Theorem climge0 11578

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.

Step	Hyp	Ref	Expression
1		climrecl.1	. . . . . 6 |- Z = ( ZZ>= ` M )
2		climrecl.2	. . . . . . 7 |- ( ph -> M e. ZZ )
3	2	adantr 276	. . . . . 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 )
6	1, 2, 4, 5	climrecl 11577	. . . . . . . . 9 |- ( ph -> A e. RR )
7	6	adantr 276	. . . . . . . 8 |- ( ( ph /\ A < 0 ) -> A e. RR )
8	7	renegcld 8451	. . . . . . 7 |- ( ( ph /\ A < 0 ) -> -u A e. RR )
9	6	lt0neg1d 8587	. . . . . . . 8 |- ( ph -> ( A < 0 <-> 0 < -u A ) )
10	9	biimpa 296	. . . . . . 7 |- ( ( ph /\ A < 0 ) -> 0 < -u A )
11	8, 10	elrpd 9814	. . . . . 6 |- ( ( ph /\ A < 0 ) -> -u A e. RR+ )
12		eqidd 2205	. . . . . 6 |- ( ( ( ph /\ A < 0 ) /\ k e. Z ) -> ( F ` k ) = ( F ` k ) )
13	4	adantr 276	. . . . . 6 |- ( ( ph /\ A < 0 ) -> F ~~> A )
14	1, 3, 11, 12, 13	climi2 11541	. . . . 5 |- ( ( ph /\ A < 0 ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < -u A )
15	1	r19.2uz 11246	. . . . 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 )
16	14, 15	syl 14	. . . 4 |- ( ( ph /\ A < 0 ) -> E. k e. Z ( abs ` ( ( F ` k ) - A ) ) < -u A )
17		simprr 531	. . . . . . . 8 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> ( abs ` ( ( F ` k ) - A ) ) < -u A )
18	5	ad2ant2r 509	. . . . . . . . 9 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> ( F ` k ) e. RR )
19	7	adantr 276	. . . . . . . . 9 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> A e. RR )
20	8	adantr 276	. . . . . . . . 9 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> -u A e. RR )
21	18, 19, 20	absdifltd 11431	. . . . . . . 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 ) ) ) )
22	17, 21	mpbid 147	. . . . . . 7 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> ( ( A - -u A ) < ( F ` k ) /\ ( F ` k ) < ( A + -u A ) ) )
23	22	simprd 114	. . . . . 6 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> ( F ` k ) < ( A + -u A ) )
24	19	recnd 8100	. . . . . . 7 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> A e. CC )
25	24	negidd 8372	. . . . . 6 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> ( A + -u A ) = 0 )
26	23, 25	breqtrd 4069	. . . . 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 ) )
28	27	ad2ant2r 509	. . . . . 6 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> 0 <_ ( F ` k ) )
29		0red 8072	. . . . . . 7 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> 0 e. RR )
30	29, 18	lenltd 8189	. . . . . 6 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> ( 0 <_ ( F ` k ) <-> -. ( F ` k ) < 0 ) )
31	28, 30	mpbid 147	. . . . 5 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> -. ( F ` k ) < 0 )
32	26, 31	pm2.21fal 1392	. . . 4 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> F. )
33	16, 32	rexlimddv 2627	. . 3 |- ( ( ph /\ A < 0 ) -> F. )
34	33	inegd 1391	. 2 |- ( ph -> -. A < 0 )
35		0re 8071	. . 3 |- 0 e. RR
36		lenlt 8147	. . 3 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 <_ A <-> -. A < 0 ) )
37	35, 6, 36	sylancr 414	. 2 |- ( ph -> ( 0 <_ A <-> -. A < 0 ) )
38	34, 37	mpbird 167	1 |- ( ph -> 0 <_ A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   F. wfal 1377   e. wcel 2175   A.wral 2483   E.wrex 2484   class class class wbr 4043   ` cfv 5270  (class class class)co 5943   RRcr 7923   0cc0 7924   + caddc 7927   < clt 8106   <_ cle 8107   - cmin 8242   -ucneg 8243   ZZcz 9371   ZZ>=cuz 9647   abscabs 11250   ~~> cli 11531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-z 9372  df-uz 9648  df-rp 9775  df-seqfrec 10591  df-exp 10682  df-cj 11095  df-re 11096  df-im 11097  df-rsqrt 11251  df-abs 11252  df-clim 11532

This theorem is referenced by:  climle  11587