Theorem seq3z 10914

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 10375	. . 3 |- ( K e. ( M ... N ) -> N e. ( ZZ>= ` K ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( ZZ>= ` K ) )
4		fveqeq2 5684	. . . 4 |- ( w = K -> ( ( seq M ( .+ , F ) ` w ) = Z <-> ( seq M ( .+ , F ) ` K ) = Z ) )
5	4	imbi2d 230	. . 3 |- ( w = K -> ( ( ph -> ( seq M ( .+ , F ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F ) ` K ) = Z ) ) )
6		fveqeq2 5684	. . . 4 |- ( w = k -> ( ( seq M ( .+ , F ) ` w ) = Z <-> ( seq M ( .+ , F ) ` k ) = Z ) )
7	6	imbi2d 230	. . 3 |- ( w = k -> ( ( ph -> ( seq M ( .+ , F ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F ) ` k ) = Z ) ) )
8		fveqeq2 5684	. . . 4 |- ( w = ( k + 1 ) -> ( ( seq M ( .+ , F ) ` w ) = Z <-> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z ) )
9	8	imbi2d 230	. . 3 |- ( w = ( k + 1 ) -> ( ( ph -> ( seq M ( .+ , F ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z ) ) )
10		fveqeq2 5684	. . . 4 |- ( w = N -> ( ( seq M ( .+ , F ) ` w ) = Z <-> ( seq M ( .+ , F ) ` N ) = Z ) )
11	10	imbi2d 230	. . 3 |- ( w = N -> ( ( ph -> ( seq M ( .+ , F ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F ) ` N ) = Z ) ) )
12		elfzuz 10374	. . . . . . . . . 10 |- ( K e. ( M ... N ) -> K e. ( ZZ>= ` M ) )
13	1, 12	syl 14	. . . . . . . . 9 |- ( ph -> K e. ( ZZ>= ` M ) )
14		eluzelz 9881	. . . . . . . . 9 |- ( K e. ( ZZ>= ` M ) -> K e. ZZ )
15	13, 14	syl 14	. . . . . . . 8 |- ( ph -> K e. ZZ )
16		simpr 110	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> x e. ( ZZ>= ` K ) )
17	13	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> K e. ( ZZ>= ` M ) )
18		uztrn 9889	. . . . . . . . . 10 |- ( ( x e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> x e. ( ZZ>= ` M ) )
19	16, 17, 18	syl2anc 411	. . . . . . . . 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 282	. . . . . . . 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 10848	. . . . . . 7 |- ( ph -> ( seq K ( .+ , F ) ` K ) = ( F ` K ) )
24		seqz.7	. . . . . . 7 |- ( ph -> ( F ` K ) = Z )
25	23, 24	eqtrd 2267	. . . . . 6 |- ( ph -> ( seq K ( .+ , F ) ` K ) = Z )
26		seqeq1 10836	. . . . . . . 8 |- ( K = M -> seq K ( .+ , F ) = seq M ( .+ , F ) )
27	26	fveq1d 5677	. . . . . . 7 |- ( K = M -> ( seq K ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` K ) )
28	27	eqeq1d 2243	. . . . . 6 |- ( K = M -> ( ( seq K ( .+ , F ) ` K ) = Z <-> ( seq M ( .+ , F ) ` K ) = Z ) )
29	25, 28	syl5ibcom 155	. . . . 5 |- ( ph -> ( K = M -> ( seq M ( .+ , F ) ` K ) = Z ) )
30		eluzel2 9876	. . . . . . . . . 10 |- ( K e. ( ZZ>= ` M ) -> M e. ZZ )
31	13, 30	syl 14	. . . . . . . . 9 |- ( ph -> M e. ZZ )
32	31	adantr 276	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> M e. ZZ )
33		simpr 110	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> K e. ( ZZ>= ` ( M + 1 ) ) )
34	20	adantlr 477	. . . . . . . 8 |- ( ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
35	22	adantlr 477	. . . . . . . 8 |- ( ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
36	32, 33, 34, 35	seq3m1 10859	. . . . . . 7 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` K ) = ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ ( F ` K ) ) )
37	24	adantr 276	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` K ) = Z )
38	37	oveq2d 6074	. . . . . . 7 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ ( F ` K ) ) = ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ Z ) )
39		oveq1 6065	. . . . . . . . 9 |- ( x = ( seq M ( .+ , F ) ` ( K - 1 ) ) -> ( x .+ Z ) = ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ Z ) )
40	39	eqeq1d 2243	. . . . . . . 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 2617	. . . . . . . . 9 |- ( ph -> A. x e. S ( x .+ Z ) = Z )
43	42	adantr 276	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> A. x e. S ( x .+ Z ) = Z )
44		eqid 2234	. . . . . . . . . 10 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
45	44, 32, 34, 35	seqf 10850	. . . . . . . . 9 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
46		eluzp1m1 9896	. . . . . . . . . 10 |- ( ( M e. ZZ /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( K - 1 ) e. ( ZZ>= ` M ) )
47	31, 46	sylan 283	. . . . . . . . 9 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( K - 1 ) e. ( ZZ>= ` M ) )
48	45, 47	ffvelcdmd 5818	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` ( K - 1 ) ) e. S )
49	40, 43, 48	rspcdva 2928	. . . . . . 7 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ Z ) = Z )
50	36, 38, 49	3eqtrd 2271	. . . . . 6 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` K ) = Z )
51	50	ex 115	. . . . 5 |- ( ph -> ( K e. ( ZZ>= ` ( M + 1 ) ) -> ( seq M ( .+ , F ) ` K ) = Z ) )
52		uzp1 9906	. . . . . 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 726	. . . 4 |- ( ph -> ( seq M ( .+ , F ) ` K ) = Z )
55	54	a1i 9	. . 3 |- ( K e. ZZ -> ( ph -> ( seq M ( .+ , F ) ` K ) = Z ) )
56		simpr 110	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> k e. ( ZZ>= ` K ) )
57	13	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> K e. ( ZZ>= ` M ) )
58		uztrn 9889	. . . . . . . . . 10 |- ( ( k e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
59	56, 57, 58	syl2anc 411	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> k e. ( ZZ>= ` M ) )
60	20	adantlr 477	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
61	22	adantlr 477	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
62	59, 60, 61	seq3p1 10851	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
63	62	adantr 276	. . . . . . 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 110	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( seq M ( .+ , F ) ` k ) = Z )
65	64	oveq1d 6073	. . . . . . 7 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( ( seq M ( .+ , F ) ` k ) .+ ( F ` ( k + 1 ) ) ) = ( Z .+ ( F ` ( k + 1 ) ) ) )
66		oveq2 6066	. . . . . . . . . 10 |- ( x = ( F ` ( k + 1 ) ) -> ( Z .+ x ) = ( Z .+ ( F ` ( k + 1 ) ) ) )
67	66	eqeq1d 2243	. . . . . . . . 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 2617	. . . . . . . . . 10 |- ( ph -> A. x e. S ( Z .+ x ) = Z )
70	69	adantr 276	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> A. x e. S ( Z .+ x ) = Z )
71		fveq2 5675	. . . . . . . . . . 11 |- ( x = ( k + 1 ) -> ( F ` x ) = ( F ` ( k + 1 ) ) )
72	71	eleq1d 2303	. . . . . . . . . 10 |- ( x = ( k + 1 ) -> ( ( F ` x ) e. S <-> ( F ` ( k + 1 ) ) e. S ) )
73	20	ralrimiva 2617	. . . . . . . . . . 11 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
74	73	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
75		peano2uz 9933	. . . . . . . . . . 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 2928	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( F ` ( k + 1 ) ) e. S )
78	67, 70, 77	rspcdva 2928	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( Z .+ ( F ` ( k + 1 ) ) ) = Z )
79	78	adantr 276	. . . . . . 7 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( Z .+ ( F ` ( k + 1 ) ) ) = Z )
80	63, 65, 79	3eqtrd 2271	. . . . . 6 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z )
81	80	ex 115	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( ( seq M ( .+ , F ) ` k ) = Z -> ( seq M ( .+ , F ) ` ( k + 1 ) ) = Z ) )
82	81	expcom 116	. . . 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 9938	. 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 104   \/ wo 716   = wceq 1398   e. wcel 2205   A.wral 2522   ` cfv 5357  (class class class)co 6058   1c1 8144   + caddc 8146   - cmin 8460   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361   seqcseq 10833

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-seqfrec 10834

This theorem is referenced by:  bcval5  11150  lgsne0  16037