Theorem seq3fveq2 10912

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 10436	. . 3 |- ( N e. ( ZZ>= ` K ) -> N e. ( K ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( K ... N ) )
4		eleq1 2301	. . . . . 6 |- ( z = K -> ( z e. ( K ... N ) <-> K e. ( K ... N ) ) )
5		fveq2 5695	. . . . . . 7 |- ( z = K -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , F ) ` K ) )
6		fveq2 5695	. . . . . . 7 |- ( z = K -> ( seq K ( .+ , G ) ` z ) = ( seq K ( .+ , G ) ` K ) )
7	5, 6	eqeq12d 2253	. . . . . 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 2301	. . . . . 6 |- ( z = w -> ( z e. ( K ... N ) <-> w e. ( K ... N ) ) )
11		fveq2 5695	. . . . . . 7 |- ( z = w -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , F ) ` w ) )
12		fveq2 5695	. . . . . . 7 |- ( z = w -> ( seq K ( .+ , G ) ` z ) = ( seq K ( .+ , G ) ` w ) )
13	11, 12	eqeq12d 2253	. . . . . 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 2301	. . . . . 6 |- ( z = ( w + 1 ) -> ( z e. ( K ... N ) <-> ( w + 1 ) e. ( K ... N ) ) )
17		fveq2 5695	. . . . . . 7 |- ( z = ( w + 1 ) -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , F ) ` ( w + 1 ) ) )
18		fveq2 5695	. . . . . . 7 |- ( z = ( w + 1 ) -> ( seq K ( .+ , G ) ` z ) = ( seq K ( .+ , G ) ` ( w + 1 ) ) )
19	17, 18	eqeq12d 2253	. . . . . 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 2301	. . . . . 6 |- ( z = N -> ( z e. ( K ... N ) <-> N e. ( K ... N ) ) )
23		fveq2 5695	. . . . . . 7 |- ( z = N -> ( seq M ( .+ , F ) ` z ) = ( seq M ( .+ , F ) ` N ) )
24		fveq2 5695	. . . . . . 7 |- ( z = N -> ( seq K ( .+ , G ) ` z ) = ( seq K ( .+ , G ) ` N ) )
25	23, 24	eqeq12d 2253	. . . . . 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 9931	. . . . . . . 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 10899	. . . . . 6 |- ( ph -> ( seq K ( .+ , G ) ` K ) = ( G ` K ) )
35	28, 34	eqtr4d 2274	. . . . 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 10441	. . . . . . . 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 6092	. . . . . 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 535	. . . . . . . . 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 9939	. . . . . . . . 9 |- ( ( w e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> w e. ( ZZ>= ` M ) )
45	42, 43, 44	syl2anc 415	. . . . . . . 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 481	. . . . . . . 8 |- ( ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
48	33	adantlr 481	. . . . . . . 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 10902	. . . . . . 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 481	. . . . . . . . 9 |- ( ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) /\ x e. ( ZZ>= ` K ) ) -> ( G ` x ) e. S )
51	42, 50, 48	seq3p1 10902	. . . . . . . 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 5695	. . . . . . . . . . 11 |- ( k = ( w + 1 ) -> ( F ` k ) = ( F ` ( w + 1 ) ) )
53		fveq2 5695	. . . . . . . . . . 11 |- ( k = ( w + 1 ) -> ( G ` k ) = ( G ` ( w + 1 ) ) )
54	52, 53	eqeq12d 2253	. . . . . . . . . 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 2623	. . . . . . . . . . 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 9948	. . . . . . . . . . . 12 |- ( w e. ( ZZ>= ` K ) -> ( w + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
59	58	ad2antrl 494	. . . . . . . . . . 11 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( w + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
60		elfzuz3 10425	. . . . . . . . . . . 12 |- ( ( w + 1 ) e. ( K ... N ) -> N e. ( ZZ>= ` ( w + 1 ) ) )
61	60	ad2antll 495	. . . . . . . . . . 11 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> N e. ( ZZ>= ` ( w + 1 ) ) )
62		elfzuzb 10422	. . . . . . . . . . 11 |- ( ( w + 1 ) e. ( ( K + 1 ) ... N ) <-> ( ( w + 1 ) e. ( ZZ>= ` ( K + 1 ) ) /\ N e. ( ZZ>= ` ( w + 1 ) ) ) )
63	59, 61, 62	sylanbrc 421	. . . . . . . . . 10 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( w + 1 ) e. ( ( K + 1 ) ... N ) )
64	54, 57, 63	rspcdva 2934	. . . . . . . . 9 |- ( ( ph /\ ( w e. ( ZZ>= ` K ) /\ ( w + 1 ) e. ( K ... N ) ) ) -> ( F ` ( w + 1 ) ) = ( G ` ( w + 1 ) ) )
65	64	oveq2d 6101	. . . . . . . 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 2274	. . . . . . 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 2253	. . . . . 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 565	. . . 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 9988	. . 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   ` 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:  seq3feq2  10913  seq3fveq  10916