Theorem iseqz 9469

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.)

Hypotheses

Ref	Expression
iseqhomo.1	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
iseqhomo.2	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
iseqhomo.s	|- ( ph -> S e. V )
iseqz.3	|- ( ( ph /\ x e. S ) -> ( Z .+ x ) = Z )
iseqz.4	|- ( ( ph /\ x e. S ) -> ( x .+ Z ) = Z )
iseqz.5	|- ( ph -> K e. ( M ... N ) )
iseqz.6	|- ( ph -> N e. V )
iseqz.7	|- ( ph -> ( F ` K ) = Z )

Assertion

Ref	Expression
iseqz	|- ( ph -> ( seq M ( .+ , F , S ) ` 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

Allowed substitution hints:   V( x, y)

Proof of Theorem iseqz

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

Step	Hyp	Ref	Expression
1		iseqz.5	. . 3 |- ( ph -> K e. ( M ... N ) )
2		elfzuz3 9042	. . 3 |- ( K e. ( M ... N ) -> N e. ( ZZ>= ` K ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( ZZ>= ` K ) )
4		fveq2 5198	. . . . 5 |- ( w = K -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` K ) )
5	4	eqeq1d 2089	. . . 4 |- ( w = K -> ( ( seq M ( .+ , F , S ) ` w ) = Z <-> ( seq M ( .+ , F , S ) ` K ) = Z ) )
6	5	imbi2d 228	. . 3 |- ( w = K -> ( ( ph -> ( seq M ( .+ , F , S ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F , S ) ` K ) = Z ) ) )
7		fveq2 5198	. . . . 5 |- ( w = k -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` k ) )
8	7	eqeq1d 2089	. . . 4 |- ( w = k -> ( ( seq M ( .+ , F , S ) ` w ) = Z <-> ( seq M ( .+ , F , S ) ` k ) = Z ) )
9	8	imbi2d 228	. . 3 |- ( w = k -> ( ( ph -> ( seq M ( .+ , F , S ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F , S ) ` k ) = Z ) ) )
10		fveq2 5198	. . . . 5 |- ( w = ( k + 1 ) -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` ( k + 1 ) ) )
11	10	eqeq1d 2089	. . . 4 |- ( w = ( k + 1 ) -> ( ( seq M ( .+ , F , S ) ` w ) = Z <-> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z ) )
12	11	imbi2d 228	. . 3 |- ( w = ( k + 1 ) -> ( ( ph -> ( seq M ( .+ , F , S ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z ) ) )
13		fveq2 5198	. . . . 5 |- ( w = N -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` N ) )
14	13	eqeq1d 2089	. . . 4 |- ( w = N -> ( ( seq M ( .+ , F , S ) ` w ) = Z <-> ( seq M ( .+ , F , S ) ` N ) = Z ) )
15	14	imbi2d 228	. . 3 |- ( w = N -> ( ( ph -> ( seq M ( .+ , F , S ) ` w ) = Z ) <-> ( ph -> ( seq M ( .+ , F , S ) ` N ) = Z ) ) )
16		elfzuz 9041	. . . . . . . . . 10 |- ( K e. ( M ... N ) -> K e. ( ZZ>= ` M ) )
17	1, 16	syl 14	. . . . . . . . 9 |- ( ph -> K e. ( ZZ>= ` M ) )
18		eluzelz 8628	. . . . . . . . 9 |- ( K e. ( ZZ>= ` M ) -> K e. ZZ )
19	17, 18	syl 14	. . . . . . . 8 |- ( ph -> K e. ZZ )
20		iseqhomo.s	. . . . . . . 8 |- ( ph -> S e. V )
21		simpr 108	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> x e. ( ZZ>= ` K ) )
22	17	adantr 270	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> K e. ( ZZ>= ` M ) )
23		uztrn 8635	. . . . . . . . . 10 |- ( ( x e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> x e. ( ZZ>= ` M ) )
24	21, 22, 23	syl2anc 403	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> x e. ( ZZ>= ` M ) )
25		iseqhomo.2	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
26	24, 25	syldan 276	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( F ` x ) e. S )
27		iseqhomo.1	. . . . . . . 8 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
28	19, 20, 26, 27	iseq1 9442	. . . . . . 7 |- ( ph -> ( seq K ( .+ , F , S ) ` K ) = ( F ` K ) )
29		iseqz.7	. . . . . . 7 |- ( ph -> ( F ` K ) = Z )
30	28, 29	eqtrd 2113	. . . . . 6 |- ( ph -> ( seq K ( .+ , F , S ) ` K ) = Z )
31		iseqeq1 9434	. . . . . . . 8 |- ( K = M -> seq K ( .+ , F , S ) = seq M ( .+ , F , S ) )
32	31	fveq1d 5200	. . . . . . 7 |- ( K = M -> ( seq K ( .+ , F , S ) ` K ) = ( seq M ( .+ , F , S ) ` K ) )
33	32	eqeq1d 2089	. . . . . 6 |- ( K = M -> ( ( seq K ( .+ , F , S ) ` K ) = Z <-> ( seq M ( .+ , F , S ) ` K ) = Z ) )
34	30, 33	syl5ibcom 153	. . . . 5 |- ( ph -> ( K = M -> ( seq M ( .+ , F , S ) ` K ) = Z ) )
35		eluzel2 8624	. . . . . . . . . 10 |- ( K e. ( ZZ>= ` M ) -> M e. ZZ )
36	17, 35	syl 14	. . . . . . . . 9 |- ( ph -> M e. ZZ )
37	36	adantr 270	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> M e. ZZ )
38		simpr 108	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> K e. ( ZZ>= ` ( M + 1 ) ) )
39	20	adantr 270	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> S e. V )
40	25	adantlr 460	. . . . . . . 8 |- ( ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
41	27	adantlr 460	. . . . . . . 8 |- ( ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
42	37, 38, 39, 40, 41	iseqm1 9447	. . . . . . 7 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F , S ) ` K ) = ( ( seq M ( .+ , F , S ) ` ( K - 1 ) ) .+ ( F ` K ) ) )
43	29	adantr 270	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` K ) = Z )
44	43	oveq2d 5548	. . . . . . 7 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F , S ) ` ( K - 1 ) ) .+ ( F ` K ) ) = ( ( seq M ( .+ , F , S ) ` ( K - 1 ) ) .+ Z ) )
45		oveq1 5539	. . . . . . . . 9 |- ( x = ( seq M ( .+ , F , S ) ` ( K - 1 ) ) -> ( x .+ Z ) = ( ( seq M ( .+ , F , S ) ` ( K - 1 ) ) .+ Z ) )
46	45	eqeq1d 2089	. . . . . . . 8 |- ( x = ( seq M ( .+ , F , S ) ` ( K - 1 ) ) -> ( ( x .+ Z ) = Z <-> ( ( seq M ( .+ , F , S ) ` ( K - 1 ) ) .+ Z ) = Z ) )
47		iseqz.4	. . . . . . . . . 10 |- ( ( ph /\ x e. S ) -> ( x .+ Z ) = Z )
48	47	ralrimiva 2434	. . . . . . . . 9 |- ( ph -> A. x e. S ( x .+ Z ) = Z )
49	48	adantr 270	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> A. x e. S ( x .+ Z ) = Z )
50		eluzp1m1 8642	. . . . . . . . . 10 |- ( ( M e. ZZ /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( K - 1 ) e. ( ZZ>= ` M ) )
51	36, 50	sylan 277	. . . . . . . . 9 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( K - 1 ) e. ( ZZ>= ` M ) )
52	51, 39, 40, 41	iseqcl 9443	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F , S ) ` ( K - 1 ) ) e. S )
53	46, 49, 52	rspcdva 2707	. . . . . . 7 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F , S ) ` ( K - 1 ) ) .+ Z ) = Z )
54	42, 44, 53	3eqtrd 2117	. . . . . 6 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F , S ) ` K ) = Z )
55	54	ex 113	. . . . 5 |- ( ph -> ( K e. ( ZZ>= ` ( M + 1 ) ) -> ( seq M ( .+ , F , S ) ` K ) = Z ) )
56		uzp1 8652	. . . . . 6 |- ( K e. ( ZZ>= ` M ) -> ( K = M \/ K e. ( ZZ>= ` ( M + 1 ) ) ) )
57	17, 56	syl 14	. . . . 5 |- ( ph -> ( K = M \/ K e. ( ZZ>= ` ( M + 1 ) ) ) )
58	34, 55, 57	mpjaod 670	. . . 4 |- ( ph -> ( seq M ( .+ , F , S ) ` K ) = Z )
59	58	a1i 9	. . 3 |- ( K e. ZZ -> ( ph -> ( seq M ( .+ , F , S ) ` K ) = Z ) )
60		simpr 108	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> k e. ( ZZ>= ` K ) )
61	17	adantr 270	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> K e. ( ZZ>= ` M ) )
62		uztrn 8635	. . . . . . . . . 10 |- ( ( k e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
63	60, 61, 62	syl2anc 403	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> k e. ( ZZ>= ` M ) )
64	20	adantr 270	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> S e. V )
65	25	adantlr 460	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
66	27	adantlr 460	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
67	63, 64, 65, 66	iseqp1 9445	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
68	67	adantr 270	. . . . . . 7 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
69		simpr 108	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( seq M ( .+ , F , S ) ` k ) = Z )
70	69	oveq1d 5547	. . . . . . 7 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) = ( Z .+ ( F ` ( k + 1 ) ) ) )
71		oveq2 5540	. . . . . . . . . 10 |- ( x = ( F ` ( k + 1 ) ) -> ( Z .+ x ) = ( Z .+ ( F ` ( k + 1 ) ) ) )
72	71	eqeq1d 2089	. . . . . . . . 9 |- ( x = ( F ` ( k + 1 ) ) -> ( ( Z .+ x ) = Z <-> ( Z .+ ( F ` ( k + 1 ) ) ) = Z ) )
73		iseqz.3	. . . . . . . . . . 11 |- ( ( ph /\ x e. S ) -> ( Z .+ x ) = Z )
74	73	ralrimiva 2434	. . . . . . . . . 10 |- ( ph -> A. x e. S ( Z .+ x ) = Z )
75	74	adantr 270	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> A. x e. S ( Z .+ x ) = Z )
76		fveq2 5198	. . . . . . . . . . 11 |- ( x = ( k + 1 ) -> ( F ` x ) = ( F ` ( k + 1 ) ) )
77	76	eleq1d 2147	. . . . . . . . . 10 |- ( x = ( k + 1 ) -> ( ( F ` x ) e. S <-> ( F ` ( k + 1 ) ) e. S ) )
78	25	ralrimiva 2434	. . . . . . . . . . 11 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
79	78	adantr 270	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
80		peano2uz 8671	. . . . . . . . . . 11 |- ( k e. ( ZZ>= ` M ) -> ( k + 1 ) e. ( ZZ>= ` M ) )
81	63, 80	syl 14	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( k + 1 ) e. ( ZZ>= ` M ) )
82	77, 79, 81	rspcdva 2707	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( F ` ( k + 1 ) ) e. S )
83	72, 75, 82	rspcdva 2707	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( Z .+ ( F ` ( k + 1 ) ) ) = Z )
84	83	adantr 270	. . . . . . 7 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( Z .+ ( F ` ( k + 1 ) ) ) = Z )
85	68, 70, 84	3eqtrd 2117	. . . . . 6 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z )
86	85	ex 113	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( ( seq M ( .+ , F , S ) ` k ) = Z -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z ) )
87	86	expcom 114	. . . 4 |- ( k e. ( ZZ>= ` K ) -> ( ph -> ( ( seq M ( .+ , F , S ) ` k ) = Z -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z ) ) )
88	87	a2d 26	. . 3 |- ( k e. ( ZZ>= ` K ) -> ( ( ph -> ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( ph -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z ) ) )
89	6, 9, 12, 15, 59, 88	uzind4 8676	. 2 |- ( N e. ( ZZ>= ` K ) -> ( ph -> ( seq M ( .+ , F , S ) ` N ) = Z ) )
90	3, 89	mpcom 36	1 |- ( ph -> ( seq M ( .+ , F , S ) ` N ) = Z )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 661   = wceq 1284   e. wcel 1433   A.wral 2348   ` cfv 4922  (class class class)co 5532   1c1 6982   + caddc 6984   - cmin 7279   ZZcz 8351   ZZ>=cuz 8619   ...cfz 9029   seqcseq 9431

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-fz 9030  df-iseq 9432

This theorem is referenced by:  ibcval5  9690