Theorem seq3id2 10834

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 10312	. . 3 |- ( N e. ( ZZ>= ` K ) -> N e. ( K ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( K ... N ) )
4		eleq1 2294	. . . . . 6 |- ( x = K -> ( x e. ( K ... N ) <-> K e. ( K ... N ) ) )
5		fveq2 5648	. . . . . . 7 |- ( x = K -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` K ) )
6	5	eqeq2d 2243	. . . . . 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 2294	. . . . . 6 |- ( x = n -> ( x e. ( K ... N ) <-> n e. ( K ... N ) ) )
10		fveq2 5648	. . . . . . 7 |- ( x = n -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` n ) )
11	10	eqeq2d 2243	. . . . . 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 2294	. . . . . 6 |- ( x = ( n + 1 ) -> ( x e. ( K ... N ) <-> ( n + 1 ) e. ( K ... N ) ) )
15		fveq2 5648	. . . . . . 7 |- ( x = ( n + 1 ) -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` ( n + 1 ) ) )
16	15	eqeq2d 2243	. . . . . 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 2294	. . . . . 6 |- ( x = N -> ( x e. ( K ... N ) <-> N e. ( K ... N ) ) )
20		fveq2 5648	. . . . . . 7 |- ( x = N -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` N ) )
21	20	eqeq2d 2243	. . . . . 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 2232	. . . . 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 10317	. . . . . . . 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 6035	. . . . . 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 5657	. . . . . . . . . 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 2606	. . . . . . . . . . 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 9826	. . . . . . . . . . . 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 10302	. . . . . . . . . . . 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 10299	. . . . . . . . . . 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 2916	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( F ` ( n + 1 ) ) = Z )
42	41	oveq2d 6044	. . . . . . . 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 6035	. . . . . . . . . . 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 2246	. . . . . . . . . 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 2606	. . . . . . . . . 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 2916	. . . . . . . . 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 2265	. . . . . . 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 9817	. . . . . . . . 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 10773	. . . . . . 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 2246	. . . . . 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 9866	. . 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 1398   e. wcel 2202   A.wral 2511   ` cfv 5333  (class class class)co 6028   1c1 8076   + caddc 8078   ZZcz 9523   ZZ>=cuz 9799   ...cfz 10288   seqcseq 10755

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-inn 9186  df-n0 9445  df-z 9524  df-uz 9800  df-fz 10289  df-seqfrec 10756

This theorem is referenced by:  seq3coll  11152  fsum3cvg  12002  fproddccvg  12196  lgsdilem2  15838