Theorem iseqz 9566

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 9118	. . 3 |- ( K e. ( M ... N ) -> N e. ( ZZ>= ` K ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( ZZ>= ` K ) )
4		fveq2 5209	. . . . 5 |- ( w = K -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` K ) )
5	4	eqeq1d 2090	. . . 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 5209	. . . . 5 |- ( w = k -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` k ) )
8	7	eqeq1d 2090	. . . 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 5209	. . . . 5 |- ( w = ( k + 1 ) -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` ( k + 1 ) ) )
11	10	eqeq1d 2090	. . . 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 5209	. . . . 5 |- ( w = N -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` N ) )
14	13	eqeq1d 2090	. . . 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 9117	. . . . . . . . . 10 |- ( K e. ( M ... N ) -> K e. ( ZZ>= ` M ) )
17	1, 16	syl 14	. . . . . . . . 9 |- ( ph -> K e. ( ZZ>= ` M ) )
18		eluzelz 8709	. . . . . . . . 9 |- ( K e. ( ZZ>= ` M ) -> K e. ZZ )
19	17, 18	syl 14	. . . . . . . 8 |- ( ph -> K e. ZZ )
20		simpr 108	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> x e. ( ZZ>= ` K ) )
21	17	adantr 270	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> K e. ( ZZ>= ` M ) )
22		uztrn 8716	. . . . . . . . . 10 |- ( ( x e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> x e. ( ZZ>= ` M ) )
23	20, 21, 22	syl2anc 403	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> x e. ( ZZ>= ` M ) )
24		iseqhomo.2	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
25	23, 24	syldan 276	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( F ` x ) e. S )
26		iseqhomo.1	. . . . . . . 8 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
27	19, 25, 26	iseq1 9533	. . . . . . 7 |- ( ph -> ( seq K ( .+ , F , S ) ` K ) = ( F ` K ) )
28		iseqz.7	. . . . . . 7 |- ( ph -> ( F ` K ) = Z )
29	27, 28	eqtrd 2114	. . . . . 6 |- ( ph -> ( seq K ( .+ , F , S ) ` K ) = Z )
30		iseqeq1 9524	. . . . . . . 8 |- ( K = M -> seq K ( .+ , F , S ) = seq M ( .+ , F , S ) )
31	30	fveq1d 5211	. . . . . . 7 |- ( K = M -> ( seq K ( .+ , F , S ) ` K ) = ( seq M ( .+ , F , S ) ` K ) )
32	31	eqeq1d 2090	. . . . . 6 |- ( K = M -> ( ( seq K ( .+ , F , S ) ` K ) = Z <-> ( seq M ( .+ , F , S ) ` K ) = Z ) )
33	29, 32	syl5ibcom 153	. . . . 5 |- ( ph -> ( K = M -> ( seq M ( .+ , F , S ) ` K ) = Z ) )
34		eluzel2 8705	. . . . . . . . . 10 |- ( K e. ( ZZ>= ` M ) -> M e. ZZ )
35	17, 34	syl 14	. . . . . . . . 9 |- ( ph -> M e. ZZ )
36	35	adantr 270	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> M e. ZZ )
37		simpr 108	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> K e. ( ZZ>= ` ( M + 1 ) ) )
38	24	adantlr 461	. . . . . . . 8 |- ( ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
39	26	adantlr 461	. . . . . . . 8 |- ( ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
40	36, 37, 38, 39	iseqm1 9543	. . . . . . 7 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F , S ) ` K ) = ( ( seq M ( .+ , F , S ) ` ( K - 1 ) ) .+ ( F ` K ) ) )
41	28	adantr 270	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` K ) = Z )
42	41	oveq2d 5559	. . . . . . 7 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F , S ) ` ( K - 1 ) ) .+ ( F ` K ) ) = ( ( seq M ( .+ , F , S ) ` ( K - 1 ) ) .+ Z ) )
43		oveq1 5550	. . . . . . . . 9 |- ( x = ( seq M ( .+ , F , S ) ` ( K - 1 ) ) -> ( x .+ Z ) = ( ( seq M ( .+ , F , S ) ` ( K - 1 ) ) .+ Z ) )
44	43	eqeq1d 2090	. . . . . . . 8 |- ( x = ( seq M ( .+ , F , S ) ` ( K - 1 ) ) -> ( ( x .+ Z ) = Z <-> ( ( seq M ( .+ , F , S ) ` ( K - 1 ) ) .+ Z ) = Z ) )
45		iseqz.4	. . . . . . . . . 10 |- ( ( ph /\ x e. S ) -> ( x .+ Z ) = Z )
46	45	ralrimiva 2435	. . . . . . . . 9 |- ( ph -> A. x e. S ( x .+ Z ) = Z )
47	46	adantr 270	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> A. x e. S ( x .+ Z ) = Z )
48		eluzp1m1 8723	. . . . . . . . . 10 |- ( ( M e. ZZ /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( K - 1 ) e. ( ZZ>= ` M ) )
49	35, 48	sylan 277	. . . . . . . . 9 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( K - 1 ) e. ( ZZ>= ` M ) )
50	49, 38, 39	iseqcl 9537	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F , S ) ` ( K - 1 ) ) e. S )
51	44, 47, 50	rspcdva 2708	. . . . . . 7 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F , S ) ` ( K - 1 ) ) .+ Z ) = Z )
52	40, 42, 51	3eqtrd 2118	. . . . . 6 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F , S ) ` K ) = Z )
53	52	ex 113	. . . . 5 |- ( ph -> ( K e. ( ZZ>= ` ( M + 1 ) ) -> ( seq M ( .+ , F , S ) ` K ) = Z ) )
54		uzp1 8733	. . . . . 6 |- ( K e. ( ZZ>= ` M ) -> ( K = M \/ K e. ( ZZ>= ` ( M + 1 ) ) ) )
55	17, 54	syl 14	. . . . 5 |- ( ph -> ( K = M \/ K e. ( ZZ>= ` ( M + 1 ) ) ) )
56	33, 53, 55	mpjaod 671	. . . 4 |- ( ph -> ( seq M ( .+ , F , S ) ` K ) = Z )
57	56	a1i 9	. . 3 |- ( K e. ZZ -> ( ph -> ( seq M ( .+ , F , S ) ` K ) = Z ) )
58		simpr 108	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> k e. ( ZZ>= ` K ) )
59	17	adantr 270	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> K e. ( ZZ>= ` M ) )
60		uztrn 8716	. . . . . . . . . 10 |- ( ( k e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
61	58, 59, 60	syl2anc 403	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> k e. ( ZZ>= ` M ) )
62	24	adantlr 461	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
63	26	adantlr 461	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
64	61, 62, 63	iseqp1 9538	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
65	64	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 ) ) ) )
66		simpr 108	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( seq M ( .+ , F , S ) ` k ) = Z )
67	66	oveq1d 5558	. . . . . . 7 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) = ( Z .+ ( F ` ( k + 1 ) ) ) )
68		oveq2 5551	. . . . . . . . . 10 |- ( x = ( F ` ( k + 1 ) ) -> ( Z .+ x ) = ( Z .+ ( F ` ( k + 1 ) ) ) )
69	68	eqeq1d 2090	. . . . . . . . 9 |- ( x = ( F ` ( k + 1 ) ) -> ( ( Z .+ x ) = Z <-> ( Z .+ ( F ` ( k + 1 ) ) ) = Z ) )
70		iseqz.3	. . . . . . . . . . 11 |- ( ( ph /\ x e. S ) -> ( Z .+ x ) = Z )
71	70	ralrimiva 2435	. . . . . . . . . 10 |- ( ph -> A. x e. S ( Z .+ x ) = Z )
72	71	adantr 270	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> A. x e. S ( Z .+ x ) = Z )
73		fveq2 5209	. . . . . . . . . . 11 |- ( x = ( k + 1 ) -> ( F ` x ) = ( F ` ( k + 1 ) ) )
74	73	eleq1d 2148	. . . . . . . . . 10 |- ( x = ( k + 1 ) -> ( ( F ` x ) e. S <-> ( F ` ( k + 1 ) ) e. S ) )
75	24	ralrimiva 2435	. . . . . . . . . . 11 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
76	75	adantr 270	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
77		peano2uz 8752	. . . . . . . . . . 11 |- ( k e. ( ZZ>= ` M ) -> ( k + 1 ) e. ( ZZ>= ` M ) )
78	61, 77	syl 14	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( k + 1 ) e. ( ZZ>= ` M ) )
79	74, 76, 78	rspcdva 2708	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( F ` ( k + 1 ) ) e. S )
80	69, 72, 79	rspcdva 2708	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( Z .+ ( F ` ( k + 1 ) ) ) = Z )
81	80	adantr 270	. . . . . . 7 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( Z .+ ( F ` ( k + 1 ) ) ) = Z )
82	65, 67, 81	3eqtrd 2118	. . . . . 6 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z )
83	82	ex 113	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( ( seq M ( .+ , F , S ) ` k ) = Z -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z ) )
84	83	expcom 114	. . . 4 |- ( k e. ( ZZ>= ` K ) -> ( ph -> ( ( seq M ( .+ , F , S ) ` k ) = Z -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z ) ) )
85	84	a2d 26	. . 3 |- ( k e. ( ZZ>= ` K ) -> ( ( ph -> ( seq M ( .+ , F , S ) ` k ) = Z ) -> ( ph -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = Z ) ) )
86	6, 9, 12, 15, 57, 85	uzind4 8757	. 2 |- ( N e. ( ZZ>= ` K ) -> ( ph -> ( seq M ( .+ , F , S ) ` N ) = Z ) )
87	3, 86	mpcom 36	1 |- ( ph -> ( seq M ( .+ , F , S ) ` N ) = Z )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 662   = wceq 1285   e. wcel 1434   A.wral 2349   ` cfv 4932  (class class class)co 5543   1c1 7044   + caddc 7046   - cmin 7346   ZZcz 8432   ZZ>=cuz 8700   ...cfz 9105   seqcseq 9521

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154

This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-fz 9106  df-iseq 9522

This theorem is referenced by:  ibcval5  9787