Theorem climge0 11903

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 11902	. . . . . . . . 9 |- ( ph -> A e. RR )
7	6	adantr 276	. . . . . . . 8 |- ( ( ph /\ A < 0 ) -> A e. RR )
8	7	renegcld 8559	. . . . . . 7 |- ( ( ph /\ A < 0 ) -> -u A e. RR )
9	6	lt0neg1d 8695	. . . . . . . 8 |- ( ph -> ( A < 0 <-> 0 < -u A ) )
10	9	biimpa 296	. . . . . . 7 |- ( ( ph /\ A < 0 ) -> 0 < -u A )
11	8, 10	elrpd 9928	. . . . . 6 |- ( ( ph /\ A < 0 ) -> -u A e. RR+ )
12		eqidd 2232	. . . . . 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 11866	. . . . 5 |- ( ( ph /\ A < 0 ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < -u A )
15	1	r19.2uz 11571	. . . . 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 533	. . . . . . . 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 11756	. . . . . . . 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 8208	. . . . . . 7 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> A e. CC )
25	24	negidd 8480	. . . . . 6 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> ( A + -u A ) = 0 )
26	23, 25	breqtrd 4114	. . . . 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 8180	. . . . . . 7 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> 0 e. RR )
30	29, 18	lenltd 8297	. . . . . 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 1417	. . . 4 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> F. )
33	16, 32	rexlimddv 2655	. . 3 |- ( ( ph /\ A < 0 ) -> F. )
34	33	inegd 1416	. 2 |- ( ph -> -. A < 0 )
35		0re 8179	. . 3 |- 0 e. RR
36		lenlt 8255	. . 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 1397   F. wfal 1402   e. wcel 2202   A.wral 2510   E.wrex 2511   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   RRcr 8031   0cc0 8032   + caddc 8035   < clt 8214   <_ cle 8215   - cmin 8350   -ucneg 8351   ZZcz 9479   ZZ>=cuz 9755   abscabs 11575   ~~> cli 11856

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-rp 9889  df-seqfrec 10711  df-exp 10802  df-cj 11420  df-re 11421  df-im 11422  df-rsqrt 11576  df-abs 11577  df-clim 11857

This theorem is referenced by:  climle  11912