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Theorem seq3feq2 9947
Description: Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
Hypotheses
Ref Expression
seq3fveq2.1  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
seq3fveq2.2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  ( G `  K
) )
seq3fveq2.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
seq3fveq2.g  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  ( G `  x )  e.  S
)
seq3fveq2.pl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
seq3feq2.4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( K  +  1 ) ) )  ->  ( F `  k )  =  ( G `  k ) )
Assertion
Ref Expression
seq3feq2  |-  ( ph  ->  (  seq M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  =  seq K
(  .+  ,  G
) )
Distinct variable groups:    x, k, y, F    k, G, x, y    k, K, x, y    ph, k, x, y   
k, M, x, y    .+ , k, x, y    S, k, x, y

Proof of Theorem seq3feq2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqid 2089 . . . . 5  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 seq3fveq2.1 . . . . . 6  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
3 eluzel2 9078 . . . . . 6  |-  ( K  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
42, 3syl 14 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
5 seq3fveq2.f . . . . 5  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
6 seq3fveq2.pl . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
71, 4, 5, 6seqf 9934 . . . 4  |-  ( ph  ->  seq M (  .+  ,  F ) : (
ZZ>= `  M ) --> S )
87ffnd 5175 . . 3  |-  ( ph  ->  seq M (  .+  ,  F )  Fn  ( ZZ>=
`  M ) )
9 uzss 9093 . . . 4  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  K )  C_  ( ZZ>=
`  M ) )
102, 9syl 14 . . 3  |-  ( ph  ->  ( ZZ>= `  K )  C_  ( ZZ>= `  M )
)
11 fnssres 5140 . . 3  |-  ( (  seq M (  .+  ,  F )  Fn  ( ZZ>=
`  M )  /\  ( ZZ>= `  K )  C_  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F )  |`  ( ZZ>=
`  K ) )  Fn  ( ZZ>= `  K
) )
128, 10, 11syl2anc 404 . 2  |-  ( ph  ->  (  seq M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  Fn  ( ZZ>= `  K ) )
13 eqid 2089 . . . 4  |-  ( ZZ>= `  K )  =  (
ZZ>= `  K )
14 eluzelz 9082 . . . . 5  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
152, 14syl 14 . . . 4  |-  ( ph  ->  K  e.  ZZ )
16 seq3fveq2.g . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  ( G `  x )  e.  S
)
1713, 15, 16, 6seqf 9934 . . 3  |-  ( ph  ->  seq K (  .+  ,  G ) : (
ZZ>= `  K ) --> S )
1817ffnd 5175 . 2  |-  ( ph  ->  seq K (  .+  ,  G )  Fn  ( ZZ>=
`  K ) )
19 fvres 5342 . . . 4  |-  ( z  e.  ( ZZ>= `  K
)  ->  ( (  seq M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 z )  =  (  seq M ( 
.+  ,  F ) `
 z ) )
2019adantl 272 . . 3  |-  ( (
ph  /\  z  e.  ( ZZ>= `  K )
)  ->  ( (  seq M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 z )  =  (  seq M ( 
.+  ,  F ) `
 z ) )
212adantr 271 . . . 4  |-  ( (
ph  /\  z  e.  ( ZZ>= `  K )
)  ->  K  e.  ( ZZ>= `  M )
)
22 seq3fveq2.2 . . . . 5  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  ( G `  K
) )
2322adantr 271 . . . 4  |-  ( (
ph  /\  z  e.  ( ZZ>= `  K )
)  ->  (  seq M (  .+  ,  F ) `  K
)  =  ( G `
 K ) )
245adantlr 462 . . . 4  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  K )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
2516adantlr 462 . . . 4  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  K )
)  /\  x  e.  ( ZZ>= `  K )
)  ->  ( G `  x )  e.  S
)
266adantlr 462 . . . 4  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  K )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
27 simpr 109 . . . 4  |-  ( (
ph  /\  z  e.  ( ZZ>= `  K )
)  ->  z  e.  ( ZZ>= `  K )
)
28 elfzuz 9490 . . . . . 6  |-  ( k  e.  ( ( K  +  1 ) ... z )  ->  k  e.  ( ZZ>= `  ( K  +  1 ) ) )
29 seq3feq2.4 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( K  +  1 ) ) )  ->  ( F `  k )  =  ( G `  k ) )
3028, 29sylan2 281 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( K  + 
1 ) ... z
) )  ->  ( F `  k )  =  ( G `  k ) )
3130adantlr 462 . . . 4  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  K )
)  /\  k  e.  ( ( K  + 
1 ) ... z
) )  ->  ( F `  k )  =  ( G `  k ) )
3221, 23, 24, 25, 26, 27, 31seq3fveq2 9946 . . 3  |-  ( (
ph  /\  z  e.  ( ZZ>= `  K )
)  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
) )
3320, 32eqtrd 2121 . 2  |-  ( (
ph  /\  z  e.  ( ZZ>= `  K )
)  ->  ( (  seq M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 z )  =  (  seq K ( 
.+  ,  G ) `
 z ) )
3412, 18, 33eqfnfvd 5414 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  =  seq K
(  .+  ,  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1290    e. wcel 1439    C_ wss 3000    |` cres 4453    Fn wfn 5023   ` cfv 5028  (class class class)co 5666   1c1 7405    + caddc 7407   ZZcz 8804   ZZ>=cuz 9073   ...cfz 9478    seqcseq 9906
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-addcom 7499  ax-addass 7501  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-0id 7507  ax-rnegex 7508  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-ltadd 7515
This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-frec 6170  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-inn 8477  df-n0 8728  df-z 8805  df-uz 9074  df-fz 9479  df-iseq 9907  df-seq3 9908
This theorem is referenced by: (None)
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