MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seqfeq2 Structured version   Unicode version

Theorem seqfeq2 11384
Description: Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (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
) )
seqfeq2.4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( K  +  1 ) ) )  ->  ( F `  k )  =  ( G `  k ) )
Assertion
Ref Expression
seqfeq2  |-  ( ph  ->  (  seq  M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  =  seq  K
(  .+  ,  G
) )
Distinct variable groups:    k, F    k, G    k, K    ph, k
Allowed substitution hints:    .+ ( k)    M( k)

Proof of Theorem seqfeq2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 seqfveq2.1 . . . 4  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
2 eluzel2 10531 . . . 4  |-  ( K  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
3 seqfn 11373 . . . 4  |-  ( M  e.  ZZ  ->  seq  M (  .+  ,  F
)  Fn  ( ZZ>= `  M ) )
41, 2, 33syl 19 . . 3  |-  ( ph  ->  seq  M (  .+  ,  F )  Fn  ( ZZ>=
`  M ) )
5 uzss 10544 . . . 4  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  K )  C_  ( ZZ>=
`  M ) )
61, 5syl 16 . . 3  |-  ( ph  ->  ( ZZ>= `  K )  C_  ( ZZ>= `  M )
)
7 fnssres 5593 . . 3  |-  ( (  seq  M (  .+  ,  F )  Fn  ( ZZ>=
`  M )  /\  ( ZZ>= `  K )  C_  ( ZZ>= `  M )
)  ->  (  seq  M (  .+  ,  F
)  |`  ( ZZ>= `  K
) )  Fn  ( ZZ>=
`  K ) )
84, 6, 7syl2anc 644 . 2  |-  ( ph  ->  (  seq  M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  Fn  ( ZZ>= `  K ) )
9 eluzelz 10534 . . 3  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
10 seqfn 11373 . . 3  |-  ( K  e.  ZZ  ->  seq  K (  .+  ,  G
)  Fn  ( ZZ>= `  K ) )
111, 9, 103syl 19 . 2  |-  ( ph  ->  seq  K (  .+  ,  G )  Fn  ( ZZ>=
`  K ) )
12 fvres 5776 . . . 4  |-  ( x  e.  ( ZZ>= `  K
)  ->  ( (  seq  M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 x )  =  (  seq  M ( 
.+  ,  F ) `
 x ) )
1312adantl 454 . . 3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  ( (  seq  M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 x )  =  (  seq  M ( 
.+  ,  F ) `
 x ) )
141adantr 453 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  K  e.  ( ZZ>= `  M )
)
15 seqfveq2.2 . . . . 5  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 K )  =  ( G `  K
) )
1615adantr 453 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  (  seq  M (  .+  ,  F
) `  K )  =  ( G `  K ) )
17 simpr 449 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  x  e.  ( ZZ>= `  K )
)
18 elfzuz 11093 . . . . . 6  |-  ( k  e.  ( ( K  +  1 ) ... x )  ->  k  e.  ( ZZ>= `  ( K  +  1 ) ) )
19 seqfeq2.4 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( K  +  1 ) ) )  ->  ( F `  k )  =  ( G `  k ) )
2018, 19sylan2 462 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( K  + 
1 ) ... x
) )  ->  ( F `  k )  =  ( G `  k ) )
2120adantlr 697 . . . 4  |-  ( ( ( ph  /\  x  e.  ( ZZ>= `  K )
)  /\  k  e.  ( ( K  + 
1 ) ... x
) )  ->  ( F `  k )  =  ( G `  k ) )
2214, 16, 17, 21seqfveq2 11383 . . 3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  (  seq  M (  .+  ,  F
) `  x )  =  (  seq  K ( 
.+  ,  G ) `
 x ) )
2313, 22eqtrd 2475 . 2  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  ( (  seq  M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 x )  =  (  seq  K ( 
.+  ,  G ) `
 x ) )
248, 11, 23eqfnfvd 5866 1  |-  ( ph  ->  (  seq  M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  =  seq  K
(  .+  ,  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1654    e. wcel 1728    C_ wss 3309    |` cres 4915    Fn wfn 5484   ` cfv 5489  (class class class)co 6117   1c1 9029    + caddc 9031   ZZcz 10320   ZZ>=cuz 10526   ...cfz 11081    seq cseq 11361
This theorem is referenced by:  seqid  11406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-cnex 9084  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104  ax-pre-mulgt0 9105
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-riota 6585  df-recs 6669  df-rdg 6704  df-er 6941  df-en 7146  df-dom 7147  df-sdom 7148  df-pnf 9160  df-mnf 9161  df-xr 9162  df-ltxr 9163  df-le 9164  df-sub 9331  df-neg 9332  df-nn 10039  df-n0 10260  df-z 10321  df-uz 10527  df-fz 11082  df-seq 11362
  Copyright terms: Public domain W3C validator