Theorem iseqfeq 9451

Description: Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.)

Hypotheses

Ref	Expression
iseqfeq.1	|- ( ph -> M e. ZZ )
iseqfeq.s	|- ( ph -> S e. V )
iseqfeq.f	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
iseqfeq.2	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( G ` k ) )
iseqfeq.pl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )

Assertion

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

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

Allowed substitution hints:   V( x, y, k)

Proof of Theorem iseqfeq

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqfeq.1	. . 3 |- ( ph -> M e. ZZ )
2		iseqfeq.s	. . 3 |- ( ph -> S e. V )
3		iseqfeq.f	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
4		iseqfeq.pl	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
5	1, 2, 3, 4	iseqfn 9441	. 2 |- ( ph -> seq M ( .+ , F , S ) Fn ( ZZ>= ` M ) )
6		iseqfeq.2	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( G ` k ) )
7	6	ralrimiva 2434	. . . . 5 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) = ( G ` k ) )
8		fveq2 5198	. . . . . . 7 |- ( k = x -> ( F ` k ) = ( F ` x ) )
9		fveq2 5198	. . . . . . 7 |- ( k = x -> ( G ` k ) = ( G ` x ) )
10	8, 9	eqeq12d 2095	. . . . . 6 |- ( k = x -> ( ( F ` k ) = ( G ` k ) <-> ( F ` x ) = ( G ` x ) ) )
11	10	rspcv 2697	. . . . 5 |- ( x e. ( ZZ>= ` M ) -> ( A. k e. ( ZZ>= ` M ) ( F ` k ) = ( G ` k ) -> ( F ` x ) = ( G ` x ) ) )
12	7, 11	mpan9 275	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) = ( G ` x ) )
13	12, 3	eqeltrrd 2156	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )
14	1, 2, 13, 4	iseqfn 9441	. 2 |- ( ph -> seq M ( .+ , G , S ) Fn ( ZZ>= ` M ) )
15		simpr 108	. . 3 |- ( ( ph /\ z e. ( ZZ>= ` M ) ) -> z e. ( ZZ>= ` M ) )
16		elfzuz 9041	. . . . 5 |- ( k e. ( M ... z ) -> k e. ( ZZ>= ` M ) )
17	16, 6	sylan2 280	. . . 4 |- ( ( ph /\ k e. ( M ... z ) ) -> ( F ` k ) = ( G ` k ) )
18	17	adantlr 460	. . 3 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ k e. ( M ... z ) ) -> ( F ` k ) = ( G ` k ) )
19	2	adantr 270	. . 3 |- ( ( ph /\ z e. ( ZZ>= ` M ) ) -> S e. V )
20	3	adantlr 460	. . 3 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
21	13	adantlr 460	. . 3 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )
22	4	adantlr 460	. . 3 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
23	15, 18, 19, 20, 21, 22	iseqfveq 9450	. 2 |- ( ( ph /\ z e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , S ) ` z ) = ( seq M ( .+ , G , S ) ` z ) )
24	5, 14, 23	eqfnfvd 5289	1 |- ( ph -> seq M ( .+ , F , S ) = seq M ( .+ , G , S ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   A.wral 2348   ` cfv 4922  (class class class)co 5532   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: (None)