeq2d 2177	. . . . . 6 |- ( x = n -> ( ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) <-> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` n ) ) )
12	9, 11	imbi12d 233	. . . . 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 229	. . . 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 2229	. . . . . 6 |- ( x = ( n + 1 ) -> ( x e. ( K ... N ) <-> ( n + 1 ) e. ( K ... N ) ) )
15		fveq2 5486	. . . . . . 7 |- ( x = ( n + 1 ) -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` ( n + 1 ) ) )
16	15	eqeq2d 2177	. . . . . 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 233	. . . . 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 229	. . . 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 2229	. . . . . 6 |- ( x = N -> ( x e. ( K ... N ) <-> N e. ( K ... N ) ) )
20		fveq2 5486	. . . . . . 7 |- ( x = N -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` N ) )
21	20	eqeq2d 2177	. . . . . 6 |- ( x = N -> ( ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` x ) <-> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` N ) ) )
22	19, 21	imbi12d 233	. . . . 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 229	. . . 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 2166	. . . . 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 9972	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) -> n e. ( K ... N ) )
27	26	adantl 275	. . . . . . 7 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> n e. ( K ... N ) )
28	27	expr 373	. . . . . 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 5849	. . . . . 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 5495	. . . . . . . . . 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 2539	. . . . . . . . . . 11 |- ( ph -> A. x e. ( ( K + 1 ) ... N ) ( F ` x ) = Z )
34	33	adantr 274	. . . . . . . . . 10 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> A. x e. ( ( K + 1 ) ... N ) ( F ` x ) = Z )
35		eluzp1p1 9491	. . . . . . . . . . . 12 |- ( n e. ( ZZ>= ` K ) -> ( n + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
36	35	ad2antrl 482	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( n + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
37		elfzuz3 9957	. . . . . . . . . . . 12 |- ( ( n + 1 ) e. ( K ... N ) -> N e. ( ZZ>= ` ( n + 1 ) ) )
38	37	ad2antll 483	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> N e. ( ZZ>= ` ( n + 1 ) ) )
39		elfzuzb 9954	. . . . . . . . . . 11 |- ( ( n + 1 ) e. ( ( K + 1 ) ... N ) <-> ( ( n + 1 ) e. ( ZZ>= ` ( K + 1 ) ) /\ N e. ( ZZ>= ` ( n + 1 ) ) ) )
40	36, 38, 39	sylanbrc 414	. . . . . . . . . 10 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( n + 1 ) e. ( ( K + 1 ) ... N ) )
41	31, 34, 40	rspcdva 2835	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( F ` ( n + 1 ) ) = Z )
42	41	oveq2d 5858	. . . . . . . 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 5849	. . . . . . . . . . 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 2180	. . . . . . . . . 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 2539	. . . . . . . . . 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 2835	. . . . . . . . 9 |- ( ph -> ( ( seq M ( .+ , F ) ` K ) .+ Z ) = ( seq M ( .+ , F ) ` K ) )
50	49	adantr 274	. . . . . . . 8 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( ( seq M ( .+ , F ) ` K ) .+ Z ) = ( seq M ( .+ , F ) ` K ) )
51	42, 50	eqtr2d 2199	. . . . . . 7 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( seq M ( .+ , F ) ` K ) = ( ( seq M ( .+ , F ) ` K ) .+ ( F ` ( n + 1 ) ) ) )
52		simprl 521	. . . . . . . . 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 274	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> K e. ( ZZ>= ` M ) )
55		uztrn 9482	. . . . . . . . 9 |- ( ( n e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> n e. ( ZZ>= ` M ) )
56	52, 54, 55	syl2anc 409	. . . . . . . 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 469	. . . . . . . 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 469	. . . . . . . 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 10397	. . . . . . 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 2180	. . . . . 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	syl5ibr 155	. . . . 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 549	. . . 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 9526	. . 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 103   = wceq 1343   e. wcel 2136   A.wral 2444   ` cfv 5188  (class class class)co 5842   1c1 7754   + caddc 7756   ZZcz 9191   ZZ>=cuz 9466   ...cfz 9944   seqcseq 10380

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-fz 9945  df-seqfrec 10381

This theorem is referenced by:  seq3coll  10755  fsum3cvg  11319  fproddccvg  11513  lgsdilem2  13577