Theorem seq3fveq2 10546

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 )
seq3fveq2.3	|- ( ph -> N e. ( ZZ>= ` K ) )
seq3fveq2.4	|- ( ( ph /\ k e. ( ( K + 1 ) ... N ) ) -> ( F ` k ) = ( G ` k ) )

Assertion

Ref	Expression
seq3fveq2	|- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , G ) ` N ) )

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

Proof of Theorem seq3fveq2

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seq3fveq2.3	. . 3 |- ( ph -> N e. ( ZZ>= ` K ) )
2		eluzfz2 10098	. . 3 |- ( N e. ( ZZ>= ` K ) -> N e. ( K ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( K ... N ) )
4		eleq1 2256	. . . . . 6 |- ( z = K -> ( z e. ( K ... N ) <-> K e. ( K ... N ) ) )
5		fveq2 5554	. . . . . . 7 |- ( z = K -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , F ) ` K ) )
6		fveq2 5554	. . . . . . 7 |- ( z = K -> ( seq K ( .+ , G ) ` z ) = ( seq K ( .+ , G ) ` K ) )
7	5, 6	eqeq12d 2208	. . . . . 6 |- ( z = K -> ( ( seq M ( .+ , F ) ` z ) = ( seq K ( .+ , G ) ` z ) <-> ( seq M ( .+ , F ) ` K ) = ( seq K ( .+ , G ) ` K ) ) )
8	4, 7	imbi12d 234	. . . . 5 |- ( z = K -> ( ( z e. ( K ... N ) -> ( seq M ( .+ , F ) ` z ) = ( seq K ( .+ , G ) ` z ) ) <-> ( K e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq K ( .+ , G ) ` K ) ) ) )
9	8	imbi2d 230	. . . 4 |- ( z = K -> ( ( ph -> ( z e. ( K ... N ) -> ( seq M ( .+ , F ) ` z ) = ( seq K ( .+ , G ) ` z ) ) ) <-> ( ph -> ( K e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq K ( .+ , G ) ` K ) ) ) ) )
10		eleq1 2256	. . . . . 6 |- ( z = w -> ( z e. ( K ... N ) <-> w e. ( K ... N ) ) )
11		fveq2 5554	. . . . . . 7 |- ( z = w -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , F ) ` w ) )
12		fveq2 5554	. . . . . . 7 |- ( z = w -> ( seq K ( .+ , G ) ` z ) = ( seq K ( .+ , G ) ` w ) )
13	11, 12	eqeq12d 2208	. . . . . 6 |- ( z = w -> ( ( seq M ( .+ , F ) ` z ) = ( seq K ( .+ , G ) ` z ) <-> ( seq M ( .+ , F ) ` w ) = ( seq K ( .+ , G ) ` w ) ) )
14	10, 13	imbi12d 234	. . . . 5 |- ( z = w -> ( ( z e. ( K ... N ) -> ( seq M ( .+ , F ) ` z ) = ( seq K ( .+ , G ) ` z ) ) <-> ( w e. ( K ... N ) -> ( seq M ( .+ , F ) ` w ) = ( seq K ( .+ , G ) ` w ) ) ) )
15	14	imbi2d 230	. . . 4 |- ( z = w -> ( ( ph -> ( z e. ( K ... N ) -> ( seq M ( .+ , F ) ` z ) = ( seq K ( .+ , G ) ` z ) ) ) <-> ( ph -> ( w e. ( K ... N ) -> ( seq M ( .+ , F ) ` w ) = ( seq K ( .+ , G ) ` w ) ) ) ) )
16		eleq1 2256	. . . . . 6 |- ( z = ( w + 1 ) -> ( z e. ( K ... N ) <-> ( w + 1 ) e. ( K ... N ) ) )
17		fveq2 5554	. . . . . . 7 |- ( z = ( w + 1 ) -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , F ) ` ( w + 1 ) ) )
18		fveq2 5554	. . . . . . 7 |- ( z = ( w + 1 ) -> ( seq K ( .+ , G ) ` z ) = ( seq K ( .+ , G ) ` ( w + 1 ) ) )
19	17, 18	eqeq12d 2208	. . . . . 6 |- ( z = ( w + 1 ) -> ( ( seq M ( .+ , F ) ` z ) = ( seq K ( .+ , G ) ` z ) <-> ( seq M ( .+ , F ) ` ( w + 1 ) ) = ( seq K ( .+ , G ) ` ( w + 1 ) ) ) )
20	16, 19	imbi12d 234	. . . . 5 |- ( z = ( w + 1 ) -> ( ( z e. ( K ... N ) -> ( seq M ( .+ , F ) ` z ) = ( seq K ( .+ , G ) ` z ) ) <-> ( ( w + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F ) ` ( w + 1 ) ) = ( seq K ( .+ , G ) ` ( w + 1 ) ) ) ) )
21	20	imbi2d 230	. . . 4 |- ( z = ( w + 1 ) -> ( ( ph -> ( z e. ( K ... N ) -> ( seq M ( .+ , F ) ` z ) = ( seq K ( .+ , G ) ` z ) ) ) <-> ( ph -> ( ( w + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F ) ` ( w + 1 ) ) = ( seq K ( .+ , G ) ` ( w + 1 ) ) ) ) ) )
22		eleq1 2256	. . . . . 6 |- ( z = N -> ( z e. ( K ... N ) <-> N e. ( K ... N ) ) )
23		fveq2 5554	. . . . . . 7 |- ( z = N -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , F ) ` N ) )
24		fveq2 5554	. . . . . . 7 |- ( z = N -> ( seq K ( .+ , G ) ` z ) = ( seq K ( .+ , G ) ` N ) )
25	23, 24	eqeq12d 2208	. . . . . 6 |- ( z = N -> ( ( seq M ( .+ , F ) ` z ) = ( seq K ( .+ , G ) ` z ) <-> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , G ) ` N ) ) )
26	22, 25	imbi12d 234	. . . . 5 |- ( z = N -> ( ( z e. ( K ... N ) -> ( seq M ( .+ , F ) ` z ) = ( seq K ( .+ , G ) ` z ) ) <-> ( N e. ( K ... N ) -> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , G ) ` N ) ) ) )
27	26	imbi2d 230	. . . 4 |- ( z = N -> ( ( ph -> ( z e. ( K ... N ) -> ( seq M ( .+ , F ) ` z ) = ( seq K ( .+ , G ) ` z ) ) ) <-> ( ph -> ( N e. ( K ... N ) -> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , G ) ` N ) ) ) ) )
28		seq3fveq2.2	. . . . . 6 |- ( ph -> ( seq M ( .+ , F ) ` K ) = ( G ` K ) )
29		seq3fveq2.1	. . . . . . . 8 |- ( ph -> K e. ( ZZ>= ` M ) )
30		eluzelz 9601	. . . . . . . 8 |- ( K e. ( ZZ>= ` M ) -> K e. ZZ )
31	29, 30	syl 14	. . . . . . 7 |- ( ph -> K e. ZZ )
32		seq3fveq2.g	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( G ` x ) e. S )
33		seq3fveq2.pl	. . . . . . 7 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
34	31, 32, 33	seq3-1 10533	. . . . . 6 |- ( ph -> ( seq K ( .+ , G ) ` K ) = ( G ` K ) )
35	28, 34	eqtr4d 2229	. . . . 5 |- ( ph -> ( seq M ( .+ , F ) ` K ) = ( seq K ( .+ , G ) ` K ) )
36	35	a1i13 24	. . . 4 |- ( K e. ZZ -> ( ph -> ( K e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq K ( .+ , G ) ` K ) ) ) )
37		peano2fzr 10103	. . . . . . . 8 |- ( ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) -> w e. ( K ... N ) )
38	37	adantl 277	. . . . . . 7 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> w e. ( K ... N ) )
39	38	expr 375	. . . . . 6 |- ( ( ph /\ w e. ( ZZ>= ` K ) ) -> ( ( w + 1 ) e. ( K ... N ) -> w e. ( K ... N ) ) )
40	39	imim1d 75	. . . . 5 |- ( ( ph /\ w e. ( ZZ>= ` K ) ) -> ( ( w e. ( K ... N ) -> ( seq M ( .+ , F ) ` w ) = ( seq K ( .+ , G ) ` w ) ) -> ( ( w + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F ) ` w ) = ( seq K ( .+ , G ) ` w ) ) ) )
41		oveq1 5925	. . . . . 6 |- ( ( seq M ( .+ , F ) ` w ) = ( seq K ( .+ , G ) ` w ) -> ( ( seq M ( .+ , F ) ` w ) .+ ( F ` ( w + 1 ) ) ) = ( ( seq K ( .+ , G ) ` w ) .+ ( F ` ( w + 1 ) ) ) )
42		simprl 529	. . . . . . . . 9 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> w e. ( ZZ>= ` K ) )
43	29	adantr 276	. . . . . . . . 9 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> K e. ( ZZ>= ` M ) )
44		uztrn 9609	. . . . . . . . 9 |- ( ( w e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> w e. ( ZZ>= ` M ) )
45	42, 43, 44	syl2anc 411	. . . . . . . 8 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> w e. ( ZZ>= ` M ) )
46		seq3fveq2.f	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
47	46	adantlr 477	. . . . . . . 8 |- ( ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
48	33	adantlr 477	. . . . . . . 8 |- ( ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
49	45, 47, 48	seq3p1 10536	. . . . . . 7 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( seq M ( .+ , F ) ` ( w + 1 ) ) = ( ( seq M ( .+ , F ) ` w ) .+ ( F ` ( w + 1 ) ) ) )
50	32	adantlr 477	. . . . . . . . 9 |- ( ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) /\ x e. ( ZZ>= ` K ) ) -> ( G ` x ) e. S )
51	42, 50, 48	seq3p1 10536	. . . . . . . 8 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( seq K ( .+ , G ) ` ( w + 1 ) ) = ( ( seq K ( .+ , G ) ` w ) .+ ( G ` ( w + 1 ) ) ) )
52		fveq2 5554	. . . . . . . . . . 11 |- ( k = ( w + 1 ) -> ( F ` k ) = ( F ` ( w + 1 ) ) )
53		fveq2 5554	. . . . . . . . . . 11 |- ( k = ( w + 1 ) -> ( G ` k ) = ( G ` ( w + 1 ) ) )
54	52, 53	eqeq12d 2208	. . . . . . . . . 10 |- ( k = ( w + 1 ) -> ( ( F ` k ) = ( G ` k ) <-> ( F ` ( w + 1 ) ) = ( G ` ( w + 1 ) ) ) )
55		seq3fveq2.4	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( ( K + 1 ) ... N ) ) -> ( F ` k ) = ( G ` k ) )
56	55	ralrimiva 2567	. . . . . . . . . . 11 |- ( ph -> A. k e. ( ( K + 1 ) ... N ) ( F ` k ) = ( G ` k ) )
57	56	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> A. k e. ( ( K + 1 ) ... N ) ( F ` k ) = ( G ` k ) )
58		eluzp1p1 9618	. . . . . . . . . . . 12 |- ( w e. ( ZZ>= ` K ) -> ( w + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
59	58	ad2antrl 490	. . . . . . . . . . 11 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( w + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
60		elfzuz3 10088	. . . . . . . . . . . 12 |- ( ( w + 1 ) e. ( K ... N ) -> N e. ( ZZ>= ` ( w + 1 ) ) )
61	60	ad2antll 491	. . . . . . . . . . 11 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> N e. ( ZZ>= ` ( w + 1 ) ) )
62		elfzuzb 10085	. . . . . . . . . . 11 |- ( ( w + 1 ) e. ( ( K + 1 ) ... N ) <-> ( ( w + 1 ) e. ( ZZ>= ` ( K + 1 ) ) /\ N e. ( ZZ>= ` ( w + 1 ) ) ) )
63	59, 61, 62	sylanbrc 417	. . . . . . . . . 10 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( w + 1 ) e. ( ( K + 1 ) ... N ) )
64	54, 57, 63	rspcdva 2869	. . . . . . . . 9 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( F ` ( w + 1 ) ) = ( G ` ( w + 1 ) ) )
65	64	oveq2d 5934	. . . . . . . 8 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( ( seq K ( .+ , G ) ` w ) .+ ( F ` ( w + 1 ) ) ) = ( ( seq K ( .+ , G ) ` w ) .+ ( G ` ( w + 1 ) ) ) )
66	51, 65	eqtr4d 2229	. . . . . . 7 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( seq K ( .+ , G ) ` ( w + 1 ) ) = ( ( seq K ( .+ , G ) ` w ) .+ ( F ` ( w + 1 ) ) ) )
67	49, 66	eqeq12d 2208	. . . . . 6 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( ( seq M ( .+ , F ) ` ( w + 1 ) ) = ( seq K ( .+ , G ) ` ( w + 1 ) ) <-> ( ( seq M ( .+ , F ) ` w ) .+ ( F ` ( w + 1 ) ) ) = ( ( seq K ( .+ , G ) ` w ) .+ ( F ` ( w + 1 ) ) ) ) )
68	41, 67	imbitrrid 156	. . . . 5 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( ( seq M ( .+ , F ) ` w ) = ( seq K ( .+ , G ) ` w ) -> ( seq M ( .+ , F ) ` ( w + 1 ) ) = ( seq K ( .+ , G ) ` ( w + 1 ) ) ) )
69	40, 68	animpimp2impd 559	. . . 4 |- ( w e. ( ZZ>= ` K ) -> ( ( ph -> ( w e. ( K ... N ) -> ( seq M ( .+ , F ) ` w ) = ( seq K ( .+ , G ) ` w ) ) ) -> ( ph -> ( ( w + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F ) ` ( w + 1 ) ) = ( seq K ( .+ , G ) ` ( w + 1 ) ) ) ) ) )
70	9, 15, 21, 27, 36, 69	uzind4 9653	. . 3 |- ( N e. ( ZZ>= ` K ) -> ( ph -> ( N e. ( K ... N ) -> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , G ) ` N ) ) ) )
71	1, 70	mpcom 36	. 2 |- ( ph -> ( N e. ( K ... N ) -> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , G ) ` N ) ) )
72	3, 71	mpd 13	1 |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , G ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   A.wral 2472   ` 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:  seq3feq2  10547  seq3fveq  10550  gsumsplit1r  12981