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Theorem climge0 10677
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 270 . . . . . 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 10676 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
76adantr 270 . . . . . . . 8  |-  ( (
ph  /\  A  <  0 )  ->  A  e.  RR )
87renegcld 7837 . . . . . . 7  |-  ( (
ph  /\  A  <  0 )  ->  -u A  e.  RR )
96lt0neg1d 7969 . . . . . . . 8  |-  ( ph  ->  ( A  <  0  <->  0  <  -u A ) )
109biimpa 290 . . . . . . 7  |-  ( (
ph  /\  A  <  0 )  ->  0  <  -u A )
118, 10elrpd 9140 . . . . . 6  |-  ( (
ph  /\  A  <  0 )  ->  -u A  e.  RR+ )
12 eqidd 2089 . . . . . 6  |-  ( ( ( ph  /\  A  <  0 )  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
134adantr 270 . . . . . 6  |-  ( (
ph  /\  A  <  0 )  ->  F  ~~>  A )
141, 3, 11, 12, 13climi2 10640 . . . . 5  |-  ( (
ph  /\  A  <  0 )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( F `  k
)  -  A ) )  <  -u A
)
151r19.2uz 10391 . . . . 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 499 . . . . . . . 8  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  ( abs `  (
( F `  k
)  -  A ) )  <  -u A
)
185ad2ant2r 493 . . . . . . . . 9  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  ( F `  k )  e.  RR )
197adantr 270 . . . . . . . . 9  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  A  e.  RR )
208adantr 270 . . . . . . . . 9  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  -u A  e.  RR )
2118, 19, 20absdifltd 10576 . . . . . . . 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 145 . . . . . . 7  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  ( ( A  -  -u A )  < 
( F `  k
)  /\  ( F `  k )  <  ( A  +  -u A ) ) )
2322simprd 112 . . . . . 6  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  ( F `  k )  <  ( A  +  -u A ) )
2419recnd 7495 . . . . . . 7  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  A  e.  CC )
2524negidd 7762 . . . . . 6  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  ( A  +  -u A )  =  0 )
2623, 25breqtrd 3861 . . . . 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 493 . . . . . 6  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  0  <_  ( F `  k )
)
29 0red 7468 . . . . . . 7  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  0  e.  RR )
3029, 18lenltd 7580 . . . . . 6  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  ( 0  <_ 
( F `  k
)  <->  -.  ( F `  k )  <  0
) )
3128, 30mpbid 145 . . . . 5  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  ->  -.  ( F `  k )  <  0
)
3226, 31pm2.21fal 1309 . . . 4  |-  ( ( ( ph  /\  A  <  0 )  /\  (
k  e.  Z  /\  ( abs `  ( ( F `  k )  -  A ) )  <  -u A ) )  -> F.  )
3316, 32rexlimddv 2493 . . 3  |-  ( (
ph  /\  A  <  0 )  -> F.  )
3433inegd 1308 . 2  |-  ( ph  ->  -.  A  <  0
)
35 0re 7467 . . 3  |-  0  e.  RR
36 lenlt 7540 . . 3  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  -.  A  <  0 ) )
3735, 6, 36sylancr 405 . 2  |-  ( ph  ->  ( 0  <_  A  <->  -.  A  <  0 ) )
3834, 37mpbird 165 1  |-  ( ph  ->  0  <_  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   F. wfal 1294    e. wcel 1438   A.wral 2359   E.wrex 2360   class class class wbr 3837   ` cfv 5002  (class class class)co 5634   RRcr 7328   0cc0 7329    + caddc 7332    < clt 7501    <_ cle 7502    - cmin 7632   -ucneg 7633   ZZcz 8720   ZZ>=cuz 8988   abscabs 10395    ~~> cli 10630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-rp 9104  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-clim 10631
This theorem is referenced by:  climle  10686
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