Theorem seq3feq2 10913

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 2238	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		seq3fveq2.1	. . . . . 6 |- ( ph -> K e. ( ZZ>= ` M ) )
3		eluzel2 9926	. . . . . 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 10901	. . . 4 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
8	7	ffnd 5534	. . 3 |- ( ph -> seq M ( .+ , F ) Fn ( ZZ>= ` M ) )
9		uzss 9943	. . . 4 |- ( K e. ( ZZ>= ` M ) -> ( ZZ>= ` K ) C_ ( ZZ>= ` M ) )
10	2, 9	syl 14	. . 3 |- ( ph -> ( ZZ>= ` K ) C_ ( ZZ>= ` M ) )
11		fnssres 5496	. . 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 415	. 2 |- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) Fn ( ZZ>= ` K ) )
13		eqid 2238	. . . 4 |- ( ZZ>= ` K ) = ( ZZ>= ` K )
14		eluzelz 9931	. . . . 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 10901	. . 3 |- ( ph -> seq K ( .+ , G ) : ( ZZ>= ` K ) --> S )
18	17	ffnd 5534	. 2 |- ( ph -> seq K ( .+ , G ) Fn ( ZZ>= ` K ) )
19		fvres 5719	. . . 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 481	. . . 4 |- ( ( ( ph /\ z e. ( ZZ>= ` K ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
25	16	adantlr 481	. . . 4 |- ( ( ( ph /\ z e. ( ZZ>= ` K ) ) /\ x e. ( ZZ>= ` K ) ) -> ( G ` x ) e. S )
26	6	adantlr 481	. . . 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 10424	. . . . . 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 481	. . . 4 |- ( ( ( ph /\ z e. ( ZZ>= ` K ) ) /\ k e. ( ( K + 1 ) ... z ) ) -> ( F ` k ) = ( G ` k ) )
32	21, 23, 24, 25, 26, 27, 31	seq3fveq2 10912	. . 3 |- ( ( ph /\ z e. ( ZZ>= ` K ) ) -> ( seq M ( .+ , F ) ` z ) = ( seq K ( .+ , G ) ` z ) )
33	20, 32	eqtrd 2271	. 2 |- ( ( ph /\ z e. ( ZZ>= ` K ) ) -> ( ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) ` z ) = ( seq K ( .+ , G ) ` z ) )
34	12, 18, 33	eqfnfvd 5809	1 |- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) = seq K ( .+ , G ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   C_ wss 3220   |` cres 4776   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   1c1 8180   + caddc 8182   ZZcz 9644   ZZ>=cuz 9921   ...cfz 10411   seqcseq 10884

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-seqfrec 10885

This theorem is used by:  seq3id  10962