Theorem seq3z 10965

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 10425	. . 3 |- ( K e. ( M ... N ) -> N e. ( ZZ>= ` K ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( ZZ>= ` K ) )
4		fveqeq2 5704	. . . 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 5704	. . . 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 5704	. . . 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 5704	. . . 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 10424	. . . . . . . . . 10 |- ( K e. ( M ... N ) -> K e. ( ZZ>= ` M ) )
13	1, 12	syl 14	. . . . . . . . 9 |- ( ph -> K e. ( ZZ>= ` M ) )
14		eluzelz 9931	. . . . . . . . 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 9939	. . . . . . . . . 10 |- ( ( x e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> x e. ( ZZ>= ` M ) )
19	16, 17, 18	syl2anc 415	. . . . . . . . 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 10899	. . . . . . 7 |- ( ph -> ( seq K ( .+ , F ) ` K ) = ( F ` K ) )
24		seqz.7	. . . . . . 7 |- ( ph -> ( F ` K ) = Z )
25	23, 24	eqtrd 2271	. . . . . 6 |- ( ph -> ( seq K ( .+ , F ) ` K ) = Z )
26		seqeq1 10887	. . . . . . . 8 |- ( K = M -> seq K ( .+ , F ) = seq M ( .+ , F ) )
27	26	fveq1d 5697	. . . . . . 7 |- ( K = M -> ( seq K ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` K ) )
28	27	eqeq1d 2247	. . . . . 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 9926	. . . . . . . . . 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 481	. . . . . . . 8 |- ( ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
35	22	adantlr 481	. . . . . . . 8 |- ( ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
36	32, 33, 34, 35	seq3m1 10910	. . . . . . 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 6101	. . . . . . 7 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ ( F ` K ) ) = ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ Z ) )
39		oveq1 6092	. . . . . . . . 9 |- ( x = ( seq M ( .+ , F ) ` ( K - 1 ) ) -> ( x .+ Z ) = ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ Z ) )
40	39	eqeq1d 2247	. . . . . . . 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 2623	. . . . . . . . 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 2238	. . . . . . . . . 10 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
45	44, 32, 34, 35	seqf 10901	. . . . . . . . 9 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
46		eluzp1m1 9946	. . . . . . . . . 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 5844	. . . . . . . 8 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` ( K - 1 ) ) e. S )
49	40, 43, 48	rspcdva 2934	. . . . . . 7 |- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ Z ) = Z )
50	36, 38, 49	3eqtrd 2275	. . . . . 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 9956	. . . . . 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 730	. . . 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 9939	. . . . . . . . . 10 |- ( ( k e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
59	56, 57, 58	syl2anc 415	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> k e. ( ZZ>= ` M ) )
60	20	adantlr 481	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
61	22	adantlr 481	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
62	59, 60, 61	seq3p1 10902	. . . . . . . 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 6100	. . . . . . 7 |- ( ( ( ph /\ k e. ( ZZ>= ` K ) ) /\ ( seq M ( .+ , F ) ` k ) = Z ) -> ( ( seq M ( .+ , F ) ` k ) .+ ( F ` ( k + 1 ) ) ) = ( Z .+ ( F ` ( k + 1 ) ) ) )
66		oveq2 6093	. . . . . . . . . 10 |- ( x = ( F ` ( k + 1 ) ) -> ( Z .+ x ) = ( Z .+ ( F ` ( k + 1 ) ) ) )
67	66	eqeq1d 2247	. . . . . . . . 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 2623	. . . . . . . . . 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 5695	. . . . . . . . . . 11 |- ( x = ( k + 1 ) -> ( F ` x ) = ( F ` ( k + 1 ) ) )
72	71	eleq1d 2307	. . . . . . . . . 10 |- ( x = ( k + 1 ) -> ( ( F ` x ) e. S <-> ( F ` ( k + 1 ) ) e. S ) )
73	20	ralrimiva 2623	. . . . . . . . . . 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 9983	. . . . . . . . . . 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 2934	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> ( F ` ( k + 1 ) ) e. S )
78	67, 70, 77	rspcdva 2934	. . . . . . . 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 2275	. . . . . 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 9988	. 2 |- ( N e. ( ZZ>= ` K ) -> ( ph -> ( seq M ( .+ , F ) ` N ) = Z ) )
85	3, 84	mpcom 36	1 |- ( ph -> ( seq M ( .+ , F ) ` N ) = Z )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   A.wral 2528   ` cfv 5377  (class class class)co 6085   1c1 8180   + caddc 8182   - cmin 8497   ZZcz 9644   ZZ>=cuz 9921   ...cfz 10411   seqcseq 10884

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-seqfrec 10885

This theorem is used by:  bcval5  11201  lgsne0  16157