Theorem seq3feq 10846

Description: Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)

Hypotheses

Ref	Expression
seq3feq.1	|- ( ph -> M e. ZZ )
seq3feq.f	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
seq3feq.2	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( G ` k ) )
seq3feq.pl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )

Assertion

Ref	Expression
seq3feq	|- ( ph -> seq M ( .+ , F ) = seq M ( .+ , G ) )

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 seq3feq

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2234	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		seq3feq.1	. . . 4 |- ( ph -> M e. ZZ )
3		seq3feq.f	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
4		seq3feq.pl	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
5	1, 2, 3, 4	seqf 10830	. . 3 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
6	5	ffnd 5511	. 2 |- ( ph -> seq M ( .+ , F ) Fn ( ZZ>= ` M ) )
7		fveq2 5672	. . . . . . 7 |- ( k = x -> ( F ` k ) = ( F ` x ) )
8		fveq2 5672	. . . . . . 7 |- ( k = x -> ( G ` k ) = ( G ` x ) )
9	7, 8	eqeq12d 2249	. . . . . 6 |- ( k = x -> ( ( F ` k ) = ( G ` k ) <-> ( F ` x ) = ( G ` x ) ) )
10		seq3feq.2	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( G ` k ) )
11	10	ralrimiva 2617	. . . . . . 7 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) = ( G ` k ) )
12	11	adantr 276	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> A. k e. ( ZZ>= ` M ) ( F ` k ) = ( G ` k ) )
13		simpr 110	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. ( ZZ>= ` M ) )
14	9, 12, 13	rspcdva 2928	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) = ( G ` x ) )
15	14, 3	eqeltrrd 2312	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )
16	1, 2, 15, 4	seqf 10830	. . 3 |- ( ph -> seq M ( .+ , G ) : ( ZZ>= ` M ) --> S )
17	16	ffnd 5511	. 2 |- ( ph -> seq M ( .+ , G ) Fn ( ZZ>= ` M ) )
18		simpr 110	. . 3 |- ( ( ph /\ z e. ( ZZ>= ` M ) ) -> z e. ( ZZ>= ` M ) )
19		simpll 527	. . . 4 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ k e. ( M ... z ) ) -> ph )
20		elfzuz 10358	. . . . 5 |- ( k e. ( M ... z ) -> k e. ( ZZ>= ` M ) )
21	20	adantl 277	. . . 4 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ k e. ( M ... z ) ) -> k e. ( ZZ>= ` M ) )
22	19, 21, 10	syl2anc 411	. . 3 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ k e. ( M ... z ) ) -> ( F ` k ) = ( G ` k ) )
23	3	adantlr 477	. . 3 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
24	15	adantlr 477	. . 3 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )
25	4	adantlr 477	. . 3 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
26	18, 22, 23, 24, 25	seq3fveq 10845	. 2 |- ( ( ph /\ z e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , G ) ` z ) )
27	6, 17, 26	eqfnfvd 5780	1 |- ( ph -> seq M ( .+ , F ) = seq M ( .+ , G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   A.wral 2522   ` cfv 5354  (class class class)co 6052   ZZcz 9579   ZZ>=cuz 9856   ...cfz 10345   seqcseq 10813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857  df-fz 10346  df-seqfrec 10814

This theorem is referenced by:  zsumdc  12074  fsum3cvg2  12084  isumshft  12180  geolim2  12202  cvgratz  12222  mertenslem2  12226  zproddc  12269