Theorem seq3id2 10506

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 10029	. . 3 |- ( N e. ( ZZ>= ` K ) -> N e. ( K ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( K ... N ) )
4		eleq1 2240	. . . . . 6 |- ( x = K -> ( x e. ( K ... N ) <-> K e. ( K ... N ) ) )
5		fveq2 5515	. . . . . . 7 |- ( x = K -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` K ) )
6	5	eqeq2d 2189	. . . . . 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 2240	. . . . . 6 |- ( x = n -> ( x e. ( K ... N ) <-> n e. ( K ... N ) ) )
10		fveq2 5515	. . . . . . 7 |- ( x = n -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` n ) )
11	10	eqeq2d 2189	. . . . . 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 2240	. . . . . 6 |- ( x = ( n + 1 ) -> ( x e. ( K ... N ) <-> ( n + 1 ) e. ( K ... N ) ) )
15		fveq2 5515	. . . . . . 7 |- ( x = ( n + 1 ) -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` ( n + 1 ) ) )
16	15	eqeq2d 2189	. . . . . 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 2240	. . . . . 6 |- ( x = N -> ( x e. ( K ... N ) <-> N e. ( K ... N ) ) )
20		fveq2 5515	. . . . . . 7 |- ( x = N -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` N ) )
21	20	eqeq2d 2189	. . . . . 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 2178	. . . . 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 10034	. . . . . . . 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 5881	. . . . . 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 5524	. . . . . . . . . 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 2550	. . . . . . . . . . 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 9551	. . . . . . . . . . . 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 10019	. . . . . . . . . . . 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 10016	. . . . . . . . . . 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 2846	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( F ` ( n + 1 ) ) = Z )
42	41	oveq2d 5890	. . . . . . . 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 5881	. . . . . . . . . . 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 2192	. . . . . . . . . 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 2550	. . . . . . . . . 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 2846	. . . . . . . . 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 2211	. . . . . . 7 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( seq M ( .+ , F ) ` K ) = ( ( seq M ( .+ , F ) ` K ) .+ ( F ` ( n + 1 ) ) ) )
52		simprl 529	. . . . . . . . 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 9542	. . . . . . . . 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 10459	. . . . . . 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 2192	. . . . . 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 559	. . . 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 9586	. . 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 1353   e. wcel 2148   A.wral 2455   ` cfv 5216  (class class class)co 5874   1c1 7811   + caddc 7813   ZZcz 9251   ZZ>=cuz 9526   ...cfz 10006   seqcseq 10442

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-inn 8918  df-n0 9175  df-z 9252  df-uz 9527  df-fz 10007  df-seqfrec 10443

This theorem is referenced by:  seq3coll  10817  fsum3cvg  11381  fproddccvg  11575  lgsdilem2  14368