Theorem seq3feq2 10469

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 2177	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		seq3fveq2.1	. . . . . 6 |- ( ph -> K e. ( ZZ>= ` M ) )
3		eluzel2 9532	. . . . . 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 10460	. . . 4 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
8	7	ffnd 5366	. . 3 |- ( ph -> seq M ( .+ , F ) Fn ( ZZ>= ` M ) )
9		uzss 9547	. . . 4 |- ( K e. ( ZZ>= ` M ) -> ( ZZ>= ` K ) C_ ( ZZ>= ` M ) )
10	2, 9	syl 14	. . 3 |- ( ph -> ( ZZ>= ` K ) C_ ( ZZ>= ` M ) )
11		fnssres 5329	. . 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 411	. 2 |- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) Fn ( ZZ>= ` K ) )
13		eqid 2177	. . . 4 |- ( ZZ>= ` K ) = ( ZZ>= ` K )
14		eluzelz 9536	. . . . 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 10460	. . 3 |- ( ph -> seq K ( .+ , G ) : ( ZZ>= ` K ) --> S )
18	17	ffnd 5366	. 2 |- ( ph -> seq K ( .+ , G ) Fn ( ZZ>= ` K ) )
19		fvres 5539	. . . 4 |- ( z e. ( ZZ>= ` K ) -> ( ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) ` z ) = ( seq M ( .+ , F ) ` z ) )
20	19	adantl 277	. . 3 |- ( ( ph /\ z e. ( ZZ>= ` K ) ) -> ( ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) ` z ) = ( seq M ( .+ , F ) ` z ) )
21	2	adantr 276	. . . 4 |- ( ( ph /\ z e. ( ZZ>= ` K ) ) -> K e. ( ZZ>= ` M ) )
22		seq3fveq2.2	. . . . 5 |- ( ph -> ( seq M ( .+ , F ) ` K ) = ( G ` K ) )
23	22	adantr 276	. . . 4 |- ( ( ph /\ z e. ( ZZ>= ` K ) ) -> ( seq M ( .+ , F ) ` K ) = ( G ` K ) )
24	5	adantlr 477	. . . 4 |- ( ( ( ph /\ z e. ( ZZ>= ` K ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
25	16	adantlr 477	. . . 4 |- ( ( ( ph /\ z e. ( ZZ>= ` K ) ) /\ x e. ( ZZ>= ` K ) ) -> ( G ` x ) e. S )
26	6	adantlr 477	. . . 4 |- ( ( ( ph /\ z e. ( ZZ>= ` K ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
27		simpr 110	. . . 4 |- ( ( ph /\ z e. ( ZZ>= ` K ) ) -> z e. ( ZZ>= ` K ) )
28		elfzuz 10020	. . . . . 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 286	. . . . 5 |- ( ( ph /\ k e. ( ( K + 1 ) ... z ) ) -> ( F ` k ) = ( G ` k ) )
31	30	adantlr 477	. . . 4 |- ( ( ( ph /\ z e. ( ZZ>= ` K ) ) /\ k e. ( ( K + 1 ) ... z ) ) -> ( F ` k ) = ( G ` k ) )
32	21, 23, 24, 25, 26, 27, 31	seq3fveq2 10468	. . 3 |- ( ( ph /\ z e. ( ZZ>= ` K ) ) -> ( seq M ( .+ , F ) ` z ) = ( seq K ( .+ , G ) ` z ) )
33	20, 32	eqtrd 2210	. 2 |- ( ( ph /\ z e. ( ZZ>= ` K ) ) -> ( ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) ` z ) = ( seq K ( .+ , G ) ` z ) )
34	12, 18, 33	eqfnfvd 5616	1 |- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) = seq K ( .+ , G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   C_ wss 3129   |` cres 4628   Fn wfn 5211   ` cfv 5216  (class class class)co 5874   1c1 7811   + caddc 7813   ZZcz 9252   ZZ>=cuz 9527   ...cfz 10007   seqcseq 10444

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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-inn 8919  df-n0 9176  df-z 9253  df-uz 9528  df-fz 10008  df-seqfrec 10445

This theorem is referenced by:  seq3id  10507