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Theorem climconst 12368
Description: An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
Hypotheses
Ref Expression
climconst.1  |-  Z  =  ( ZZ>= `  M )
climconst.2  |-  ( ph  ->  M  e.  ZZ )
climconst.3  |-  ( ph  ->  F  e.  V )
climconst.4  |-  ( ph  ->  A  e.  CC )
climconst.5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
Assertion
Ref Expression
climconst  |-  ( ph  ->  F  ~~>  A )
Distinct variable groups:    A, k    k, F    ph, k    k, Z
Allowed substitution hints:    M( k)    V( k)

Proof of Theorem climconst
Dummy variables  j  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climconst.2 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
2 uzid 10531 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
31, 2syl 16 . . . . . 6  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
4 climconst.1 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
53, 4syl6eleqr 2533 . . . . 5  |-  ( ph  ->  M  e.  Z )
65adantr 453 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  M  e.  Z )
7 climconst.4 . . . . . . . . . 10  |-  ( ph  ->  A  e.  CC )
87subidd 9430 . . . . . . . . 9  |-  ( ph  ->  ( A  -  A
)  =  0 )
98fveq2d 5761 . . . . . . . 8  |-  ( ph  ->  ( abs `  ( A  -  A )
)  =  ( abs `  0 ) )
10 abs0 12121 . . . . . . . 8  |-  ( abs `  0 )  =  0
119, 10syl6eq 2490 . . . . . . 7  |-  ( ph  ->  ( abs `  ( A  -  A )
)  =  0 )
1211adantr 453 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs `  ( A  -  A
) )  =  0 )
13 rpgt0 10654 . . . . . . 7  |-  ( x  e.  RR+  ->  0  < 
x )
1413adantl 454 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  0  <  x )
1512, 14eqbrtrd 4257 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs `  ( A  -  A
) )  <  x
)
1615ralrimivw 2796 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  A. k  e.  Z  ( abs `  ( A  -  A
) )  <  x
)
17 fveq2 5757 . . . . . . 7  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  M )
)
1817, 4syl6eqr 2492 . . . . . 6  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  Z )
1918raleqdv 2916 . . . . 5  |-  ( j  =  M  ->  ( A. k  e.  ( ZZ>=
`  j ) ( abs `  ( A  -  A ) )  <  x  <->  A. k  e.  Z  ( abs `  ( A  -  A
) )  <  x
) )
2019rspcev 3058 . . . 4  |-  ( ( M  e.  Z  /\  A. k  e.  Z  ( abs `  ( A  -  A ) )  <  x )  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( A  -  A ) )  <  x )
216, 16, 20syl2anc 644 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  -  A )
)  <  x )
2221ralrimiva 2795 . 2  |-  ( ph  ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( A  -  A
) )  <  x
)
23 climconst.3 . . 3  |-  ( ph  ->  F  e.  V )
24 climconst.5 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
257adantr 453 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
264, 1, 23, 24, 7, 25clim2c 12330 . 2  |-  ( ph  ->  ( F  ~~>  A  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  -  A )
)  <  x )
)
2722, 26mpbird 225 1  |-  ( ph  ->  F  ~~>  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727   A.wral 2711   E.wrex 2712   class class class wbr 4237   ` cfv 5483  (class class class)co 6110   CCcc 9019   0cc0 9021    < clt 9151    - cmin 9322   ZZcz 10313   ZZ>=cuz 10519   RR+crp 10643   abscabs 12070    ~~> cli 12309
This theorem is referenced by:  climconst2  12373  fsumcvg  12537  expcnv  12674  ntrivcvgfvn0  25258  fprodcvg  25287  fprodntriv  25299  faclim2  25398  clim1fr1  27741  climneg  27750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-2 10089  df-n0 10253  df-z 10314  df-uz 10520  df-rp 10644  df-seq 11355  df-exp 11414  df-cj 11935  df-re 11936  df-im 11937  df-sqr 12071  df-abs 12072  df-clim 12313
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