Theorem seq3fveq2 10657

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 10189	. . 3 |- ( N e. ( ZZ>= ` K ) -> N e. ( K ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( K ... N ) )
4		eleq1 2270	. . . . . 6 |- ( z = K -> ( z e. ( K ... N ) <-> K e. ( K ... N ) ) )
5		fveq2 5599	. . . . . . 7 |- ( z = K -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , F ) ` K ) )
6		fveq2 5599	. . . . . . 7 |- ( z = K -> ( seq K ( .+ , G ) ` z ) = ( seq K ( .+ , G ) ` K ) )
7	5, 6	eqeq12d 2222	. . . . . 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 2270	. . . . . 6 |- ( z = w -> ( z e. ( K ... N ) <-> w e. ( K ... N ) ) )
11		fveq2 5599	. . . . . . 7 |- ( z = w -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , F ) ` w ) )
12		fveq2 5599	. . . . . . 7 |- ( z = w -> ( seq K ( .+ , G ) ` z ) = ( seq K ( .+ , G ) ` w ) )
13	11, 12	eqeq12d 2222	. . . . . 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 2270	. . . . . 6 |- ( z = ( w + 1 ) -> ( z e. ( K ... N ) <-> ( w + 1 ) e. ( K ... N ) ) )
17		fveq2 5599	. . . . . . 7 |- ( z = ( w + 1 ) -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , F ) ` ( w + 1 ) ) )
18		fveq2 5599	. . . . . . 7 |- ( z = ( w + 1 ) -> ( seq K ( .+ , G ) ` z ) = ( seq K ( .+ , G ) ` ( w + 1 ) ) )
19	17, 18	eqeq12d 2222	. . . . . 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 2270	. . . . . 6 |- ( z = N -> ( z e. ( K ... N ) <-> N e. ( K ... N ) ) )
23		fveq2 5599	. . . . . . 7 |- ( z = N -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , F ) ` N ) )
24		fveq2 5599	. . . . . . 7 |- ( z = N -> ( seq K ( .+ , G ) ` z ) = ( seq K ( .+ , G ) ` N ) )
25	23, 24	eqeq12d 2222	. . . . . 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 9692	. . . . . . . 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 10644	. . . . . 6 |- ( ph -> ( seq K ( .+ , G ) ` K ) = ( G ` K ) )
35	28, 34	eqtr4d 2243	. . . . 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 10194	. . . . . . . 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 5974	. . . . . 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 9700	. . . . . . . . 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 10647	. . . . . . 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 10647	. . . . . . . 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 5599	. . . . . . . . . . 11 |- ( k = ( w + 1 ) -> ( F ` k ) = ( F ` ( w + 1 ) ) )
53		fveq2 5599	. . . . . . . . . . 11 |- ( k = ( w + 1 ) -> ( G ` k ) = ( G ` ( w + 1 ) ) )
54	52, 53	eqeq12d 2222	. . . . . . . . . 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 2581	. . . . . . . . . . 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 9709	. . . . . . . . . . . 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 10179	. . . . . . . . . . . 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 10176	. . . . . . . . . . 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 2889	. . . . . . . . 9 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( F ` ( w + 1 ) ) = ( G ` ( w + 1 ) ) )
65	64	oveq2d 5983	. . . . . . . 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 2243	. . . . . . 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 2222	. . . . . 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 9744	. . 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 1373   e. wcel 2178   A.wral 2486   ` cfv 5290  (class class class)co 5967   1c1 7961   + caddc 7963   ZZcz 9407   ZZ>=cuz 9683   ...cfz 10165   seqcseq 10629

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166  df-seqfrec 10630

This theorem is referenced by:  seq3feq2  10658  seq3fveq  10661  gsumsplit1r  13345