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Theorem seq3feq 10238
Description: Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
Hypotheses
Ref Expression
seq3feq.1  |-  ( ph  ->  M  e.  ZZ )
seq3feq.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
seq3feq.2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( G `  k ) )
seq3feq.pl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
Assertion
Ref Expression
seq3feq  |-  ( ph  ->  seq M (  .+  ,  F )  =  seq M (  .+  ,  G ) )
Distinct variable groups:    .+ , k, x, y    k, F, x, y    k, G, x, y    k, M, x, y    S, k, x, y    ph, k, x, y

Proof of Theorem seq3feq
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqid 2137 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 seq3feq.1 . . . 4  |-  ( ph  ->  M  e.  ZZ )
3 seq3feq.f . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
4 seq3feq.pl . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
51, 2, 3, 4seqf 10227 . . 3  |-  ( ph  ->  seq M (  .+  ,  F ) : (
ZZ>= `  M ) --> S )
65ffnd 5268 . 2  |-  ( ph  ->  seq M (  .+  ,  F )  Fn  ( ZZ>=
`  M ) )
7 fveq2 5414 . . . . . . 7  |-  ( k  =  x  ->  ( F `  k )  =  ( F `  x ) )
8 fveq2 5414 . . . . . . 7  |-  ( k  =  x  ->  ( G `  k )  =  ( G `  x ) )
97, 8eqeq12d 2152 . . . . . 6  |-  ( k  =  x  ->  (
( F `  k
)  =  ( G `
 k )  <->  ( F `  x )  =  ( G `  x ) ) )
10 seq3feq.2 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( G `  k ) )
1110ralrimiva 2503 . . . . . . 7  |-  ( ph  ->  A. k  e.  (
ZZ>= `  M ) ( F `  k )  =  ( G `  k ) )
1211adantr 274 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  A. k  e.  ( ZZ>= `  M )
( F `  k
)  =  ( G `
 k ) )
13 simpr 109 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  x  e.  ( ZZ>= `  M )
)
149, 12, 13rspcdva 2789 . . . . 5  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  =  ( G `  x ) )
1514, 3eqeltrrd 2215 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( G `  x )  e.  S
)
161, 2, 15, 4seqf 10227 . . 3  |-  ( ph  ->  seq M (  .+  ,  G ) : (
ZZ>= `  M ) --> S )
1716ffnd 5268 . 2  |-  ( ph  ->  seq M (  .+  ,  G )  Fn  ( ZZ>=
`  M ) )
18 simpr 109 . . 3  |-  ( (
ph  /\  z  e.  ( ZZ>= `  M )
)  ->  z  e.  ( ZZ>= `  M )
)
19 simpll 518 . . . 4  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... z ) )  ->  ph )
20 elfzuz 9795 . . . . 5  |-  ( k  e.  ( M ... z )  ->  k  e.  ( ZZ>= `  M )
)
2120adantl 275 . . . 4  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... z ) )  ->  k  e.  ( ZZ>= `  M )
)
2219, 21, 10syl2anc 408 . . 3  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... z ) )  ->  ( F `  k )  =  ( G `  k ) )
233adantlr 468 . . 3  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  M )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
2415adantlr 468 . . 3  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  M )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( G `  x )  e.  S
)
254adantlr 468 . . 3  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
2618, 22, 23, 24, 25seq3fveq 10237 . 2  |-  ( (
ph  /\  z  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq M (  .+  ,  G ) `  z
) )
276, 17, 26eqfnfvd 5514 1  |-  ( ph  ->  seq M (  .+  ,  F )  =  seq M (  .+  ,  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   A.wral 2414   ` cfv 5118  (class class class)co 5767   ZZcz 9047   ZZ>=cuz 9319   ...cfz 9783    seqcseq 10211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-ltadd 7729
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-fz 9784  df-seqfrec 10212
This theorem is referenced by:  zsumdc  11146  fsum3cvg2  11156  isumshft  11252  geolim2  11274  cvgratz  11294  mertenslem2  11298
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