Theorem seqfveq2g 10892

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 10415	. . 3 |- ( N e. ( ZZ>= ` K ) -> N e. ( K ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( K ... N ) )
4		eleq1 2301	. . . . . 6 |- ( x = K -> ( x e. ( K ... N ) <-> K e. ( K ... N ) ) )
5		fveq2 5690	. . . . . . 7 |- ( x = K -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` K ) )
6		fveq2 5690	. . . . . . 7 |- ( x = K -> ( seq K ( .+ , G ) ` x ) = ( seq K ( .+ , G ) ` K ) )
7	5, 6	eqeq12d 2253	. . . . . 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 2301	. . . . . 6 |- ( x = n -> ( x e. ( K ... N ) <-> n e. ( K ... N ) ) )
11		fveq2 5690	. . . . . . 7 |- ( x = n -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` n ) )
12		fveq2 5690	. . . . . . 7 |- ( x = n -> ( seq K ( .+ , G ) ` x ) = ( seq K ( .+ , G ) ` n ) )
13	11, 12	eqeq12d 2253	. . . . . 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 2301	. . . . . 6 |- ( x = ( n + 1 ) -> ( x e. ( K ... N ) <-> ( n + 1 ) e. ( K ... N ) ) )
17		fveq2 5690	. . . . . . 7 |- ( x = ( n + 1 ) -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` ( n + 1 ) ) )
18		fveq2 5690	. . . . . . 7 |- ( x = ( n + 1 ) -> ( seq K ( .+ , G ) ` x ) = ( seq K ( .+ , G ) ` ( n + 1 ) ) )
19	17, 18	eqeq12d 2253	. . . . . 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 2301	. . . . . 6 |- ( x = N -> ( x e. ( K ... N ) <-> N e. ( K ... N ) ) )
23		fveq2 5690	. . . . . . 7 |- ( x = N -> ( seq M ( .+ , F ) ` x ) = ( seq M ( .+ , F ) ` N ) )
24		fveq2 5690	. . . . . . 7 |- ( x = N -> ( seq K ( .+ , G ) ` x ) = ( seq K ( .+ , G ) ` N ) )
25	23, 24	eqeq12d 2253	. . . . . 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 9910	. . . . . . . 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 10878	. . . . . . 7 |- ( ( K e. ZZ /\ G e. X /\ .+ e. V ) -> ( seq K ( .+ , G ) ` K ) = ( G ` K ) )
35	31, 32, 33, 34	syl3anc 1278	. . . . . 6 |- ( ph -> ( seq K ( .+ , G ) ` K ) = ( G ` K ) )
36	28, 35	eqtr4d 2274	. . . . 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 10420	. . . . . . . 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 6082	. . . . . 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 9918	. . . . . . . . 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 10881	. . . . . . . 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 1278	. . . . . . 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 10881	. . . . . . . . 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 1278	. . . . . . . 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 5690	. . . . . . . . . . 11 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
56		fveq2 5690	. . . . . . . . . . 11 |- ( k = ( n + 1 ) -> ( G ` k ) = ( G ` ( n + 1 ) ) )
57	55, 56	eqeq12d 2253	. . . . . . . . . 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 2623	. . . . . . . . . . 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 9927	. . . . . . . . . . . 12 |- ( n e. ( ZZ>= ` K ) -> ( n + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
62	61	ad2antrl 494	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( n + 1 ) e. ( ZZ>= ` ( K + 1 ) ) )
63		elfzuz3 10404	. . . . . . . . . . . 12 |- ( ( n + 1 ) e. ( K ... N ) -> N e. ( ZZ>= ` ( n + 1 ) ) )
64	63	ad2antll 495	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> N e. ( ZZ>= ` ( n + 1 ) ) )
65		elfzuzb 10401	. . . . . . . . . . 11 |- ( ( n + 1 ) e. ( ( K + 1 ) ... N ) <-> ( ( n + 1 ) e. ( ZZ>= ` ( K + 1 ) ) /\ N e. ( ZZ>= ` ( n + 1 ) ) ) )
66	62, 64, 65	sylanbrc 421	. . . . . . . . . 10 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( n + 1 ) e. ( ( K + 1 ) ... N ) )
67	57, 60, 66	rspcdva 2934	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( ZZ>= ` K ) /\ ( n + 1 ) e. ( K ... N ) ) ) -> ( F ` ( n + 1 ) ) = ( G ` ( n + 1 ) ) )
68	67	oveq2d 6091	. . . . . . . 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 2274	. . . . . . 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 2253	. . . . . 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 565	. . . 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 9971	. . 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 1402   e. wcel 2209   A.wral 2528   ` cfv 5372  (class class class)co 6075   1c1 8170   + caddc 8172   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   seqcseq 10862

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-seqfrec 10863

This theorem is referenced by:  seqfveqg  10893  gzsumsplit1r  13692