Theorem iseqoveq 9592

Description: Equality theorem for the sequence builder operation. This is similar to iseqeq2 9577 but .+ and Q only have to agree on elements of S. (Contributed by Jim Kingdon, 21-Apr-2022.)

Hypotheses

Ref	Expression
iseqoveq.m	|- ( ph -> M e. ZZ )
iseqoveq.f	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
iseqoveq.eq	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( x Q y ) )
iseqoveq.plcl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
iseqoveq.qcl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )

Assertion

Ref	Expression
iseqoveq	|- ( ph -> seq M ( .+ , F , S ) = seq M ( Q , F , S ) )

Distinct variable groups:   x, .+ , y   x, F, y   x, M, y   x, Q, y   x, S, y   ph, x, y

Proof of Theorem iseqoveq

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

Step	Hyp	Ref	Expression
1		eqid 2083	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		iseqoveq.m	. . . 4 |- ( ph -> M e. ZZ )
3		iseqoveq.f	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
4		iseqoveq.plcl	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
5	1, 2, 3, 4	iseqfcl 9587	. . 3 |- ( ph -> seq M ( .+ , F , S ) : ( ZZ>= ` M ) --> S )
6		ffn 5097	. . 3 |- ( seq M ( .+ , F , S ) : ( ZZ>= ` M ) --> S -> seq M ( .+ , F , S ) Fn ( ZZ>= ` M ) )
7	5, 6	syl 14	. 2 |- ( ph -> seq M ( .+ , F , S ) Fn ( ZZ>= ` M ) )
8		iseqoveq.qcl	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )
9	1, 2, 3, 8	iseqfcl 9587	. . 3 |- ( ph -> seq M ( Q , F , S ) : ( ZZ>= ` M ) --> S )
10		ffn 5097	. . 3 |- ( seq M ( Q , F , S ) : ( ZZ>= ` M ) --> S -> seq M ( Q , F , S ) Fn ( ZZ>= ` M ) )
11	9, 10	syl 14	. 2 |- ( ph -> seq M ( Q , F , S ) Fn ( ZZ>= ` M ) )
12		fveq2 5229	. . . . . 6 |- ( w = M -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` M ) )
13		fveq2 5229	. . . . . 6 |- ( w = M -> ( seq M ( Q , F , S ) ` w ) = ( seq M ( Q , F , S ) ` M ) )
14	12, 13	eqeq12d 2097	. . . . 5 |- ( w = M -> ( ( seq M ( .+ , F , S ) ` w ) = ( seq M ( Q , F , S ) ` w ) <-> ( seq M ( .+ , F , S ) ` M ) = ( seq M ( Q , F , S ) ` M ) ) )
15	14	imbi2d 228	. . . 4 |- ( w = M -> ( ( ph -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( Q , F , S ) ` w ) ) <-> ( ph -> ( seq M ( .+ , F , S ) ` M ) = ( seq M ( Q , F , S ) ` M ) ) ) )
16		fveq2 5229	. . . . . 6 |- ( w = k -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` k ) )
17		fveq2 5229	. . . . . 6 |- ( w = k -> ( seq M ( Q , F , S ) ` w ) = ( seq M ( Q , F , S ) ` k ) )
18	16, 17	eqeq12d 2097	. . . . 5 |- ( w = k -> ( ( seq M ( .+ , F , S ) ` w ) = ( seq M ( Q , F , S ) ` w ) <-> ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) ) )
19	18	imbi2d 228	. . . 4 |- ( w = k -> ( ( ph -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( Q , F , S ) ` w ) ) <-> ( ph -> ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) ) ) )
20		fveq2 5229	. . . . . 6 |- ( w = ( k + 1 ) -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` ( k + 1 ) ) )
21		fveq2 5229	. . . . . 6 |- ( w = ( k + 1 ) -> ( seq M ( Q , F , S ) ` w ) = ( seq M ( Q , F , S ) ` ( k + 1 ) ) )
22	20, 21	eqeq12d 2097	. . . . 5 |- ( w = ( k + 1 ) -> ( ( seq M ( .+ , F , S ) ` w ) = ( seq M ( Q , F , S ) ` w ) <-> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( seq M ( Q , F , S ) ` ( k + 1 ) ) ) )
23	22	imbi2d 228	. . . 4 |- ( w = ( k + 1 ) -> ( ( ph -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( Q , F , S ) ` w ) ) <-> ( ph -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( seq M ( Q , F , S ) ` ( k + 1 ) ) ) ) )
24		fveq2 5229	. . . . . 6 |- ( w = n -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( .+ , F , S ) ` n ) )
25		fveq2 5229	. . . . . 6 |- ( w = n -> ( seq M ( Q , F , S ) ` w ) = ( seq M ( Q , F , S ) ` n ) )
26	24, 25	eqeq12d 2097	. . . . 5 |- ( w = n -> ( ( seq M ( .+ , F , S ) ` w ) = ( seq M ( Q , F , S ) ` w ) <-> ( seq M ( .+ , F , S ) ` n ) = ( seq M ( Q , F , S ) ` n ) ) )
27	26	imbi2d 228	. . . 4 |- ( w = n -> ( ( ph -> ( seq M ( .+ , F , S ) ` w ) = ( seq M ( Q , F , S ) ` w ) ) <-> ( ph -> ( seq M ( .+ , F , S ) ` n ) = ( seq M ( Q , F , S ) ` n ) ) ) )
28		simpr 108	. . . . . . 7 |- ( ( ph /\ M e. ZZ ) -> M e. ZZ )
29	3	adantlr 461	. . . . . . 7 |- ( ( ( ph /\ M e. ZZ ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
30	4	adantlr 461	. . . . . . 7 |- ( ( ( ph /\ M e. ZZ ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
31	28, 29, 30	iseq1 9585	. . . . . 6 |- ( ( ph /\ M e. ZZ ) -> ( seq M ( .+ , F , S ) ` M ) = ( F ` M ) )
32	8	adantlr 461	. . . . . . 7 |- ( ( ( ph /\ M e. ZZ ) /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )
33	28, 29, 32	iseq1 9585	. . . . . 6 |- ( ( ph /\ M e. ZZ ) -> ( seq M ( Q , F , S ) ` M ) = ( F ` M ) )
34	31, 33	eqtr4d 2118	. . . . 5 |- ( ( ph /\ M e. ZZ ) -> ( seq M ( .+ , F , S ) ` M ) = ( seq M ( Q , F , S ) ` M ) )
35	34	expcom 114	. . . 4 |- ( M e. ZZ -> ( ph -> ( seq M ( .+ , F , S ) ` M ) = ( seq M ( Q , F , S ) ` M ) ) )
36		simpr 108	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
37	36	adantr 270	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) ) -> k e. ( ZZ>= ` M ) )
38	3	adantlr 461	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
39	38	adantlr 461	. . . . . . . . . . 11 |- ( ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
40	4	adantlr 461	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
41	40	adantlr 461	. . . . . . . . . . 11 |- ( ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
42	37, 39, 41	iseqcl 9589	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) ) -> ( seq M ( .+ , F , S ) ` k ) e. S )
43		fveq2 5229	. . . . . . . . . . . 12 |- ( x = ( k + 1 ) -> ( F ` x ) = ( F ` ( k + 1 ) ) )
44	43	eleq1d 2151	. . . . . . . . . . 11 |- ( x = ( k + 1 ) -> ( ( F ` x ) e. S <-> ( F ` ( k + 1 ) ) e. S ) )
45	3	ralrimiva 2439	. . . . . . . . . . . 12 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
46	45	ad2antrr 472	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) ) -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. S )
47		peano2uz 8804	. . . . . . . . . . . 12 |- ( k e. ( ZZ>= ` M ) -> ( k + 1 ) e. ( ZZ>= ` M ) )
48	47	ad2antlr 473	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) ) -> ( k + 1 ) e. ( ZZ>= ` M ) )
49	44, 46, 48	rspcdva 2715	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) ) -> ( F ` ( k + 1 ) ) e. S )
50		iseqoveq.eq	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( x Q y ) )
51	50	ralrimivva 2448	. . . . . . . . . . 11 |- ( ph -> A. x e. S A. y e. S ( x .+ y ) = ( x Q y ) )
52	51	ad2antrr 472	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) ) -> A. x e. S A. y e. S ( x .+ y ) = ( x Q y ) )
53		oveq1 5570	. . . . . . . . . . . 12 |- ( x = ( seq M ( .+ , F , S ) ` k ) -> ( x .+ y ) = ( ( seq M ( .+ , F , S ) ` k ) .+ y ) )
54		oveq1 5570	. . . . . . . . . . . 12 |- ( x = ( seq M ( .+ , F , S ) ` k ) -> ( x Q y ) = ( ( seq M ( .+ , F , S ) ` k ) Q y ) )
55	53, 54	eqeq12d 2097	. . . . . . . . . . 11 |- ( x = ( seq M ( .+ , F , S ) ` k ) -> ( ( x .+ y ) = ( x Q y ) <-> ( ( seq M ( .+ , F , S ) ` k ) .+ y ) = ( ( seq M ( .+ , F , S ) ` k ) Q y ) ) )
56		oveq2 5571	. . . . . . . . . . . 12 |- ( y = ( F ` ( k + 1 ) ) -> ( ( seq M ( .+ , F , S ) ` k ) .+ y ) = ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
57		oveq2 5571	. . . . . . . . . . . 12 |- ( y = ( F ` ( k + 1 ) ) -> ( ( seq M ( .+ , F , S ) ` k ) Q y ) = ( ( seq M ( .+ , F , S ) ` k ) Q ( F ` ( k + 1 ) ) ) )
58	56, 57	eqeq12d 2097	. . . . . . . . . . 11 |- ( y = ( F ` ( k + 1 ) ) -> ( ( ( seq M ( .+ , F , S ) ` k ) .+ y ) = ( ( seq M ( .+ , F , S ) ` k ) Q y ) <-> ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) = ( ( seq M ( .+ , F , S ) ` k ) Q ( F ` ( k + 1 ) ) ) ) )
59	55, 58	rspc2va 2722	. . . . . . . . . 10 |- ( ( ( ( seq M ( .+ , F , S ) ` k ) e. S /\ ( F ` ( k + 1 ) ) e. S ) /\ A. x e. S A. y e. S ( x .+ y ) = ( x Q y ) ) -> ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) = ( ( seq M ( .+ , F , S ) ` k ) Q ( F ` ( k + 1 ) ) ) )
60	42, 49, 52, 59	syl21anc 1169	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) ) -> ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) = ( ( seq M ( .+ , F , S ) ` k ) Q ( F ` ( k + 1 ) ) ) )
61		simpr 108	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) ) -> ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) )
62	61	oveq1d 5578	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) ) -> ( ( seq M ( .+ , F , S ) ` k ) Q ( F ` ( k + 1 ) ) ) = ( ( seq M ( Q , F , S ) ` k ) Q ( F ` ( k + 1 ) ) ) )
63	60, 62	eqtrd 2115	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) ) -> ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) = ( ( seq M ( Q , F , S ) ` k ) Q ( F ` ( k + 1 ) ) ) )
64	36, 38, 40	iseqp1 9590	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) )
65	8	adantlr 461	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )
66	36, 38, 65	iseqp1 9590	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( seq M ( Q , F , S ) ` ( k + 1 ) ) = ( ( seq M ( Q , F , S ) ` k ) Q ( F ` ( k + 1 ) ) ) )
67	64, 66	eqeq12d 2097	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( seq M ( Q , F , S ) ` ( k + 1 ) ) <-> ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) = ( ( seq M ( Q , F , S ) ` k ) Q ( F ` ( k + 1 ) ) ) ) )
68	67	adantr 270	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) ) -> ( ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( seq M ( Q , F , S ) ` ( k + 1 ) ) <-> ( ( seq M ( .+ , F , S ) ` k ) .+ ( F ` ( k + 1 ) ) ) = ( ( seq M ( Q , F , S ) ` k ) Q ( F ` ( k + 1 ) ) ) ) )
69	63, 68	mpbird 165	. . . . . . 7 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( seq M ( Q , F , S ) ` ( k + 1 ) ) )
70	69	ex 113	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( seq M ( Q , F , S ) ` ( k + 1 ) ) ) )
71	70	expcom 114	. . . . 5 |- ( k e. ( ZZ>= ` M ) -> ( ph -> ( ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( seq M ( Q , F , S ) ` ( k + 1 ) ) ) ) )
72	71	a2d 26	. . . 4 |- ( k e. ( ZZ>= ` M ) -> ( ( ph -> ( seq M ( .+ , F , S ) ` k ) = ( seq M ( Q , F , S ) ` k ) ) -> ( ph -> ( seq M ( .+ , F , S ) ` ( k + 1 ) ) = ( seq M ( Q , F , S ) ` ( k + 1 ) ) ) ) )
73	15, 19, 23, 27, 35, 72	uzind4 8809	. . 3 |- ( n e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( .+ , F , S ) ` n ) = ( seq M ( Q , F , S ) ` n ) ) )
74	73	impcom 123	. 2 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , S ) ` n ) = ( seq M ( Q , F , S ) ` n ) )
75	7, 11, 74	eqfnfvd 5320	1 |- ( ph -> seq M ( .+ , F , S ) = seq M ( Q , F , S ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   e. wcel 1434   A.wral 2353   Fn wfn 4947   -->wf 4948   ` cfv 4952  (class class class)co 5563   1c1 7096   + caddc 7098   ZZcz 8484   ZZ>=cuz 8752   seqcseq 9573

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 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-ltadd 7206

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 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485  df-uz 8753  df-iseq 9574

This theorem is referenced by: (None)