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Theorem lmnn 18689
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.
StepHypRef Expression
1 lmnn.4 . 2  |-  ( ph  ->  P  e.  X )
2 rpreccl 10377 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
32adantl 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR+ )
43rpred 10390 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR )
53rpge0d 10394 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  0  <_  ( 1  /  x ) )
6 flge0nn0 10948 . . . . . 6  |-  ( ( ( 1  /  x
)  e.  RR  /\  0  <_  ( 1  /  x ) )  -> 
( |_ `  (
1  /  x ) )  e.  NN0 )
74, 5, 6syl2anc 642 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( |_ `  ( 1  /  x
) )  e.  NN0 )
8 nn0p1nn 10003 . . . . 5  |-  ( ( |_ `  ( 1  /  x ) )  e.  NN0  ->  ( ( |_ `  ( 1  /  x ) )  +  1 )  e.  NN )
97, 8syl 15 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( |_ `  ( 1  /  x ) )  +  1 )  e.  NN )
10 lmnn.3 . . . . . . . 8  |-  ( ph  ->  D  e.  ( * Met `  X ) )
1110ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  D  e.  ( * Met `  X
) )
12 lmnn.5 . . . . . . . . 9  |-  ( ph  ->  F : NN --> X )
1312ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  F : NN --> X )
14 nnuz 10263 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
1514uztrn2 10245 . . . . . . . . 9  |-  ( ( ( ( |_ `  ( 1  /  x
) )  +  1 )  e.  NN  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x
) )  +  1 ) ) )  -> 
k  e.  NN )
169, 15sylan 457 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  k  e.  NN )
17 ffvelrn 5663 . . . . . . . 8  |-  ( ( F : NN --> X  /\  k  e.  NN )  ->  ( F `  k
)  e.  X )
1813, 16, 17syl2anc 642 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( F `  k )  e.  X
)
191ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  P  e.  X
)
20 xmetcl 17896 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  ( F `  k )  e.  X  /\  P  e.  X
)  ->  ( ( F `  k ) D P )  e.  RR* )
2111, 18, 19, 20syl3anc 1182 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( F `
 k ) D P )  e.  RR* )
2216nnrecred 9791 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  / 
k )  e.  RR )
2322rexrd 8881 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  / 
k )  e.  RR* )
24 rpxr 10361 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e. 
RR* )
2524ad2antlr 707 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  x  e.  RR* )
26 lmnn.6 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( F `  k ) D P )  < 
( 1  /  k
) )
2726adantlr 695 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  NN )  ->  (
( F `  k
) D P )  <  ( 1  / 
k ) )
2816, 27syldan 456 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( F `
 k ) D P )  <  (
1  /  k ) )
294adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  /  x )  e.  RR )
309nnred 9761 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( |_ `  ( 1  /  x ) )  +  1 )  e.  RR )
3130adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( |_
`  ( 1  /  x ) )  +  1 )  e.  RR )
3216nnred 9761 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  k  e.  RR )
33 flltp1 10932 . . . . . . . . 9  |-  ( ( 1  /  x )  e.  RR  ->  (
1  /  x )  <  ( ( |_
`  ( 1  /  x ) )  +  1 ) )
3429, 33syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  /  x )  <  (
( |_ `  (
1  /  x ) )  +  1 ) )
35 eluzle 10240 . . . . . . . . 9  |-  ( k  e.  ( ZZ>= `  (
( |_ `  (
1  /  x ) )  +  1 ) )  ->  ( ( |_ `  ( 1  /  x ) )  +  1 )  <_  k
)
3635adantl 452 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( |_
`  ( 1  /  x ) )  +  1 )  <_  k
)
3729, 31, 32, 34, 36ltletrd 8976 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  /  x )  <  k
)
38 simplr 731 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  x  e.  RR+ )
39 rpregt0 10367 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <  x ) )
40 nnrp 10363 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR+ )
4140rpregt0d 10396 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
42 ltrec1 9643 . . . . . . . . 9  |-  ( ( ( x  e.  RR  /\  0  <  x )  /\  ( k  e.  RR  /\  0  < 
k ) )  -> 
( ( 1  /  x )  <  k  <->  ( 1  /  k )  <  x ) )
4339, 41, 42syl2an 463 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  k  e.  NN )  ->  (
( 1  /  x
)  <  k  <->  ( 1  /  k )  < 
x ) )
4438, 16, 43syl2anc 642 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( 1  /  x )  < 
k  <->  ( 1  / 
k )  <  x
) )
4537, 44mpbid 201 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  / 
k )  <  x
)
4621, 23, 25, 28, 45xrlttrd 10490 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( F `
 k ) D P )  <  x
)
4746ralrimiva 2626 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  A. k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) ( ( F `  k
) D P )  <  x )
48 fveq2 5525 . . . . . 6  |-  ( j  =  ( ( |_
`  ( 1  /  x ) )  +  1 )  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )
4948raleqdv 2742 . . . . 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 ) )
5049rspcev 2884 . . . 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 )
519, 47, 50syl2anc 642 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( ( F `  k ) D P )  <  x )
5251ralrimiva 2626 . 2  |-  ( ph  ->  A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( ( F `  k ) D P )  < 
x )
53 lmnn.2 . . 3  |-  J  =  ( MetOpen `  D )
54 1z 10053 . . . 4  |-  1  e.  ZZ
5554a1i 10 . . 3  |-  ( ph  ->  1  e.  ZZ )
56 eqidd 2284 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  =  ( F `  k
) )
5753, 10, 14, 55, 56, 12lmmbrf 18688 . 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 ) ) )
581, 52, 57mpbir2and 888 1  |-  ( ph  ->  F ( ~~> t `  J ) P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740   RR*cxr 8866    < clt 8867    <_ cle 8868    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   |_cfl 10924   * Metcxmt 16369   MetOpencmopn 16372   ~~> tclm 16956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-fl 10925  df-topgen 13344  df-xmet 16373  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-lm 16959
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