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Theorem seqfeq2 11834
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 10871 . . . 4  |-  ( K  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
3 seqfn 11823 . . . 4  |-  ( M  e.  ZZ  ->  seq M (  .+  ,  F )  Fn  ( ZZ>=
`  M ) )
41, 2, 33syl 20 . . 3  |-  ( ph  ->  seq M (  .+  ,  F )  Fn  ( ZZ>=
`  M ) )
5 uzss 10886 . . . 4  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  K )  C_  ( ZZ>=
`  M ) )
61, 5syl 16 . . 3  |-  ( ph  ->  ( ZZ>= `  K )  C_  ( ZZ>= `  M )
)
7 fnssres 5529 . . 3  |-  ( (  seq M (  .+  ,  F )  Fn  ( ZZ>=
`  M )  /\  ( ZZ>= `  K )  C_  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F )  |`  ( ZZ>=
`  K ) )  Fn  ( ZZ>= `  K
) )
84, 6, 7syl2anc 661 . 2  |-  ( ph  ->  (  seq M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  Fn  ( ZZ>= `  K ) )
9 eluzelz 10875 . . 3  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
10 seqfn 11823 . . 3  |-  ( K  e.  ZZ  ->  seq K (  .+  ,  G )  Fn  ( ZZ>=
`  K ) )
111, 9, 103syl 20 . 2  |-  ( ph  ->  seq K (  .+  ,  G )  Fn  ( ZZ>=
`  K ) )
12 fvres 5709 . . . 4  |-  ( x  e.  ( ZZ>= `  K
)  ->  ( (  seq M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 x )  =  (  seq M ( 
.+  ,  F ) `
 x ) )
1312adantl 466 . . 3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  ( (  seq M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 x )  =  (  seq M ( 
.+  ,  F ) `
 x ) )
141adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  K  e.  ( ZZ>= `  M )
)
15 seqfveq2.2 . . . . 5  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  ( G `  K
) )
1615adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  (  seq M (  .+  ,  F ) `  K
)  =  ( G `
 K ) )
17 simpr 461 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  x  e.  ( ZZ>= `  K )
)
18 elfzuz 11454 . . . . . 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 474 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( K  + 
1 ) ... x
) )  ->  ( F `  k )  =  ( G `  k ) )
2120adantlr 714 . . . 4  |-  ( ( ( ph  /\  x  e.  ( ZZ>= `  K )
)  /\  k  e.  ( ( K  + 
1 ) ... x
) )  ->  ( F `  k )  =  ( G `  k ) )
2214, 16, 17, 21seqfveq2 11833 . . 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 5805 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  =  seq K
(  .+  ,  G
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3333    |` cres 4847    Fn wfn 5418   ` cfv 5423  (class class class)co 6096   1c1 9288    + caddc 9290   ZZcz 10651   ZZ>=cuz 10866   ...cfz 11442    seqcseq 11811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-seq 11812
This theorem is referenced by:  seqid  11856
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