Theorem seqfveq2g 10694

Description: Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Hypotheses

Ref	Expression
seqfveq2.1	|- ( ph -> K e. ( ZZ>= ` M ) )
seqfveq2.2	|- ( ph -> ( seq M ( .+ , F ) ` K ) = ( G ` K ) )
seqfveq2g.p	|- ( ph -> .+ e. V )
seqfveq2g.f	|- ( ph -> F e. W )
seqfveq2g.g	|- ( ph -> G e. X )
seqfveq2.3	|- ( ph -> N e. ( ZZ>= ` K ) )
seqfveq2.4	|- ( ( ph /\ k e. ( ( K + 1 ) ... N ) ) -> ( F ` k ) = ( G ` k ) )

Assertion

Ref	Expression
seqfveq2g	|- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , G ) ` N ) )

Distinct variable groups:   k, F   k, G   k, K   k, N   ph, k

Allowed substitution hints:   .+ ( k)   M( k)   V( k)   W( k)   X( k)

Proof of Theorem seqfveq2g

Dummy variables n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqfveq2.3	. . 3 |- ( ph -> N e. ( ZZ>= ` K ) )
2		eluzfz2 10224	. . 3 |- ( N e. ( ZZ>= ` K ) -> N e. ( K ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( K ... N ) )
4		eleq1 2292	. . . . . 6 |- ( x = K -> ( x e. ( K ... N ) <-> K e. ( K ... N ) ) )
5		fveq2 5626	. . . . . . 7 |- ( x = K -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` K ) )
6		fveq2 5626	. . . . . . 7 |- ( x = K -> ( seq K ( .+ , G ) ` x ) = ( seq K ( .+ , G ) ` K ) )
7	5, 6	eqeq12d 2244	. . . . . 6 |- ( x = K -> ( ( seq M ( .+ , F ) ` x ) = ( seq K ( .+ , G ) ` x ) <-> ( seq M ( .+ , F ) ` K ) = ( seq K ( .+ , G ) ` K ) ) )
8	4, 7	imbi12d 234	. . . . 5 |- ( x = K -> ( ( x e. ( K ... N ) -> ( seq M ( .+ , F ) ` x ) = ( seq K ( .+ , G ) ` x ) ) <-> ( K e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq K ( .+ , G ) ` K ) ) ) )
9	8	imbi2d 230	. . . 4 |- ( x = K -> ( ( ph -> ( x e. ( K ... N ) -> ( seq M ( .+ , F ) ` x ) = ( seq K ( .+ , G ) ` x ) ) ) <-> ( ph -> ( K e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq K ( .+ , G ) ` K ) ) ) ) )
10		eleq1 2292	. . . . . 6 |- ( x = n -> ( x e. ( K ... N ) <-> n e. ( K ... N ) ) )
11		fveq2 5626	. . . . . . 7 |- ( x = n -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` n ) )
12		fveq2 5626	. . . . . . 7 |- ( x = n -> ( seq K ( .+ , G ) ` x ) = ( seq K ( .+ , G ) ` n ) )
13	11, 12	eqeq12d 2244	. . . . . 6 |- ( x = n -> ( ( seq M ( .+ , F ) ` x ) = ( seq K ( .+ , G ) ` x ) <-> ( seq M ( .+ , F ) ` n ) = ( seq K ( .+ , G ) ` n ) ) )
14	10, 13	imbi12d 234	. . . . 5 |- ( x = n -> ( ( x e. ( K ... N ) -> ( seq M ( .+ , F ) ` x ) = ( seq K ( .+ , G ) ` x ) ) <-> ( n e. ( K ... N ) -> ( seq M ( .+ , F ) ` n ) = ( seq K ( .+ , G ) ` n ) ) ) )
15	14	imbi2d 230	. . . 4 |- ( x = n -> ( ( ph -> ( x e. ( K ... N ) -> ( seq M ( .+ , F ) ` x ) = ( seq K ( .+ , G ) ` x ) ) ) <-> ( ph -> ( n e. ( K ... N ) -> ( seq M ( .+ , F ) ` n ) = ( seq K ( .+ , G ) ` n ) ) ) ) )
16		eleq1 2292	. . . . . 6 |- ( x = ( n + 1 ) -> ( x e. ( K ... N ) <-> ( n + 1 ) e. ( K ... N ) ) )
17		fveq2 5626	. . . . . . 7 |- ( x = ( n + 1 ) -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` ( n + 1 ) ) )
18		fveq2 5626	. . . . . . 7 |- ( x = ( n + 1 ) -> ( seq K ( .+ , G ) ` x ) = ( seq K ( .+ , G ) ` ( n + 1 ) ) )
19	17, 18	eqeq12d 2244	. . . . . 6 |- ( x = ( n + 1 ) -> ( ( seq M ( .+ , F ) ` x ) = ( seq K ( .+ , G ) ` x ) <-> ( seq M ( .+ , F ) ` ( n + 1 ) ) = ( seq K ( .+ , G ) ` ( n + 1 ) ) ) )
20	16, 19	imbi12d 234	. . . . 5 |- ( x = ( n + 1 ) -> ( ( x e. ( K ... N ) -> ( seq M ( .+ , F ) ` x ) = ( seq K ( .+ , G ) ` x ) ) <-> ( ( n + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F ) ` ( n + 1 ) ) = ( seq K ( .+ , G ) ` ( n + 1 ) ) ) ) )
21	20	imbi2d 230	. . . 4 |- ( x = ( n + 1 ) -> ( ( ph -> ( x e. ( K ... N ) -> ( seq M ( .+ , F ) ` x ) = ( seq K ( .+ , G ) ` x ) ) ) <-> ( ph -> ( ( n + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F ) ` ( n + 1 ) ) = ( seq K ( .+ , G ) ` ( n + 1 ) ) ) ) ) )
22		eleq1 2292	. . . . . 6 |- ( x = N -> ( x e. ( K ... N ) <-> N e. ( K ... N ) ) )
23		fveq2 5626	. . . . . . 7 |- ( x = N -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` N ) )
24		fveq2 5626	. . . . . . 7 |- ( x = N -> ( seq K ( .+ , G ) ` x ) = ( seq K ( .+ , G ) ` N ) )
25	23, 24	eqeq12d 2244	. . . . . 6 |- ( x = N -> ( ( seq M ( .+ , F ) ` x ) = ( seq K ( .+ , G ) ` x ) <-> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , G ) ` N ) ) )
26	22, 25	imbi12d 234	. . . . 5 |- ( x = N -> ( ( x e. ( K ... N ) -> ( seq M ( .+ , F ) ` x ) = ( seq K ( .+ , G ) ` x ) ) <-> ( N e. ( K ... N ) -> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , G ) ` N ) ) ) )
27	26	imbi2d 230	. . . 4 |- ( x = N -> ( ( ph -> ( x e. ( K ... N ) -> ( seq M ( .+ , F ) ` x ) = ( seq K ( .+ , G ) ` x ) ) ) <-> ( ph -> ( N e. ( K ... N ) -> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , G ) ` N ) ) ) ) )
28		seqfveq2.2	. . . . . 6 |- ( ph -> ( seq M ( .+ , F ) ` K ) = ( G ` K ) )
29		seqfveq2.1	. . . . . . . 8 |- ( ph -> K e. ( ZZ>= ` M ) )
30		eluzelz 9727	. . . . . . . 8 |- ( K e. ( ZZ>= ` M ) -> K e. ZZ )
31	29, 30	syl 14	. . . . . . 7 |- ( ph -> K e. ZZ )
32		seqfveq2g.g	. . . . . . 7 |- ( ph -> G e. X )
33		seqfveq2g.p	. . . . . . 7 |- ( ph -> .+ e. V )
34		seq1g 10680	. . . . . . 7 |- ( ( K e. ZZ /\ G e. X /\ .+ e. V ) -> ( seq K ( .+ , G ) ` K ) = ( G ` K ) )
35	31, 32, 33, 34	syl3anc 1271	. . . . . 6 |- ( ph -> ( seq K ( .+ , G ) ` K ) = ( G ` K ) )
36	28, 35	eqtr4d 2265	. . . . 5 |- ( ph -> ( seq M ( .+ , F ) ` K ) = ( seq K ( .+ , G ) ` K ) )
37	36	a1d 22	. . . 4 |- ( ph -> ( K e. ( K ... N ) -> ( seq M ( .+ , F ) ` K ) = ( seq K ( .+ , G ) ` K ) ) )
38		peano2fzr 10229	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) -> n e. ( K ... N ) )
39	38	adantl 277	. . . . . . 7 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> n e. ( K ... N ) )
40	39	expr 375	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` K ) ) -> ( ( n + 1 ) e. ( K ... N ) -> n e. ( K ... N ) ) )
41	40	imim1d 75	. . . . 5 |- ( ( ph /\ n e. ( ZZ>= ` K ) ) -> ( ( n e. ( K ... N ) -> ( seq M ( .+ , F ) ` n ) = ( seq K ( .+ , G ) ` n ) ) -> ( ( n + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F ) ` n ) = ( seq K ( .+ , G ) ` n ) ) ) )
42		oveq1 6007	. . . . . 6 |- ( ( seq M ( .+ , F ) ` n ) = ( seq K ( .+ , G ) ` n ) -> ( ( seq M ( .+ , F ) ` n ) .+ ( F ` ( n + 1 ) ) ) = ( ( seq K ( .+ , G ) ` n ) .+ ( F ` ( n + 1 ) ) ) )
43		simpl 109	. . . . . . . . 9 |- ( ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) -> n e. ( ZZ>= ` K ) )
44		uztrn 9735	. . . . . . . . 9 |- ( ( n e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> n e. ( ZZ>= ` M ) )
45	43, 29, 44	syl2anr 290	. . . . . . . 8 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> n e. ( ZZ>= ` M ) )
46		seqfveq2g.f	. . . . . . . . 9 |- ( ph -> F e. W )
47	46	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> F e. W )
48	33	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> .+ e. V )
49		seqp1g 10683	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` M ) /\ F e. W /\ .+ e. V ) -> ( seq M ( .+ , F ) ` ( n + 1 ) ) = ( ( seq M ( .+ , F ) ` n ) .+ ( F ` ( n + 1 ) ) ) )
50	45, 47, 48, 49	syl3anc 1271	. . . . . . 7 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( seq M ( .+ , F ) ` ( n + 1 ) ) = ( ( seq M ( .+ , F ) ` n ) .+ ( F ` ( n + 1 ) ) ) )
51	43	adantl 277	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> n e. ( ZZ>= ` K ) )
52	32	adantr 276	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> G e. X )
53		seqp1g 10683	. . . . . . . . 9 |- ( ( n e. ( ZZ>= ` K ) /\ G e. X /\ .+ e. V ) -> ( seq K ( .+ , G ) ` ( n + 1 ) ) = ( ( seq K ( .+ , G ) ` n ) .+ ( G ` ( n + 1 ) ) ) )
54	51, 52, 48, 53	syl3anc 1271	. . . . . . . 8 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( seq K ( .+ , G ) ` ( n + 1 ) ) = ( ( seq K ( .+ , G ) ` n ) .+ ( G ` ( n + 1 ) ) ) )
55		fveq2 5626	. . . . . . . . . . 11 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
56		fveq2 5626	. . . . . . . . . . 11 |- ( k = ( n + 1 ) -> ( G ` k ) = ( G ` ( n + 1 ) ) )
57	55, 56	eqeq12d 2244	. . . . . . . . . 10 |- ( k = ( n + 1 ) -> ( ( F ` k ) = ( G ` k ) <-> ( F ` ( n + 1 ) ) = ( G ` ( n + 1 ) ) ) )
58		seqfveq2.4	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( ( K + 1 ) ... N ) ) -> ( F ` k ) = ( G ` k ) )
59	58	ralrimiva 2603	. . . . . . . . . . 11 |- ( ph -> A. k e. ( ( K + 1 ) ... N ) ( F ` k ) = ( G ` k ) )
60	59	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> A. k e. ( ( K + 1 ) ... N ) ( F ` k ) = ( G ` k ) )
61		eluzp1p1 9744	. . . . . . . . . . . 12 |- ( n e. ( ZZ>= ` K ) -> ( n + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
62	61	ad2antrl 490	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( n + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
63		elfzuz3 10214	. . . . . . . . . . . 12 |- ( ( n + 1 ) e. ( K ... N ) -> N e. ( ZZ>= ` ( n + 1 ) ) )
64	63	ad2antll 491	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> N e. ( ZZ>= ` ( n + 1 ) ) )
65		elfzuzb 10211	. . . . . . . . . . 11 |- ( ( n + 1 ) e. ( ( K + 1 ) ... N ) <-> ( ( n + 1 ) e. ( ZZ>= ` ( K + 1 ) ) /\ N e. ( ZZ>= ` ( n + 1 ) ) ) )
66	62, 64, 65	sylanbrc 417	. . . . . . . . . 10 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( n + 1 ) e. ( ( K + 1 ) ... N ) )
67	57, 60, 66	rspcdva 2912	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( F ` ( n + 1 ) ) = ( G ` ( n + 1 ) ) )
68	67	oveq2d 6016	. . . . . . . 8 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( ( seq K ( .+ , G ) ` n ) .+ ( F ` ( n + 1 ) ) ) = ( ( seq K ( .+ , G ) ` n ) .+ ( G ` ( n + 1 ) ) ) )
69	54, 68	eqtr4d 2265	. . . . . . 7 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( seq K ( .+ , G ) ` ( n + 1 ) ) = ( ( seq K ( .+ , G ) ` n ) .+ ( F ` ( n + 1 ) ) ) )
70	50, 69	eqeq12d 2244	. . . . . 6 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( ( seq M ( .+ , F ) ` ( n + 1 ) ) = ( seq K ( .+ , G ) ` ( n + 1 ) ) <-> ( ( seq M ( .+ , F ) ` n ) .+ ( F ` ( n + 1 ) ) ) = ( ( seq K ( .+ , G ) ` n ) .+ ( F ` ( n + 1 ) ) ) ) )
71	42, 70	imbitrrid 156	. . . . 5 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( ( seq M ( .+ , F ) ` n ) = ( seq K ( .+ , G ) ` n ) -> ( seq M ( .+ , F ) ` ( n + 1 ) ) = ( seq K ( .+ , G ) ` ( n + 1 ) ) ) )
72	41, 71	animpimp2impd 559	. . . 4 |- ( n e. ( ZZ>= ` K ) -> ( ( ph -> ( n e. ( K ... N ) -> ( seq M ( .+ , F ) ` n ) = ( seq K ( .+ , G ) ` n ) ) ) -> ( ph -> ( ( n + 1 ) e. ( K ... N ) -> ( seq M ( .+ , F ) ` ( n + 1 ) ) = ( seq K ( .+ , G ) ` ( n + 1 ) ) ) ) ) )
73	9, 15, 21, 27, 37, 72	uzind4i 9783	. . 3 |- ( N e. ( ZZ>= ` K ) -> ( ph -> ( N e. ( K ... N ) -> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , G ) ` N ) ) ) )
74	1, 73	mpcom 36	. 2 |- ( ph -> ( N e. ( K ... N ) -> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , G ) ` N ) ) )
75	3, 74	mpd 13	1 |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , G ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   ` cfv 5317  (class class class)co 6000   1c1 7996   + caddc 7998   ZZcz 9442   ZZ>=cuz 9718   ...cfz 10200   seqcseq 10664

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-seqfrec 10665

This theorem is referenced by:  seqfveqg  10695