Theorem lmnn 23061

Description: A condition that implies convergence. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 1-May-2014.)

Hypotheses

Ref	Expression
lmnn.2	|- J = ( MetOpen ` D )
lmnn.3	|- ( ph -> D e. ( *Met ` X ) )
lmnn.4	|- ( ph -> P e. X )
lmnn.5	|- ( ph -> F : NN --> X )
lmnn.6	|- ( ( ph /\ k e. NN ) -> ( ( F ` k ) D P ) < ( 1 / k ) )

Assertion

Ref	Expression
lmnn	|- ( ph -> F ( ~~> t ` J ) P )

Distinct variable groups:   D, k   k, F   P, k   ph, k   k, X

Allowed substitution hint:   J( k)

Proof of Theorem lmnn

Dummy variables j x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmnn.4	. 2 |- ( ph -> P e. X )
2		rpreccl 11857	. . . . . . . 8 |- ( x e. RR+ -> ( 1 / x ) e. RR+ )
3	2	adantl 482	. . . . . . 7 |- ( ( ph /\ x e. RR+ ) -> ( 1 / x ) e. RR+ )
4	3	rpred 11872	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> ( 1 / x ) e. RR )
5	3	rpge0d 11876	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> 0 <_ ( 1 / x ) )
6		flge0nn0 12621	. . . . . 6 |- ( ( ( 1 / x ) e. RR /\ 0 <_ ( 1 / x ) ) -> ( |_ ` ( 1 / x ) ) e. NN0 )
7	4, 5, 6	syl2anc 693	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> ( |_ ` ( 1 / x ) ) e. NN0 )
8		nn0p1nn 11332	. . . . 5 |- ( ( |_ ` ( 1 / x ) ) e. NN0 -> ( ( |_ ` ( 1 / x ) ) + 1 ) e. NN )
9	7, 8	syl 17	. . . 4 |- ( ( ph /\ x e. RR+ ) -> ( ( |_ ` ( 1 / x ) ) + 1 ) e. NN )
10		lmnn.3	. . . . . . . 8 |- ( ph -> D e. ( *Met ` X ) )
11	10	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> D e. ( *Met ` X ) )
12		lmnn.5	. . . . . . . . 9 |- ( ph -> F : NN --> X )
13	12	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> F : NN --> X )
14		eluznn 11758	. . . . . . . . 9 |- ( ( ( ( |_ ` ( 1 / x ) ) + 1 ) e. NN /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> k e. NN )
15	9, 14	sylan 488	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> k e. NN )
16	13, 15	ffvelrnd 6360	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> ( F ` k ) e. X )
17	1	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> P e. X )
18		xmetcl 22136	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ ( F ` k ) e. X /\ P e. X ) -> ( ( F ` k ) D P ) e. RR* )
19	11, 16, 17, 18	syl3anc 1326	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> ( ( F ` k ) D P ) e. RR* )
20	15	nnrecred 11066	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> ( 1 / k ) e. RR )
21	20	rexrd 10089	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> ( 1 / k ) e. RR* )
22		rpxr 11840	. . . . . . 7 |- ( x e. RR+ -> x e. RR* )
23	22	ad2antlr 763	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> x e. RR* )
24		lmnn.6	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> ( ( F ` k ) D P ) < ( 1 / k ) )
25	24	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ k e. NN ) -> ( ( F ` k ) D P ) < ( 1 / k ) )
26	15, 25	syldan 487	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> ( ( F ` k ) D P ) < ( 1 / k ) )
27	4	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> ( 1 / x ) e. RR )
28	9	nnred 11035	. . . . . . . . 9 |- ( ( ph /\ x e. RR+ ) -> ( ( |_ ` ( 1 / x ) ) + 1 ) e. RR )
29	28	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> ( ( |_ ` ( 1 / x ) ) + 1 ) e. RR )
30	15	nnred 11035	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> k e. RR )
31		flltp1 12601	. . . . . . . . 9 |- ( ( 1 / x ) e. RR -> ( 1 / x ) < ( ( |_ ` ( 1 / x ) ) + 1 ) )
32	27, 31	syl 17	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> ( 1 / x ) < ( ( |_ ` ( 1 / x ) ) + 1 ) )
33		eluzle 11700	. . . . . . . . 9 |- ( k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) -> ( ( |_ ` ( 1 / x ) ) + 1 ) <_ k )
34	33	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> ( ( |_ ` ( 1 / x ) ) + 1 ) <_ k )
35	27, 29, 30, 32, 34	ltletrd 10197	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> ( 1 / x ) < k )
36		simplr 792	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> x e. RR+ )
37		rpregt0 11846	. . . . . . . . 9 |- ( x e. RR+ -> ( x e. RR /\ 0 < x ) )
38		nnrp 11842	. . . . . . . . . 10 |- ( k e. NN -> k e. RR+ )
39	38	rpregt0d 11878	. . . . . . . . 9 |- ( k e. NN -> ( k e. RR /\ 0 < k ) )
40		ltrec1 10910	. . . . . . . . 9 |- ( ( ( x e. RR /\ 0 < x ) /\ ( k e. RR /\ 0 < k ) ) -> ( ( 1 / x ) < k <-> ( 1 / k ) < x ) )
41	37, 39, 40	syl2an 494	. . . . . . . 8 |- ( ( x e. RR+ /\ k e. NN ) -> ( ( 1 / x ) < k <-> ( 1 / k ) < x ) )
42	36, 15, 41	syl2anc 693	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> ( ( 1 / x ) < k <-> ( 1 / k ) < x ) )
43	35, 42	mpbid 222	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> ( 1 / k ) < x )
44	19, 21, 23, 26, 43	xrlttrd 11990	. . . . 5 |- ( ( ( ph /\ x e. RR+ ) /\ k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ) -> ( ( F ` k ) D P ) < x )
45	44	ralrimiva 2966	. . . 4 |- ( ( ph /\ x e. RR+ ) -> A. k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ( ( F ` k ) D P ) < x )
46		fveq2 6191	. . . . . 6 |- ( j = ( ( |_ ` ( 1 / x ) ) + 1 ) -> ( ZZ>= ` j ) = ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) )
47	46	raleqdv 3144	. . . . 5 |- ( j = ( ( |_ ` ( 1 / x ) ) + 1 ) -> ( A. k e. ( ZZ>= ` j ) ( ( F ` k ) D P ) < x <-> A. k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ( ( F ` k ) D P ) < x ) )
48	47	rspcev 3309	. . . 4 |- ( ( ( ( |_ ` ( 1 / x ) ) + 1 ) e. NN /\ A. k e. ( ZZ>= ` ( ( |_ ` ( 1 / x ) ) + 1 ) ) ( ( F ` k ) D P ) < x ) -> E. j e. NN A. k e. ( ZZ>= ` j ) ( ( F ` k ) D P ) < x )
49	9, 45, 48	syl2anc 693	. . 3 |- ( ( ph /\ x e. RR+ ) -> E. j e. NN A. k e. ( ZZ>= ` j ) ( ( F ` k ) D P ) < x )
50	49	ralrimiva 2966	. 2 |- ( ph -> A. x e. RR+ E. j e. NN A. k e. ( ZZ>= ` j ) ( ( F ` k ) D P ) < x )
51		lmnn.2	. . 3 |- J = ( MetOpen ` D )
52		nnuz 11723	. . 3 |- NN = ( ZZ>= ` 1 )
53		1zzd 11408	. . 3 |- ( ph -> 1 e. ZZ )
54		eqidd 2623	. . 3 |- ( ( ph /\ k e. NN ) -> ( F ` k ) = ( F ` k ) )
55	51, 10, 52, 53, 54, 12	lmmbrf 23060	. 2 |- ( ph -> ( F ( ~~> t ` J ) P <-> ( P e. X /\ A. x e. RR+ E. j e. NN A. k e. ( ZZ>= ` j ) ( ( F ` k ) D P ) < x ) ) )
56	1, 50, 55	mpbir2and 957	1 |- ( ph -> F ( ~~> t ` J ) P )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZ>=cuz 11687   RR+crp 11832   |_cfl 12591   *Metcxmt 19731   MetOpencmopn 19736   ~~> tclm 21030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-fl 12593  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-lm 21033

This theorem is referenced by: (None)