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Theorem climconst 13341
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 11106 . . . . . . 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 2566 . . . . 5  |-  ( ph  ->  M  e.  Z )
65adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  M  e.  Z )
7 climconst.4 . . . . . . . . . 10  |-  ( ph  ->  A  e.  CC )
87subidd 9928 . . . . . . . . 9  |-  ( ph  ->  ( A  -  A
)  =  0 )
98fveq2d 5875 . . . . . . . 8  |-  ( ph  ->  ( abs `  ( A  -  A )
)  =  ( abs `  0 ) )
10 abs0 13093 . . . . . . . 8  |-  ( abs `  0 )  =  0
119, 10syl6eq 2524 . . . . . . 7  |-  ( ph  ->  ( abs `  ( A  -  A )
)  =  0 )
1211adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs `  ( A  -  A
) )  =  0 )
13 rpgt0 11241 . . . . . . 7  |-  ( x  e.  RR+  ->  0  < 
x )
1413adantl 466 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  0  <  x )
1512, 14eqbrtrd 4472 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs `  ( A  -  A
) )  <  x
)
1615ralrimivw 2882 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  A. k  e.  Z  ( abs `  ( A  -  A
) )  <  x
)
17 fveq2 5871 . . . . . . 7  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  M )
)
1817, 4syl6eqr 2526 . . . . . 6  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  Z )
1918raleqdv 3069 . . . . 5  |-  ( j  =  M  ->  ( A. k  e.  ( ZZ>=
`  j ) ( abs `  ( A  -  A ) )  <  x  <->  A. k  e.  Z  ( abs `  ( A  -  A
) )  <  x
) )
2019rspcev 3219 . . . 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 661 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  -  A )
)  <  x )
2221ralrimiva 2881 . 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 465 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
264, 1, 23, 24, 7, 25clim2c 13303 . 2  |-  ( ph  ->  ( F  ~~>  A  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  -  A )
)  <  x )
)
2722, 26mpbird 232 1  |-  ( ph  ->  F  ~~>  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   class class class wbr 4452   ` cfv 5593  (class class class)co 6294   CCcc 9500   0cc0 9502    < clt 9638    - cmin 9815   ZZcz 10874   ZZ>=cuz 11092   RR+crp 11230   abscabs 13042    ~~> cli 13282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-2nd 6795  df-recs 7052  df-rdg 7086  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-n0 10806  df-z 10875  df-uz 11093  df-rp 11231  df-seq 12086  df-exp 12145  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-clim 13286
This theorem is referenced by:  climconst2  13346  fsumcvg  13509  expcnv  13650  ntrivcvgfvn0  28928  fprodcvg  28957  fprodntriv  28969  faclim2  29068  clim1fr1  31434  climneg  31443  ioodvbdlimc1lem2  31553  ioodvbdlimc2lem  31555  fourierdlem103  31801  fourierdlem104  31802
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