Theorem seq3fveq2 10569

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 10109	. . 3 |- ( N e. ( ZZ>= ` K ) -> N e. ( K ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( K ... N ) )
4		eleq1 2259	. . . . . 6 |- ( z = K -> ( z e. ( K ... N ) <-> K e. ( K ... N ) ) )
5		fveq2 5559	. . . . . . 7 |- ( z = K -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , F ) ` K ) )
6		fveq2 5559	. . . . . . 7 |- ( z = K -> ( seq K ( .+ , G ) ` z ) = ( seq K ( .+ , G ) ` K ) )
7	5, 6	eqeq12d 2211	. . . . . 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 2259	. . . . . 6 |- ( z = w -> ( z e. ( K ... N ) <-> w e. ( K ... N ) ) )
11		fveq2 5559	. . . . . . 7 |- ( z = w -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , F ) ` w ) )
12		fveq2 5559	. . . . . . 7 |- ( z = w -> ( seq K ( .+ , G ) ` z ) = ( seq K ( .+ , G ) ` w ) )
13	11, 12	eqeq12d 2211	. . . . . 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 2259	. . . . . 6 |- ( z = ( w + 1 ) -> ( z e. ( K ... N ) <-> ( w + 1 ) e. ( K ... N ) ) )
17		fveq2 5559	. . . . . . 7 |- ( z = ( w + 1 ) -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , F ) ` ( w + 1 ) ) )
18		fveq2 5559	. . . . . . 7 |- ( z = ( w + 1 ) -> ( seq K ( .+ , G ) ` z ) = ( seq K ( .+ , G ) ` ( w + 1 ) ) )
19	17, 18	eqeq12d 2211	. . . . . 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 2259	. . . . . 6 |- ( z = N -> ( z e. ( K ... N ) <-> N e. ( K ... N ) ) )
23		fveq2 5559	. . . . . . 7 |- ( z = N -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , F ) ` N ) )
24		fveq2 5559	. . . . . . 7 |- ( z = N -> ( seq K ( .+ , G ) ` z ) = ( seq K ( .+ , G ) ` N ) )
25	23, 24	eqeq12d 2211	. . . . . 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 9612	. . . . . . . 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 10556	. . . . . 6 |- ( ph -> ( seq K ( .+ , G ) ` K ) = ( G ` K ) )
35	28, 34	eqtr4d 2232	. . . . 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 10114	. . . . . . . 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 5930	. . . . . 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 9620	. . . . . . . . 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 10559	. . . . . . 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 10559	. . . . . . . 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 5559	. . . . . . . . . . 11 |- ( k = ( w + 1 ) -> ( F ` k ) = ( F ` ( w + 1 ) ) )
53		fveq2 5559	. . . . . . . . . . 11 |- ( k = ( w + 1 ) -> ( G ` k ) = ( G ` ( w + 1 ) ) )
54	52, 53	eqeq12d 2211	. . . . . . . . . 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 2570	. . . . . . . . . . 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 9629	. . . . . . . . . . . 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 10099	. . . . . . . . . . . 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 10096	. . . . . . . . . . 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 2873	. . . . . . . . 9 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( F ` ( w + 1 ) ) = ( G ` ( w + 1 ) ) )
65	64	oveq2d 5939	. . . . . . . 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 2232	. . . . . . 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 2211	. . . . . 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 9664	. . 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 2167   A.wral 2475   ` cfv 5259  (class class class)co 5923   1c1 7882   + caddc 7884   ZZcz 9328   ZZ>=cuz 9603   ...cfz 10085   seqcseq 10541

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-1re 7975  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-addcom 7981  ax-addass 7983  ax-distr 7985  ax-i2m1 7986  ax-0lt1 7987  ax-0id 7989  ax-rnegex 7990  ax-cnre 7992  ax-pre-ltirr 7993  ax-pre-ltwlin 7994  ax-pre-lttrn 7995  ax-pre-ltadd 7997

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-1st 6199  df-2nd 6200  df-recs 6364  df-frec 6450  df-pnf 8065  df-mnf 8066  df-xr 8067  df-ltxr 8068  df-le 8069  df-sub 8201  df-neg 8202  df-inn 8993  df-n0 9252  df-z 9329  df-uz 9604  df-fz 10086  df-seqfrec 10542

This theorem is referenced by:  seq3feq2  10570  seq3fveq  10573  gsumsplit1r  13051