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Theorem seq3feq2 10130
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 2113 . . . . 5  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 seq3fveq2.1 . . . . . 6  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
3 eluzel2 9227 . . . . . 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 10121 . . . 4  |-  ( ph  ->  seq M (  .+  ,  F ) : (
ZZ>= `  M ) --> S )
87ffnd 5229 . . 3  |-  ( ph  ->  seq M (  .+  ,  F )  Fn  ( ZZ>=
`  M ) )
9 uzss 9242 . . . 4  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  K )  C_  ( ZZ>=
`  M ) )
102, 9syl 14 . . 3  |-  ( ph  ->  ( ZZ>= `  K )  C_  ( ZZ>= `  M )
)
11 fnssres 5192 . . 3  |-  ( (  seq M (  .+  ,  F )  Fn  ( ZZ>=
`  M )  /\  ( ZZ>= `  K )  C_  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F )  |`  ( ZZ>=
`  K ) )  Fn  ( ZZ>= `  K
) )
128, 10, 11syl2anc 406 . 2  |-  ( ph  ->  (  seq M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  Fn  ( ZZ>= `  K ) )
13 eqid 2113 . . . 4  |-  ( ZZ>= `  K )  =  (
ZZ>= `  K )
14 eluzelz 9231 . . . . 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 10121 . . 3  |-  ( ph  ->  seq K (  .+  ,  G ) : (
ZZ>= `  K ) --> S )
1817ffnd 5229 . 2  |-  ( ph  ->  seq K (  .+  ,  G )  Fn  ( ZZ>=
`  K ) )
19 fvres 5397 . . . 4  |-  ( z  e.  ( ZZ>= `  K
)  ->  ( (  seq M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 z )  =  (  seq M ( 
.+  ,  F ) `
 z ) )
2019adantl 273 . . 3  |-  ( (
ph  /\  z  e.  ( ZZ>= `  K )
)  ->  ( (  seq M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 z )  =  (  seq M ( 
.+  ,  F ) `
 z ) )
212adantr 272 . . . 4  |-  ( (
ph  /\  z  e.  ( ZZ>= `  K )
)  ->  K  e.  ( ZZ>= `  M )
)
22 seq3fveq2.2 . . . . 5  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  ( G `  K
) )
2322adantr 272 . . . 4  |-  ( (
ph  /\  z  e.  ( ZZ>= `  K )
)  ->  (  seq M (  .+  ,  F ) `  K
)  =  ( G `
 K ) )
245adantlr 466 . . . 4  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  K )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
2516adantlr 466 . . . 4  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  K )
)  /\  x  e.  ( ZZ>= `  K )
)  ->  ( G `  x )  e.  S
)
266adantlr 466 . . . 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 9689 . . . . . 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 282 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( K  + 
1 ) ... z
) )  ->  ( F `  k )  =  ( G `  k ) )
3130adantlr 466 . . . 4  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  K )
)  /\  k  e.  ( ( K  + 
1 ) ... z
) )  ->  ( F `  k )  =  ( G `  k ) )
3221, 23, 24, 25, 26, 27, 31seq3fveq2 10129 . . 3  |-  ( (
ph  /\  z  e.  ( ZZ>= `  K )
)  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
) )
3320, 32eqtrd 2145 . 2  |-  ( (
ph  /\  z  e.  ( ZZ>= `  K )
)  ->  ( (  seq M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 z )  =  (  seq K ( 
.+  ,  G ) `
 z ) )
3412, 18, 33eqfnfvd 5473 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  =  seq K
(  .+  ,  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1312    e. wcel 1461    C_ wss 3035    |` cres 4499    Fn wfn 5074   ` cfv 5079  (class class class)co 5726   1c1 7542    + caddc 7544   ZZcz 8952   ZZ>=cuz 9222   ...cfz 9677    seqcseq 10105
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 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-addcom 7639  ax-addass 7641  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-0id 7647  ax-rnegex 7648  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-ltadd 7655
This theorem depends on definitions:  df-bi 116  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-ilim 4249  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-frec 6240  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-inn 8625  df-n0 8876  df-z 8953  df-uz 9223  df-fz 9678  df-seqfrec 10106
This theorem is referenced by:  seq3id  10168
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