Theorem seq3feq 10627

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 2205	. . . 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 10611	. . 3 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
6	5	ffnd 5428	. 2 |- ( ph -> seq M ( .+ , F ) Fn ( ZZ>= ` M ) )
7		fveq2 5578	. . . . . . 7 |- ( k = x -> ( F ` k ) = ( F ` x ) )
8		fveq2 5578	. . . . . . 7 |- ( k = x -> ( G ` k ) = ( G ` x ) )
9	7, 8	eqeq12d 2220	. . . . . 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 2579	. . . . . . 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 2882	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) = ( G ` x ) )
15	14, 3	eqeltrrd 2283	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )
16	1, 2, 15, 4	seqf 10611	. . 3 |- ( ph -> seq M ( .+ , G ) : ( ZZ>= ` M ) --> S )
17	16	ffnd 5428	. 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 10145	. . . . 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 10626	. 2 |- ( ( ph /\ z e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , G ) ` z ) )
27	6, 17, 26	eqfnfvd 5682	1 |- ( ph -> seq M ( .+ , F ) = seq M ( .+ , G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   A.wral 2484   ` cfv 5272  (class class class)co 5946   ZZcz 9374   ZZ>=cuz 9650   ...cfz 10132   seqcseq 10594

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-ltadd 8043

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-inn 9039  df-n0 9298  df-z 9375  df-uz 9651  df-fz 10133  df-seqfrec 10595

This theorem is referenced by:  zsumdc  11728  fsum3cvg2  11738  isumshft  11834  geolim2  11856  cvgratz  11876  mertenslem2  11880  zproddc  11923