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Theorem seq3feq 10428
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 2170 . . . 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 10417 . . 3  |-  ( ph  ->  seq M (  .+  ,  F ) : (
ZZ>= `  M ) --> S )
65ffnd 5348 . 2  |-  ( ph  ->  seq M (  .+  ,  F )  Fn  ( ZZ>=
`  M ) )
7 fveq2 5496 . . . . . . 7  |-  ( k  =  x  ->  ( F `  k )  =  ( F `  x ) )
8 fveq2 5496 . . . . . . 7  |-  ( k  =  x  ->  ( G `  k )  =  ( G `  x ) )
97, 8eqeq12d 2185 . . . . . 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 2543 . . . . . . 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 2839 . . . . 5  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  =  ( G `  x ) )
1514, 3eqeltrrd 2248 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( G `  x )  e.  S
)
161, 2, 15, 4seqf 10417 . . 3  |-  ( ph  ->  seq M (  .+  ,  G ) : (
ZZ>= `  M ) --> S )
1716ffnd 5348 . 2  |-  ( ph  ->  seq M (  .+  ,  G )  Fn  ( ZZ>=
`  M ) )
18 simpr 109 . . 3  |-  ( (
ph  /\  z  e.  ( ZZ>= `  M )
)  ->  z  e.  ( ZZ>= `  M )
)
19 simpll 524 . . . 4  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... z ) )  ->  ph )
20 elfzuz 9977 . . . . 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 409 . . 3  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... z ) )  ->  ( F `  k )  =  ( G `  k ) )
233adantlr 474 . . 3  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  M )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
2415adantlr 474 . . 3  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  M )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( G `  x )  e.  S
)
254adantlr 474 . . 3  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
2618, 22, 23, 24, 25seq3fveq 10427 . 2  |-  ( (
ph  /\  z  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq M (  .+  ,  G ) `  z
) )
276, 17, 26eqfnfvd 5596 1  |-  ( ph  ->  seq M (  .+  ,  F )  =  seq M (  .+  ,  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   A.wral 2448   ` cfv 5198  (class class class)co 5853   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965    seqcseq 10401
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966  df-seqfrec 10402
This theorem is referenced by:  zsumdc  11347  fsum3cvg2  11357  isumshft  11453  geolim2  11475  cvgratz  11495  mertenslem2  11499  zproddc  11542
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