Theorem seq3z 10125

Description: If the operation .+ has an absorbing element Z (a.k.a. zero element), then any sequence containing a Z evaluates to Z. (Contributed by Mario Carneiro, 27-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)

Hypotheses

Ref	Expression
seq3homo.1	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
seq3homo.2	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
seqz.3	|- ( ( ph /\ x e. S ) -> ( Z .+ x ) = Z )
seqz.4	|- ( ( ph /\ x e. S ) -> ( x .+ Z ) = Z )
seqz.5	|- ( ph -> K e. ( M ... N ) )
seqz.7	|- ( ph -> ( F ` K ) = Z )

Assertion

Ref	Expression
seq3z	|- ( ph -> ( seq M ( .+ , F ) ` N ) = Z )

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

Proof of Theorem seq3z

Dummy variables k w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqz.5	. . 3 |- ( ph -> K e. ( M ... N ) )
2		elfzuz3 9644	. . 3 |- ( K e. ( M ... N ) -> N e. ( ZZ>= ` K ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( ZZ>= ` K ) )
4		fveqeq2 5362	. . . 4 |- ( w = K -> ( ( seq M ( .+ , F ) ` w ) = Z <-> ( seq M ( .+ , F ) ` K ) = Z ) )
5	4	imbi2d 229	. . 3 |- ( w = K -> ( ( ph -> ( seq M ( .+ , F ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F ) ` K ) = Z ) ) )
6		fveqeq2 5362	. . . 4 |- ( w = k -> ( ( seq M ( .+ , F ) ` w ) = Z <-> ( seq M ( .+ , F ) ` k ) = Z ) )
7	6	imbi2d 229	. . 3 |- ( w = k -> ( ( ph -> ( seq M ( .+ , F ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F ) ` k ) = Z ) ) )
8		fveqeq2 5362	. . . 4 |- ( w = ( k + 1 ) -> ( ( seq M ( .+ , F ) ` w ) = Z <-> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z ) )
9	8	imbi2d 229	. . 3 |- ( w = ( k + 1 ) -> ( ( ph -> ( seq M ( .+ , F ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z ) ) )
10		fveqeq2 5362	. . . 4 |- ( w = N -> ( ( seq M ( .+ , F ) ` w ) = Z <-> ( seq M ( .+ , F ) ` N ) = Z ) )
11	10	imbi2d 229	. . 3 |- ( w = N -> ( ( ph -> ( seq M ( .+ , F ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F ) ` N ) = Z ) ) )
12		elfzuz 9643	. . . . . . . . . 10 |- ( K e. ( M ... N ) -> K e. ( ZZ>= ` M ) )
13	1, 12	syl 14	. . . . . . . . 9 |- ( ph -> K e. ( ZZ>= ` M ) )
14		eluzelz 9185	. . . . . . . . 9 |- ( K e. ( ZZ>= ` M ) -> K e. ZZ )
15	13, 14	syl 14	. . . . . . . 8 |- ( ph -> K e. ZZ )
16		simpr 109	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> x e. ( ZZ>= ` K ) )
17	13	adantr 272	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> K e. ( ZZ>= ` M ) )
18		uztrn 9192	. . . . . . . . . 10 |- ( ( x e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> x e. ( ZZ>= ` M ) )
19	16, 17, 18	syl2anc 406	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> x e. ( ZZ>= ` M ) )
20		seq3homo.2	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
21	19, 20	syldan 278	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( F ` x ) e. S )
22		seq3homo.1	. . . . . . . 8 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
23	15, 21, 22	seq3-1 10074	. . . . . . 7 |- ( ph -> ( seq K ( .+ , F ) ` K ) = ( F ` K ) )
24		seqz.7	. . . . . . 7 |- ( ph -> ( F ` K ) = Z )
25	23, 24	eqtrd 2132	. . . . . 6 |- ( ph -> ( seq K ( .+ , F ) ` K ) = Z )
26		seqeq1 10062	. . . . . . . 8 |- ( K = M -> seq K ( .+ , F ) = seq M ( .+ , F ) )
27	26	fveq1d 5355	. . . . . . 7 |- ( K = M -> ( seq K ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` K ) )
28	27	eqeq1d 2108	. . . . . 6 |- ( K = M -> ( ( seq K ( .+ , F ) ` K ) = Z <-> ( seq M ( .+ , F ) ` K ) = Z ) )
29	25, 28	syl5ibcom 154	. . . . 5 |- ( ph -> ( K = M -> ( seq M ( .+ , F ) ` K ) = Z ) )
30		eluzel2 9181	. . . . . . . . . 10 |- ( K e. ( ZZ>= ` M ) -> M e. ZZ )
31	13, 30	syl 14	. . . . . . . . 9 |- ( ph -> M e. ZZ )
32	31	adantr 272	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> M e. ZZ )
33		simpr 109	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> K e. ( ZZ>= ` ( M + 1 ) ) )
34	20	adantlr 464	. . . . . . . 8 |- ( ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
35	22	adantlr 464	. . . . . . . 8 |- ( ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
36	32, 33, 34, 35	seq3m1 10082	. . . . . . 7 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` K ) = ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ ( F ` K ) ) )
37	24	adantr 272	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` K ) = Z )
38	37	oveq2d 5722	. . . . . . 7 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ ( F ` K ) ) = ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ Z ) )
39		oveq1 5713	. . . . . . . . 9 |- ( x = ( seq M ( .+ , F ) ` ( K - 1 ) ) -> ( x .+ Z ) = ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ Z ) )
40	39	eqeq1d 2108	. . . . . . . 8 |- ( x = ( seq M ( .+ , F ) ` ( K - 1 ) ) -> ( ( x .+ Z ) = Z <-> ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ Z ) = Z ) )
41		seqz.4	. . . . . . . . . 10 |- ( ( ph /\ x e. S ) -> ( x .+ Z ) = Z )
42	41	ralrimiva 2464	. . . . . . . . 9 |- ( ph -> A. x e. S ( x .+ Z ) = Z )
43	42	adantr 272	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> A. x e. S ( x .+ Z ) = Z )
44		eqid 2100	. . . . . . . . . 10 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
45	44, 32, 34, 35	seqf 10075	. . . . . . . . 9 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
46		eluzp1m1 9199	. . . . . . . . . 10 |- ( ( M e. ZZ /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( K - 1 ) e. ( ZZ>= ` M ) )
47	31, 46	sylan 279	. . . . . . . . 9 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( K - 1 ) e. ( ZZ>= ` M ) )
48	45, 47	ffvelrnd 5488	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` ( K - 1 ) ) e. S )
49	40, 43, 48	rspcdva 2749	. . . . . . 7 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ Z ) = Z )
50	36, 38, 49	3eqtrd 2136	. . . . . 6 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` K ) = Z )
51	50	ex 114	. . . . 5 |- ( ph -> ( K e. ( ZZ>= ` ( M + 1 ) ) -> ( seq M ( .+ , F ) ` K ) = Z ) )
52		uzp1 9209	. . . . . 6 |- ( K e. ( ZZ>= ` M ) -> ( K = M \/ K e. ( ZZ>= ` ( M + 1 ) ) ) )
53	13, 52	syl 14	. . . . 5 |- ( ph -> ( K = M \/ K e. ( ZZ>= ` ( M + 1 ) ) ) )
54	29, 51, 53	mpjaod 679	. . . 4 |- ( ph -> ( seq M ( .+ , F ) ` K ) = Z )
55	54	a1i 9	. . 3 |- ( K e. ZZ -> ( ph -> ( seq M ( .+ , F ) ` K ) = Z ) )
56		simpr 109	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> k e. ( ZZ>= ` K ) )
57	13	adantr 272	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> K e. ( ZZ>= ` M ) )
58		uztrn 9192	. . . . . . . . . 10 |- ( ( k e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
59	56, 57, 58	syl2anc 406	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> k e. ( ZZ>= ` M ) )
60	20	adantlr 464	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
61	22	adantlr 464	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
62	59, 60, 61	seq3p1 10076	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
63	62	adantr 272	. . . . . . 7 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
64		simpr 109	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( seq M ( .+ , F ) ` k ) = Z )
65	64	oveq1d 5721	. . . . . . 7 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( ( seq M ( .+ , F ) ` k ) .+ ( F ` ( k + 1 ) ) ) = ( Z .+ ( F ` ( k + 1 ) ) ) )
66		oveq2 5714	. . . . . . . . . 10 |- ( x = ( F ` ( k + 1 ) ) -> ( Z .+ x ) = ( Z .+ ( F ` ( k + 1 ) ) ) )
67	66	eqeq1d 2108	. . . . . . . . 9 |- ( x = ( F ` ( k + 1 ) ) -> ( ( Z .+ x ) = Z <-> ( Z .+ ( F ` ( k + 1 ) ) ) = Z ) )
68		seqz.3	. . . . . . . . . . 11 |- ( ( ph /\ x e. S ) -> ( Z .+ x ) = Z )
69	68	ralrimiva 2464	. . . . . . . . . 10 |- ( ph -> A. x e. S ( Z .+ x ) = Z )
70	69	adantr 272	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> A. x e. S ( Z .+ x ) = Z )
71		fveq2 5353	. . . . . . . . . . 11 |- ( x = ( k + 1 ) -> ( F ` x ) = ( F ` ( k + 1 ) ) )
72	71	eleq1d 2168	. . . . . . . . . 10 |- ( x = ( k + 1 ) -> ( ( F ` x ) e. S <-> ( F ` ( k + 1 ) ) e. S ) )
73	20	ralrimiva 2464	. . . . . . . . . . 11 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
74	73	adantr 272	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
75		peano2uz 9228	. . . . . . . . . . 11 |- ( k e. ( ZZ>= ` M ) -> ( k + 1 ) e. ( ZZ>= ` M ) )
76	59, 75	syl 14	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( k + 1 ) e. ( ZZ>= ` M ) )
77	72, 74, 76	rspcdva 2749	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( F ` ( k + 1 ) ) e. S )
78	67, 70, 77	rspcdva 2749	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( Z .+ ( F ` ( k + 1 ) ) ) = Z )
79	78	adantr 272	. . . . . . 7 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( Z .+ ( F ` ( k + 1 ) ) ) = Z )
80	63, 65, 79	3eqtrd 2136	. . . . . 6 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z )
81	80	ex 114	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( ( seq M ( .+ , F ) ` k ) = Z -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z ) )
82	81	expcom 115	. . . 4 |- ( k e. ( ZZ>= ` K ) -> ( ph -> ( ( seq M ( .+ , F ) ` k ) = Z -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z ) ) )
83	82	a2d 26	. . 3 |- ( k e. ( ZZ>= ` K ) -> ( ( ph -> ( seq M ( .+ , F ) ` k ) = Z ) -> ( ph -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z ) ) )
84	5, 7, 9, 11, 55, 83	uzind4 9233	. 2 |- ( N e. ( ZZ>= ` K ) -> ( ph -> ( seq M ( .+ , F ) ` N ) = Z ) )
85	3, 84	mpcom 36	1 |- ( ph -> ( seq M ( .+ , F ) ` N ) = Z )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 670   = wceq 1299   e. wcel 1448   A.wral 2375   ` cfv 5059  (class class class)co 5706   1c1 7501   + caddc 7503   - cmin 7804   ZZcz 8906   ZZ>=cuz 9176   ...cfz 9631   seqcseq 10059

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-ltadd 7611

This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-n0 8830  df-z 8907  df-uz 9177  df-fz 9632  df-seqfrec 10060

This theorem is referenced by:  bcval5  10350