Theorem iseqfeq 9617

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

Hypotheses

Ref	Expression
iseqfeq.1	|- ( ph -> M e. ZZ )
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

Proof of Theorem iseqfeq

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2083	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		iseqfeq.1	. . . 4 |- ( ph -> M e. ZZ )
3		iseqfeq.f	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
4		iseqfeq.pl	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
5	1, 2, 3, 4	iseqfcl 9605	. . 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		iseqfeq.2	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( G ` k ) )
9	8	ralrimiva 2439	. . . . . 6 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) = ( G ` k ) )
10		fveq2 5230	. . . . . . . 8 |- ( k = x -> ( F ` k ) = ( F ` x ) )
11		fveq2 5230	. . . . . . . 8 |- ( k = x -> ( G ` k ) = ( G ` x ) )
12	10, 11	eqeq12d 2097	. . . . . . 7 |- ( k = x -> ( ( F ` k ) = ( G ` k ) <-> ( F ` x ) = ( G ` x ) ) )
13	12	rspcv 2706	. . . . . 6 |- ( x e. ( ZZ>= ` M ) -> ( A. k e. ( ZZ>= ` M ) ( F ` k ) = ( G ` k ) -> ( F ` x ) = ( G ` x ) ) )
14	9, 13	mpan9 275	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) = ( G ` x ) )
15	14, 3	eqeltrrd 2160	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )
16	1, 2, 15, 4	iseqfcl 9605	. . 3 |- ( ph -> seq M ( .+ , G , S ) : ( ZZ>= ` M ) --> S )
17		ffn 5097	. . 3 |- ( seq M ( .+ , G , S ) : ( ZZ>= ` M ) --> S -> seq M ( .+ , G , S ) Fn ( ZZ>= ` M ) )
18	16, 17	syl 14	. 2 |- ( ph -> seq M ( .+ , G , S ) Fn ( ZZ>= ` M ) )
19		simpr 108	. . 3 |- ( ( ph /\ z e. ( ZZ>= ` M ) ) -> z e. ( ZZ>= ` M ) )
20		elfzuz 9187	. . . . 5 |- ( k e. ( M ... z ) -> k e. ( ZZ>= ` M ) )
21	20, 8	sylan2 280	. . . 4 |- ( ( ph /\ k e. ( M ... z ) ) -> ( F ` k ) = ( G ` k ) )
22	21	adantlr 461	. . 3 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ k e. ( M ... z ) ) -> ( F ` k ) = ( G ` k ) )
23	3	adantlr 461	. . 3 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
24	15	adantlr 461	. . 3 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )
25	4	adantlr 461	. . 3 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
26	19, 22, 23, 24, 25	iseqfveq 9616	. 2 |- ( ( ph /\ z e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F , S ) ` z ) = ( seq M ( .+ , G , S ) ` z ) )
27	7, 18, 26	eqfnfvd 5321	1 |- ( ph -> seq M ( .+ , F , S ) = seq M ( .+ , G , S ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   e. wcel 1434   A.wral 2353   Fn wfn 4947   -->wf 4948   ` cfv 4952  (class class class)co 5564   ZZcz 8502   ZZ>=cuz 8770   ...cfz 9175   seqcseq 9591

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 7199  ax-resscn 7200  ax-1cn 7201  ax-1re 7202  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-addcom 7208  ax-addass 7210  ax-distr 7212  ax-i2m1 7213  ax-0lt1 7214  ax-0id 7216  ax-rnegex 7217  ax-cnre 7219  ax-pre-ltirr 7220  ax-pre-ltwlin 7221  ax-pre-lttrn 7222  ax-pre-ltadd 7224

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 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-pnf 7287  df-mnf 7288  df-xr 7289  df-ltxr 7290  df-le 7291  df-sub 7418  df-neg 7419  df-inn 8177  df-n0 8426  df-z 8503  df-uz 8771  df-fz 9176  df-iseq 9592

This theorem is referenced by: (None)