Theorem seqfveqg 10839

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

Hypotheses

Ref	Expression
seqfveq.1	|- ( ph -> N e. ( ZZ>= ` M ) )
seqfveq.2	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) = ( G ` k ) )
seqfveqg.p	|- ( ph -> .+ e. V )
seqfveqg.f	|- ( ph -> F e. W )
seqfveqg.g	|- ( ph -> G e. X )

Assertion

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

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

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

Proof of Theorem seqfveqg

Step	Hyp	Ref	Expression
1		seqfveq.1	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
2		eluzel2 9857	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
3	1, 2	syl 14	. . 3 |- ( ph -> M e. ZZ )
4	3	uzidd 9868	. 2 |- ( ph -> M e. ( ZZ>= ` M ) )
5		seqfveqg.f	. . . 4 |- ( ph -> F e. W )
6		seqfveqg.p	. . . 4 |- ( ph -> .+ e. V )
7		seq1g 10824	. . . 4 |- ( ( M e. ZZ /\ F e. W /\ .+ e. V ) -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
8	3, 5, 6, 7	syl3anc 1274	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
9		fveq2 5669	. . . . 5 |- ( k = M -> ( F ` k ) = ( F ` M ) )
10		fveq2 5669	. . . . 5 |- ( k = M -> ( G ` k ) = ( G ` M ) )
11	9, 10	eqeq12d 2247	. . . 4 |- ( k = M -> ( ( F ` k ) = ( G ` k ) <-> ( F ` M ) = ( G ` M ) ) )
12		seqfveq.2	. . . . 5 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) = ( G ` k ) )
13	12	ralrimiva 2615	. . . 4 |- ( ph -> A. k e. ( M ... N ) ( F ` k ) = ( G ` k ) )
14		eluzfz1 10364	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
15	1, 14	syl 14	. . . 4 |- ( ph -> M e. ( M ... N ) )
16	11, 13, 15	rspcdva 2925	. . 3 |- ( ph -> ( F ` M ) = ( G ` M ) )
17	8, 16	eqtrd 2265	. 2 |- ( ph -> ( seq M ( .+ , F ) ` M ) = ( G ` M ) )
18		seqfveqg.g	. 2 |- ( ph -> G e. X )
19		fzp1ss 10406	. . . . 5 |- ( M e. ZZ -> ( ( M + 1 ) ... N ) C_ ( M ... N ) )
20	3, 19	syl 14	. . . 4 |- ( ph -> ( ( M + 1 ) ... N ) C_ ( M ... N ) )
21	20	sselda 3237	. . 3 |- ( ( ph /\ k e. ( ( M + 1 ) ... N ) ) -> k e. ( M ... N ) )
22	21, 12	syldan 282	. 2 |- ( ( ph /\ k e. ( ( M + 1 ) ... N ) ) -> ( F ` k ) = ( G ` k ) )
23	4, 17, 6, 5, 18, 1, 22	seqfveq2g 10838	1 |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq M ( .+ , G ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   C_ wss 3210   ` cfv 5351  (class class class)co 6049   1c1 8127   + caddc 8129   ZZcz 9576   ZZ>=cuz 9852   ...cfz 10341   seqcseq 10808

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-ltadd 8242

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-inn 9237  df-n0 9496  df-z 9577  df-uz 9853  df-fz 10342  df-seqfrec 10809

This theorem is referenced by:  seqf1oglem2  10881  seqf1og  10882