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Theorem seq3feq 10366
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 2157 . . . 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 10355 . . 3  |-  ( ph  ->  seq M (  .+  ,  F ) : (
ZZ>= `  M ) --> S )
65ffnd 5319 . 2  |-  ( ph  ->  seq M (  .+  ,  F )  Fn  ( ZZ>=
`  M ) )
7 fveq2 5467 . . . . . . 7  |-  ( k  =  x  ->  ( F `  k )  =  ( F `  x ) )
8 fveq2 5467 . . . . . . 7  |-  ( k  =  x  ->  ( G `  k )  =  ( G `  x ) )
97, 8eqeq12d 2172 . . . . . 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 2530 . . . . . . 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 2821 . . . . 5  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  =  ( G `  x ) )
1514, 3eqeltrrd 2235 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( G `  x )  e.  S
)
161, 2, 15, 4seqf 10355 . . 3  |-  ( ph  ->  seq M (  .+  ,  G ) : (
ZZ>= `  M ) --> S )
1716ffnd 5319 . 2  |-  ( ph  ->  seq M (  .+  ,  G )  Fn  ( ZZ>=
`  M ) )
18 simpr 109 . . 3  |-  ( (
ph  /\  z  e.  ( ZZ>= `  M )
)  ->  z  e.  ( ZZ>= `  M )
)
19 simpll 519 . . . 4  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... z ) )  ->  ph )
20 elfzuz 9919 . . . . 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 469 . . 3  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  M )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
2415adantlr 469 . . 3  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  M )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( G `  x )  e.  S
)
254adantlr 469 . . 3  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
2618, 22, 23, 24, 25seq3fveq 10365 . 2  |-  ( (
ph  /\  z  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq M (  .+  ,  G ) `  z
) )
276, 17, 26eqfnfvd 5567 1  |-  ( ph  ->  seq M (  .+  ,  F )  =  seq M (  .+  ,  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128   A.wral 2435   ` cfv 5169  (class class class)co 5821   ZZcz 9162   ZZ>=cuz 9434   ...cfz 9907    seqcseq 10339
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546  ax-cnex 7818  ax-resscn 7819  ax-1cn 7820  ax-1re 7821  ax-icn 7822  ax-addcl 7823  ax-addrcl 7824  ax-mulcl 7825  ax-addcom 7827  ax-addass 7829  ax-distr 7831  ax-i2m1 7832  ax-0lt1 7833  ax-0id 7835  ax-rnegex 7836  ax-cnre 7838  ax-pre-ltirr 7839  ax-pre-ltwlin 7840  ax-pre-lttrn 7841  ax-pre-ltadd 7843
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-frec 6335  df-pnf 7909  df-mnf 7910  df-xr 7911  df-ltxr 7912  df-le 7913  df-sub 8043  df-neg 8044  df-inn 8829  df-n0 9086  df-z 9163  df-uz 9435  df-fz 9908  df-seqfrec 10340
This theorem is referenced by:  zsumdc  11276  fsum3cvg2  11286  isumshft  11382  geolim2  11404  cvgratz  11424  mertenslem2  11428  zproddc  11471
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