Theorem climge0 11126

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 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 )
6	1, 2, 4, 5	climrecl 11125	. . . . . . . . 9 |- ( ph -> A e. RR )
7	6	adantr 274	. . . . . . . 8 |- ( ( ph /\ A < 0 ) -> A e. RR )
8	7	renegcld 8166	. . . . . . 7 |- ( ( ph /\ A < 0 ) -> -u A e. RR )
9	6	lt0neg1d 8301	. . . . . . . 8 |- ( ph -> ( A < 0 <-> 0 < -u A ) )
10	9	biimpa 294	. . . . . . 7 |- ( ( ph /\ A < 0 ) -> 0 < -u A )
11	8, 10	elrpd 9510	. . . . . 6 |- ( ( ph /\ A < 0 ) -> -u A e. RR+ )
12		eqidd 2141	. . . . . 6 |- ( ( ( ph /\ A < 0 ) /\ k e. Z ) -> ( F ` k ) = ( F ` k ) )
13	4	adantr 274	. . . . . 6 |- ( ( ph /\ A < 0 ) -> F ~~> A )
14	1, 3, 11, 12, 13	climi2 11089	. . . . 5 |- ( ( ph /\ A < 0 ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < -u A )
15	1	r19.2uz 10797	. . . . 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 522	. . . . . . . 8 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> ( abs ` ( ( F ` k ) - A ) ) < -u A )
18	5	ad2ant2r 501	. . . . . . . . 9 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> ( F ` k ) e. RR )
19	7	adantr 274	. . . . . . . . 9 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> A e. RR )
20	8	adantr 274	. . . . . . . . 9 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> -u A e. RR )
21	18, 19, 20	absdifltd 10982	. . . . . . . 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 146	. . . . . . 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 113	. . . . . 6 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> ( F ` k ) < ( A + -u A ) )
24	19	recnd 7818	. . . . . . 7 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> A e. CC )
25	24	negidd 8087	. . . . . 6 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> ( A + -u A ) = 0 )
26	23, 25	breqtrd 3962	. . . . 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 501	. . . . . 6 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> 0 <_ ( F ` k ) )
29		0red 7791	. . . . . . 7 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> 0 e. RR )
30	29, 18	lenltd 7904	. . . . . 6 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> ( 0 <_ ( F ` k ) <-> -. ( F ` k ) < 0 ) )
31	28, 30	mpbid 146	. . . . 5 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> -. ( F ` k ) < 0 )
32	26, 31	pm2.21fal 1352	. . . 4 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> F. )
33	16, 32	rexlimddv 2557	. . 3 |- ( ( ph /\ A < 0 ) -> F. )
34	33	inegd 1351	. 2 |- ( ph -> -. A < 0 )
35		0re 7790	. . 3 |- 0 e. RR
36		lenlt 7864	. . 3 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 <_ A <-> -. A < 0 ) )
37	35, 6, 36	sylancr 411	. 2 |- ( ph -> ( 0 <_ A <-> -. A < 0 ) )
38	34, 37	mpbird 166	1 |- ( ph -> 0 <_ A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   F. wfal 1337   e. wcel 1481   A.wral 2417   E.wrex 2418   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   RRcr 7643   0cc0 7644   + caddc 7647   < clt 7824   <_ cle 7825   - cmin 7957   -ucneg 7958   ZZcz 9078   ZZ>=cuz 9350   abscabs 10801   ~~> cli 11079

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-rp 9471  df-seqfrec 10250  df-exp 10324  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-clim 11080

This theorem is referenced by:  climle  11135