Theorem climge0 11275

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 11274	. . . . . . . . 9 |- ( ph -> A e. RR )
7	6	adantr 274	. . . . . . . 8 |- ( ( ph /\ A < 0 ) -> A e. RR )
8	7	renegcld 8286	. . . . . . 7 |- ( ( ph /\ A < 0 ) -> -u A e. RR )
9	6	lt0neg1d 8421	. . . . . . . 8 |- ( ph -> ( A < 0 <-> 0 < -u A ) )
10	9	biimpa 294	. . . . . . 7 |- ( ( ph /\ A < 0 ) -> 0 < -u A )
11	8, 10	elrpd 9637	. . . . . 6 |- ( ( ph /\ A < 0 ) -> -u A e. RR+ )
12		eqidd 2171	. . . . . 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 11238	. . . . 5 |- ( ( ph /\ A < 0 ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < -u A )
15	1	r19.2uz 10944	. . . . 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 527	. . . . . . . 8 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> ( abs ` ( ( F ` k ) - A ) ) < -u A )
18	5	ad2ant2r 506	. . . . . . . . 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 11129	. . . . . . . 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 7935	. . . . . . 7 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> A e. CC )
25	24	negidd 8207	. . . . . 6 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> ( A + -u A ) = 0 )
26	23, 25	breqtrd 4013	. . . . 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 506	. . . . . 6 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> 0 <_ ( F ` k ) )
29		0red 7908	. . . . . . 7 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> 0 e. RR )
30	29, 18	lenltd 8024	. . . . . 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 1368	. . . 4 |- ( ( ( ph /\ A < 0 ) /\ ( k e. Z /\ ( abs ` ( ( F ` k ) - A ) ) < -u A ) ) -> F. )
33	16, 32	rexlimddv 2592	. . 3 |- ( ( ph /\ A < 0 ) -> F. )
34	33	inegd 1367	. 2 |- ( ph -> -. A < 0 )
35		0re 7907	. . 3 |- 0 e. RR
36		lenlt 7982	. . 3 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 <_ A <-> -. A < 0 ) )
37	35, 6, 36	sylancr 412	. 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 1348   F. wfal 1353   e. wcel 2141   A.wral 2448   E.wrex 2449   class class class wbr 3987   ` cfv 5196  (class class class)co 5850   RRcr 7760   0cc0 7761   + caddc 7764   < clt 7941   <_ cle 7942   - cmin 8077   -ucneg 8078   ZZcz 9199   ZZ>=cuz 9474   abscabs 10948   ~~> cli 11228

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880  ax-caucvg 7881

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-3 8925  df-4 8926  df-n0 9123  df-z 9200  df-uz 9475  df-rp 9598  df-seqfrec 10389  df-exp 10463  df-cj 10793  df-re 10794  df-im 10795  df-rsqrt 10949  df-abs 10950  df-clim 11229

This theorem is referenced by:  climle  11284