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Theorem climconst 11066
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 9347 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
31, 2syl 14 . . . . . 6  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
4 climconst.1 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
53, 4eleqtrrdi 2233 . . . . 5  |-  ( ph  ->  M  e.  Z )
65adantr 274 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  M  e.  Z )
7 climconst.4 . . . . . . . . . 10  |-  ( ph  ->  A  e.  CC )
87subidd 8068 . . . . . . . . 9  |-  ( ph  ->  ( A  -  A
)  =  0 )
98fveq2d 5425 . . . . . . . 8  |-  ( ph  ->  ( abs `  ( A  -  A )
)  =  ( abs `  0 ) )
10 abs0 10837 . . . . . . . 8  |-  ( abs `  0 )  =  0
119, 10syl6eq 2188 . . . . . . 7  |-  ( ph  ->  ( abs `  ( A  -  A )
)  =  0 )
1211adantr 274 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs `  ( A  -  A
) )  =  0 )
13 rpgt0 9460 . . . . . . 7  |-  ( x  e.  RR+  ->  0  < 
x )
1413adantl 275 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  0  <  x )
1512, 14eqbrtrd 3950 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs `  ( A  -  A
) )  <  x
)
1615ralrimivw 2506 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  A. k  e.  Z  ( abs `  ( A  -  A
) )  <  x
)
17 fveq2 5421 . . . . . . 7  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  M )
)
1817, 4syl6eqr 2190 . . . . . 6  |-  ( j  =  M  ->  ( ZZ>=
`  j )  =  Z )
1918raleqdv 2632 . . . . 5  |-  ( j  =  M  ->  ( A. k  e.  ( ZZ>=
`  j ) ( abs `  ( A  -  A ) )  <  x  <->  A. k  e.  Z  ( abs `  ( A  -  A
) )  <  x
) )
2019rspcev 2789 . . . 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 408 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  -  A )
)  <  x )
2221ralrimiva 2505 . 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 274 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
264, 1, 23, 24, 7, 25clim2c 11060 . 2  |-  ( ph  ->  ( F  ~~>  A  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  -  A )
)  <  x )
)
2722, 26mpbird 166 1  |-  ( ph  ->  F  ~~>  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   A.wral 2416   E.wrex 2417   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7625   0cc0 7627    < clt 7807    - cmin 7940   ZZcz 9061   ZZ>=cuz 9333   RR+crp 9448   abscabs 10776    ~~> cli 11054
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-n0 8985  df-z 9062  df-uz 9334  df-rp 9449  df-seqfrec 10226  df-exp 10300  df-cj 10621  df-rsqrt 10777  df-abs 10778  df-clim 11055
This theorem is referenced by:  climconst2  11067  fsum3cvg  11154  fproddccvg  11348
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