Theorem seq3feq2 10547

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 2193	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		seq3fveq2.1	. . . . . 6 |- ( ph -> K e. ( ZZ>= ` M ) )
3		eluzel2 9597	. . . . . 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 10535	. . . 4 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
8	7	ffnd 5404	. . 3 |- ( ph -> seq M ( .+ , F ) Fn ( ZZ>= ` M ) )
9		uzss 9613	. . . 4 |- ( K e. ( ZZ>= ` M ) -> ( ZZ>= ` K ) C_ ( ZZ>= ` M ) )
10	2, 9	syl 14	. . 3 |- ( ph -> ( ZZ>= ` K ) C_ ( ZZ>= ` M ) )
11		fnssres 5367	. . 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 2193	. . . 4 |- ( ZZ>= ` K ) = ( ZZ>= ` K )
14		eluzelz 9601	. . . . 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 10535	. . 3 |- ( ph -> seq K ( .+ , G ) : ( ZZ>= ` K ) --> S )
18	17	ffnd 5404	. 2 |- ( ph -> seq K ( .+ , G ) Fn ( ZZ>= ` K ) )
19		fvres 5578	. . . 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 10087	. . . . . 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 10546	. . 3 |- ( ( ph /\ z e. ( ZZ>= ` K ) ) -> ( seq M ( .+ , F ) ` z ) = ( seq K ( .+ , G ) ` z ) )
33	20, 32	eqtrd 2226	. 2 |- ( ( ph /\ z e. ( ZZ>= ` K ) ) -> ( ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) ` z ) = ( seq K ( .+ , G ) ` z ) )
34	12, 18, 33	eqfnfvd 5658	1 |- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) = seq K ( .+ , G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   C_ wss 3153   |` cres 4661   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   1c1 7873   + caddc 7875   ZZcz 9317   ZZ>=cuz 9592   ...cfz 10074   seqcseq 10518

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-seqfrec 10519

This theorem is referenced by:  seq3id  10596