Theorem seq3feq 10428

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 2170	. . . 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 10417	. . 3 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
6	5	ffnd 5348	. 2 |- ( ph -> seq M ( .+ , F ) Fn ( ZZ>= ` M ) )
7		fveq2 5496	. . . . . . 7 |- ( k = x -> ( F ` k ) = ( F ` x ) )
8		fveq2 5496	. . . . . . 7 |- ( k = x -> ( G ` k ) = ( G ` x ) )
9	7, 8	eqeq12d 2185	. . . . . 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 2543	. . . . . . 7 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) = ( G ` k ) )
12	11	adantr 274	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> A. k e. ( ZZ>= ` M ) ( F ` k ) = ( G ` k ) )
13		simpr 109	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. ( ZZ>= ` M ) )
14	9, 12, 13	rspcdva 2839	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) = ( G ` x ) )
15	14, 3	eqeltrrd 2248	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )
16	1, 2, 15, 4	seqf 10417	. . 3 |- ( ph -> seq M ( .+ , G ) : ( ZZ>= ` M ) --> S )
17	16	ffnd 5348	. 2 |- ( ph -> seq M ( .+ , G ) Fn ( ZZ>= ` M ) )
18		simpr 109	. . 3 |- ( ( ph /\ z e. ( ZZ>= ` M ) ) -> z e. ( ZZ>= ` M ) )
19		simpll 524	. . . 4 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ k e. ( M ... z ) ) -> ph )
20		elfzuz 9977	. . . . 5 |- ( k e. ( M ... z ) -> k e. ( ZZ>= ` M ) )
21	20	adantl 275	. . . 4 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ k e. ( M ... z ) ) -> k e. ( ZZ>= ` M ) )
22	19, 21, 10	syl2anc 409	. . 3 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ k e. ( M ... z ) ) -> ( F ` k ) = ( G ` k ) )
23	3	adantlr 474	. . 3 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
24	15	adantlr 474	. . 3 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )
25	4	adantlr 474	. . 3 |- ( ( ( ph /\ z e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
26	18, 22, 23, 24, 25	seq3fveq 10427	. 2 |- ( ( ph /\ z e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , G ) ` z ) )
27	6, 17, 26	eqfnfvd 5596	1 |- ( ph -> seq M ( .+ , F ) = seq M ( .+ , G ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   A.wral 2448   ` cfv 5198  (class class class)co 5853   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965   seqcseq 10401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966  df-seqfrec 10402

This theorem is referenced by:  zsumdc  11347  fsum3cvg2  11357  isumshft  11453  geolim2  11475  cvgratz  11495  mertenslem2  11499  zproddc  11542