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Theorem climconst 12013
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 10238 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
31, 2syl 15 . . . . . 6  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
4 climconst.1 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
53, 4syl6eleqr 2375 . . . . 5  |-  ( ph  ->  M  e.  Z )
65adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  M  e.  Z )
7 climconst.4 . . . . . . . . . 10  |-  ( ph  ->  A  e.  CC )
87subidd 9141 . . . . . . . . 9  |-  ( ph  ->  ( A  -  A
)  =  0 )
98fveq2d 5490 . . . . . . . 8  |-  ( ph  ->  ( abs `  ( A  -  A )
)  =  ( abs `  0 ) )
10 abs0 11766 . . . . . . . 8  |-  ( abs `  0 )  =  0
119, 10syl6eq 2332 . . . . . . 7  |-  ( ph  ->  ( abs `  ( A  -  A )
)  =  0 )
1211adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs `  ( A  -  A
) )  =  0 )
13 rpgt0 10361 . . . . . . 7  |-  ( x  e.  RR+  ->  0  < 
x )
1413adantl 452 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  0  <  x )
1512, 14eqbrtrd 4044 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs `  ( A  -  A
) )  <  x
)
1615ralrimivw 2628 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  A. k  e.  Z  ( abs `  ( A  -  A
) )  <  x
)
17 fveq2 5486 . . . . . . 7  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  M )
)
1817, 4syl6eqr 2334 . . . . . 6  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  Z )
1918raleqdv 2743 . . . . 5  |-  ( j  =  M  ->  ( A. k  e.  ( ZZ>=
`  j ) ( abs `  ( A  -  A ) )  <  x  <->  A. k  e.  Z  ( abs `  ( A  -  A
) )  <  x
) )
2019rspcev 2885 . . . 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 642 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  -  A )
)  <  x )
2221ralrimiva 2627 . 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 451 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
264, 1, 23, 24, 7, 25clim2c 11975 . 2  |-  ( ph  ->  ( F  ~~>  A  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  -  A )
)  <  x )
)
2722, 26mpbird 223 1  |-  ( ph  ->  F  ~~>  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1685   A.wral 2544   E.wrex 2545   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   CCcc 8731   0cc0 8733    < clt 8863    - cmin 9033   ZZcz 10020   ZZ>=cuz 10226   RR+crp 10350   abscabs 11715    ~~> cli 11954
This theorem is referenced by:  climconst2  12018  fsumcvg  12181  expcnv  12318  clim1fr1  27138  climneg  27147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-seq 11043  df-exp 11101  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-clim 11958
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