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Theorem climconst 12296
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 10460 . . . . . . 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 2499 . . . . 5  |-  ( ph  ->  M  e.  Z )
65adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  M  e.  Z )
7 climconst.4 . . . . . . . . . 10  |-  ( ph  ->  A  e.  CC )
87subidd 9359 . . . . . . . . 9  |-  ( ph  ->  ( A  -  A
)  =  0 )
98fveq2d 5695 . . . . . . . 8  |-  ( ph  ->  ( abs `  ( A  -  A )
)  =  ( abs `  0 ) )
10 abs0 12049 . . . . . . . 8  |-  ( abs `  0 )  =  0
119, 10syl6eq 2456 . . . . . . 7  |-  ( ph  ->  ( abs `  ( A  -  A )
)  =  0 )
1211adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs `  ( A  -  A
) )  =  0 )
13 rpgt0 10583 . . . . . . 7  |-  ( x  e.  RR+  ->  0  < 
x )
1413adantl 453 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  0  <  x )
1512, 14eqbrtrd 4196 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs `  ( A  -  A
) )  <  x
)
1615ralrimivw 2754 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  A. k  e.  Z  ( abs `  ( A  -  A
) )  <  x
)
17 fveq2 5691 . . . . . . 7  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  M )
)
1817, 4syl6eqr 2458 . . . . . 6  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  Z )
1918raleqdv 2874 . . . . 5  |-  ( j  =  M  ->  ( A. k  e.  ( ZZ>=
`  j ) ( abs `  ( A  -  A ) )  <  x  <->  A. k  e.  Z  ( abs `  ( A  -  A
) )  <  x
) )
2019rspcev 3016 . . . 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 643 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  -  A )
)  <  x )
2221ralrimiva 2753 . 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 452 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
264, 1, 23, 24, 7, 25clim2c 12258 . 2  |-  ( ph  ->  ( F  ~~>  A  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  -  A )
)  <  x )
)
2722, 26mpbird 224 1  |-  ( ph  ->  F  ~~>  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2670   E.wrex 2671   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   CCcc 8948   0cc0 8950    < clt 9080    - cmin 9251   ZZcz 10242   ZZ>=cuz 10448   RR+crp 10572   abscabs 11998    ~~> cli 12237
This theorem is referenced by:  climconst2  12301  fsumcvg  12465  expcnv  12602  ntrivcvgfvn0  25184  fprodcvg  25213  fprodntriv  25225  faclim2  25319  clim1fr1  27598  climneg  27607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-seq 11283  df-exp 11342  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241
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