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Theorem seqfveq 11070
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 ) )
Assertion
Ref Expression
seqfveq  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 N )  =  (  seq  M ( 
.+  ,  G ) `
 N ) )
Distinct variable groups:    k, F    k, G    k, M    k, N    ph, k
Allowed substitution hint:    .+ ( k)

Proof of Theorem seqfveq
StepHypRef Expression
1 seqfveq.1 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 eluzel2 10235 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
31, 2syl 15 . . 3  |-  ( ph  ->  M  e.  ZZ )
4 uzid 10242 . . 3  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
53, 4syl 15 . 2  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
6 seq1 11059 . . . 4  |-  ( M  e.  ZZ  ->  (  seq  M (  .+  ,  F ) `  M
)  =  ( F `
 M ) )
73, 6syl 15 . . 3  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 M )  =  ( F `  M
) )
8 eluzfz1 10803 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
91, 8syl 15 . . . 4  |-  ( ph  ->  M  e.  ( M ... N ) )
10 seqfveq.2 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  =  ( G `  k ) )
1110ralrimiva 2626 . . . 4  |-  ( ph  ->  A. k  e.  ( M ... N ) ( F `  k
)  =  ( G `
 k ) )
12 fveq2 5525 . . . . . 6  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
13 fveq2 5525 . . . . . 6  |-  ( k  =  M  ->  ( G `  k )  =  ( G `  M ) )
1412, 13eqeq12d 2297 . . . . 5  |-  ( k  =  M  ->  (
( F `  k
)  =  ( G `
 k )  <->  ( F `  M )  =  ( G `  M ) ) )
1514rspcv 2880 . . . 4  |-  ( M  e.  ( M ... N )  ->  ( A. k  e.  ( M ... N ) ( F `  k )  =  ( G `  k )  ->  ( F `  M )  =  ( G `  M ) ) )
169, 11, 15sylc 56 . . 3  |-  ( ph  ->  ( F `  M
)  =  ( G `
 M ) )
177, 16eqtrd 2315 . 2  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 M )  =  ( G `  M
) )
18 fzp1ss 10837 . . . . 5  |-  ( M  e.  ZZ  ->  (
( M  +  1 ) ... N ) 
C_  ( M ... N ) )
193, 18syl 15 . . . 4  |-  ( ph  ->  ( ( M  + 
1 ) ... N
)  C_  ( M ... N ) )
2019sselda 3180 . . 3  |-  ( (
ph  /\  k  e.  ( ( M  + 
1 ) ... N
) )  ->  k  e.  ( M ... N
) )
2120, 10syldan 456 . 2  |-  ( (
ph  /\  k  e.  ( ( M  + 
1 ) ... N
) )  ->  ( F `  k )  =  ( G `  k ) )
225, 17, 1, 21seqfveq2 11068 1  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 N )  =  (  seq  M ( 
.+  ,  G ) `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ` cfv 5255  (class class class)co 5858   1c1 8738    + caddc 8740   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046
This theorem is referenced by:  seqfeq  11071  seqf1olem2  11086  seqf1o  11087  sumeq2ii  12166  fsum  12193  fsumser  12203  gsumccat  14464  gsumzaddlem  15203  gsumconst  15209  wilthlem3  20308  prodeq3ii  24720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047
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