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Theorem climge0 11062
Description: A nonnegative sequence converges to a nonnegative number. (Contributed by NM, 11-Sep-2005.)
Hypotheses
Ref Expression
climrecl.1  |-  Z  =  ( ZZ>= `  M )
climrecl.2  |-  ( ph  ->  M  e.  ZZ )
climrecl.3  |-  ( ph  ->  F  ~~>  A )
climrecl.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
climge0.5  |-  ( (
ph  /\  k  e.  Z )  ->  0  <_  ( F `  k
) )
Assertion
Ref Expression
climge0  |-  ( ph  ->  0  <_  A )
Distinct variable groups:    k, F    k, M    ph, k    k, Z    A, k

Proof of Theorem climge0
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 climrecl.1 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
2 climrecl.2 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
32adantr 274 . . . . . 6  |-  ( (
ph  /\  A  <  0 )  ->  M  e.  ZZ )
4 climrecl.3 . . . . . . . . . 10  |-  ( ph  ->  F  ~~>  A )
5 climrecl.4 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
61, 2, 4, 5climrecl 11061 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
76adantr 274 . . . . . . . 8  |-  ( (
ph  /\  A  <  0 )  ->  A  e.  RR )
87renegcld 8110 . . . . . . 7  |-  ( (
ph  /\  A  <  0 )  ->  -u A  e.  RR )
96lt0neg1d 8245 . . . . . . . 8  |-  ( ph  ->  ( A  <  0  <->  0  <  -u A ) )
109biimpa 294 . . . . . . 7  |-  ( (
ph  /\  A  <  0 )  ->  0  <  -u A )
118, 10elrpd 9449 . . . . . 6  |-  ( (
ph  /\  A  <  0 )  ->  -u A  e.  RR+ )
12 eqidd 2118 . . . . . 6  |-  ( ( ( ph  /\  A  <  0 )  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
134adantr 274 . . . . . 6  |-  ( (
ph  /\  A  <  0 )  ->  F  ~~>  A )
141, 3, 11, 12, 13climi2 11025 . . . . 5  |-  ( (
ph  /\  A  <  0 )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( F `  k
)  -  A ) )  <  -u A
)
151r19.2uz 10733 . . . . 5  |-  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( ( F `  k )  -  A
) )  <  -u A  ->  E. k  e.  Z  ( abs `  ( ( F `  k )  -  A ) )  <  -u A )
1614, 15syl 14 . . . 4  |-  ( (
ph  /\  A  <  0 )  ->  E. k  e.  Z  ( abs `  ( ( F `  k )  -  A
) )  <  -u A
)
17 simprr 506 . . . . . . . 8  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  ( abs `  (
( F `  k
)  -  A ) )  <  -u A
)
185ad2ant2r 500 . . . . . . . . 9  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  ( F `  k )  e.  RR )
197adantr 274 . . . . . . . . 9  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  A  e.  RR )
208adantr 274 . . . . . . . . 9  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  -u A  e.  RR )
2118, 19, 20absdifltd 10918 . . . . . . . 8  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  ( ( abs `  ( ( F `  k )  -  A
) )  <  -u A  <->  ( ( A  -  -u A
)  <  ( F `  k )  /\  ( F `  k )  <  ( A  +  -u A ) ) ) )
2217, 21mpbid 146 . . . . . . 7  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  ( ( A  -  -u A )  < 
( F `  k
)  /\  ( F `  k )  <  ( A  +  -u A ) ) )
2322simprd 113 . . . . . 6  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  ( F `  k )  <  ( A  +  -u A ) )
2419recnd 7762 . . . . . . 7  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  A  e.  CC )
2524negidd 8031 . . . . . 6  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  ( A  +  -u A )  =  0 )
2623, 25breqtrd 3924 . . . . 5  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  ( F `  k )  <  0
)
27 climge0.5 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  0  <_  ( F `  k
) )
2827ad2ant2r 500 . . . . . 6  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  0  <_  ( F `  k )
)
29 0red 7735 . . . . . . 7  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  0  e.  RR )
3029, 18lenltd 7848 . . . . . 6  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  ( 0  <_ 
( F `  k
)  <->  -.  ( F `  k )  <  0
) )
3128, 30mpbid 146 . . . . 5  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  -.  ( F `  k )  <  0
)
3226, 31pm2.21fal 1336 . . . 4  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  -> F.  )
3316, 32rexlimddv 2531 . . 3  |-  ( (
ph  /\  A  <  0 )  -> F.  )
3433inegd 1335 . 2  |-  ( ph  ->  -.  A  <  0
)
35 0re 7734 . . 3  |-  0  e.  RR
36 lenlt 7808 . . 3  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  -.  A  <  0 ) )
3735, 6, 36sylancr 410 . 2  |-  ( ph  ->  ( 0  <_  A  <->  -.  A  <  0 ) )
3834, 37mpbird 166 1  |-  ( ph  ->  0  <_  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316   F. wfal 1321    e. wcel 1465   A.wral 2393   E.wrex 2394   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   RRcr 7587   0cc0 7588    + caddc 7591    < clt 7768    <_ cle 7769    - cmin 7901   -ucneg 7902   ZZcz 9022   ZZ>=cuz 9294   abscabs 10737    ~~> cli 11015
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-n0 8946  df-z 9023  df-uz 9295  df-rp 9410  df-seqfrec 10187  df-exp 10261  df-cj 10582  df-re 10583  df-im 10584  df-rsqrt 10738  df-abs 10739  df-clim 11016
This theorem is referenced by:  climle  11071
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