Theorem seq3feq 10474

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 2177	. . . 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 10463	. . 3 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
6	5	ffnd 5368	. 2 |- ( ph -> seq M ( .+ , F ) Fn ( ZZ>= ` M ) )
7		fveq2 5517	. . . . . . 7 |- ( k = x -> ( F ` k ) = ( F ` x ) )
8		fveq2 5517	. . . . . . 7 |- ( k = x -> ( G ` k ) = ( G ` x ) )
9	7, 8	eqeq12d 2192	. . . . . 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 2550	. . . . . . 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 2848	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) = ( G ` x ) )
15	14, 3	eqeltrrd 2255	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )
16	1, 2, 15, 4	seqf 10463	. . 3 |- ( ph -> seq M ( .+ , G ) : ( ZZ>= ` M ) --> S )
17	16	ffnd 5368	. 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 10023	. . . . 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 10473	. 2 |- ( ( ph /\ z e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , G ) ` z ) )
27	6, 17, 26	eqfnfvd 5618	1 |- ( ph -> seq M ( .+ , F ) = seq M ( .+ , G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   A.wral 2455   ` cfv 5218  (class class class)co 5877   ZZcz 9255   ZZ>=cuz 9530   ...cfz 10010   seqcseq 10447

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-fz 10011  df-seqfrec 10448

This theorem is referenced by:  zsumdc  11394  fsum3cvg2  11404  isumshft  11500  geolim2  11522  cvgratz  11542  mertenslem2  11546  zproddc  11589