Theorem seq3feq2 9947

Description: Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)

Hypotheses

Ref	Expression
seq3fveq2.1	|- ( ph -> K e. ( ZZ>= ` M ) )
seq3fveq2.2	|- ( ph -> ( seq M ( .+ , F ) ` K ) = ( G ` K ) )
seq3fveq2.f	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
seq3fveq2.g	|- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( G ` x ) e. S )
seq3fveq2.pl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
seq3feq2.4	|- ( ( ph /\ k e. ( ZZ>= ` ( K + 1 ) ) ) -> ( F ` k ) = ( G ` k ) )

Assertion

Ref	Expression
seq3feq2	|- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) = seq K ( .+ , G ) )

Distinct variable groups:   x, k, y, F   k, G, x, y   k, K, x, y   ph, k, x, y   k, M, x, y   .+ , k, x, y   S, k, x, y

Proof of Theorem seq3feq2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2089	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		seq3fveq2.1	. . . . . 6 |- ( ph -> K e. ( ZZ>= ` M ) )
3		eluzel2 9078	. . . . . 6 |- ( K e. ( ZZ>= ` M ) -> M e. ZZ )
4	2, 3	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
5		seq3fveq2.f	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
6		seq3fveq2.pl	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
7	1, 4, 5, 6	seqf 9934	. . . 4 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
8	7	ffnd 5175	. . 3 |- ( ph -> seq M ( .+ , F ) Fn ( ZZ>= ` M ) )
9		uzss 9093	. . . 4 |- ( K e. ( ZZ>= ` M ) -> ( ZZ>= ` K ) C_ ( ZZ>= ` M ) )
10	2, 9	syl 14	. . 3 |- ( ph -> ( ZZ>= ` K ) C_ ( ZZ>= ` M ) )
11		fnssres 5140	. . 3 |- ( ( seq M ( .+ , F ) Fn ( ZZ>= ` M ) /\ ( ZZ>= ` K ) C_ ( ZZ>= ` M ) ) -> ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) Fn ( ZZ>= ` K ) )
12	8, 10, 11	syl2anc 404	. 2 |- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) Fn ( ZZ>= ` K ) )
13		eqid 2089	. . . 4 |- ( ZZ>= ` K ) = ( ZZ>= ` K )
14		eluzelz 9082	. . . . 5 |- ( K e. ( ZZ>= ` M ) -> K e. ZZ )
15	2, 14	syl 14	. . . 4 |- ( ph -> K e. ZZ )
16		seq3fveq2.g	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( G ` x ) e. S )
17	13, 15, 16, 6	seqf 9934	. . 3 |- ( ph -> seq K ( .+ , G ) : ( ZZ>= ` K ) --> S )
18	17	ffnd 5175	. 2 |- ( ph -> seq K ( .+ , G ) Fn ( ZZ>= ` K ) )
19		fvres 5342	. . . 4 |- ( z e. ( ZZ>= ` K ) -> ( ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) ` z ) = ( seq M ( .+ , F ) ` z ) )
20	19	adantl 272	. . 3 |- ( ( ph /\ z e. ( ZZ>= ` K ) ) -> ( ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) ` z ) = ( seq M ( .+ , F ) ` z ) )
21	2	adantr 271	. . . 4 |- ( ( ph /\ z e. ( ZZ>= ` K ) ) -> K e. ( ZZ>= ` M ) )
22		seq3fveq2.2	. . . . 5 |- ( ph -> ( seq M ( .+ , F ) ` K ) = ( G ` K ) )
23	22	adantr 271	. . . 4 |- ( ( ph /\ z e. ( ZZ>= ` K ) ) -> ( seq M ( .+ , F ) ` K ) = ( G ` K ) )
24	5	adantlr 462	. . . 4 |- ( ( ( ph /\ z e. ( ZZ>= ` K ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
25	16	adantlr 462	. . . 4 |- ( ( ( ph /\ z e. ( ZZ>= ` K ) ) /\ x e. ( ZZ>= ` K ) ) -> ( G ` x ) e. S )
26	6	adantlr 462	. . . 4 |- ( ( ( ph /\ z e. ( ZZ>= ` K ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
27		simpr 109	. . . 4 |- ( ( ph /\ z e. ( ZZ>= ` K ) ) -> z e. ( ZZ>= ` K ) )
28		elfzuz 9490	. . . . . 6 |- ( k e. ( ( K + 1 ) ... z ) -> k e. ( ZZ>= ` ( K + 1 ) ) )
29		seq3feq2.4	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` ( K + 1 ) ) ) -> ( F ` k ) = ( G ` k ) )
30	28, 29	sylan2 281	. . . . 5 |- ( ( ph /\ k e. ( ( K + 1 ) ... z ) ) -> ( F ` k ) = ( G ` k ) )
31	30	adantlr 462	. . . 4 |- ( ( ( ph /\ z e. ( ZZ>= ` K ) ) /\ k e. ( ( K + 1 ) ... z ) ) -> ( F ` k ) = ( G ` k ) )
32	21, 23, 24, 25, 26, 27, 31	seq3fveq2 9946	. . 3 |- ( ( ph /\ z e. ( ZZ>= ` K ) ) -> ( seq M ( .+ , F ) ` z ) = ( seq K ( .+ , G ) ` z ) )
33	20, 32	eqtrd 2121	. 2 |- ( ( ph /\ z e. ( ZZ>= ` K ) ) -> ( ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) ` z ) = ( seq K ( .+ , G ) ` z ) )
34	12, 18, 33	eqfnfvd 5414	1 |- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) = seq K ( .+ , G ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   e. wcel 1439   C_ wss 3000   |` cres 4453   Fn wfn 5023   ` cfv 5028  (class class class)co 5666   1c1 7405   + caddc 7407   ZZcz 8804   ZZ>=cuz 9073   ...cfz 9478   seqcseq 9906

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-addcom 7499  ax-addass 7501  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-0id 7507  ax-rnegex 7508  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-ltadd 7515

This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-frec 6170  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-inn 8477  df-n0 8728  df-z 8805  df-uz 9074  df-fz 9479  df-iseq 9907  df-seq3 9908

This theorem is referenced by: (None)