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Theorem lmnn 20733
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 11010 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
32adantl 463 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR+ )
43rpred 11023 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR )
53rpge0d 11027 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  0  <_  ( 1  /  x ) )
6 flge0nn0 11662 . . . . . 6  |-  ( ( ( 1  /  x
)  e.  RR  /\  0  <_  ( 1  /  x ) )  -> 
( |_ `  (
1  /  x ) )  e.  NN0 )
74, 5, 6syl2anc 656 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( |_ `  ( 1  /  x
) )  e.  NN0 )
8 nn0p1nn 10615 . . . . 5  |-  ( ( |_ `  ( 1  /  x ) )  e.  NN0  ->  ( ( |_ `  ( 1  /  x ) )  +  1 )  e.  NN )
97, 8syl 16 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( |_ `  ( 1  /  x ) )  +  1 )  e.  NN )
10 lmnn.3 . . . . . . . 8  |-  ( ph  ->  D  e.  ( *Met `  X ) )
1110ad2antrr 720 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  D  e.  ( *Met `  X
) )
12 lmnn.5 . . . . . . . . 9  |-  ( ph  ->  F : NN --> X )
1312ad2antrr 720 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  F : NN --> X )
14 eluznn 10921 . . . . . . . . 9  |-  ( ( ( ( |_ `  ( 1  /  x
) )  +  1 )  e.  NN  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x
) )  +  1 ) ) )  -> 
k  e.  NN )
159, 14sylan 468 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  k  e.  NN )
1613, 15ffvelrnd 5841 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( F `  k )  e.  X
)
171ad2antrr 720 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  P  e.  X
)
18 xmetcl 19865 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  ( F `  k )  e.  X  /\  P  e.  X
)  ->  ( ( F `  k ) D P )  e.  RR* )
1911, 16, 17, 18syl3anc 1213 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( F `
 k ) D P )  e.  RR* )
2015nnrecred 10363 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  / 
k )  e.  RR )
2120rexrd 9429 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  / 
k )  e.  RR* )
22 rpxr 10994 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e. 
RR* )
2322ad2antlr 721 . . . . . 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
) )
2524adantlr 709 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  NN )  ->  (
( F `  k
) D P )  <  ( 1  / 
k ) )
2615, 25syldan 467 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( F `
 k ) D P )  <  (
1  /  k ) )
274adantr 462 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  /  x )  e.  RR )
289nnred 10333 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( |_ `  ( 1  /  x ) )  +  1 )  e.  RR )
2928adantr 462 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( |_
`  ( 1  /  x ) )  +  1 )  e.  RR )
3015nnred 10333 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  k  e.  RR )
31 flltp1 11646 . . . . . . . . 9  |-  ( ( 1  /  x )  e.  RR  ->  (
1  /  x )  <  ( ( |_
`  ( 1  /  x ) )  +  1 ) )
3227, 31syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  /  x )  <  (
( |_ `  (
1  /  x ) )  +  1 ) )
33 eluzle 10869 . . . . . . . . 9  |-  ( k  e.  ( ZZ>= `  (
( |_ `  (
1  /  x ) )  +  1 ) )  ->  ( ( |_ `  ( 1  /  x ) )  +  1 )  <_  k
)
3433adantl 463 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( |_
`  ( 1  /  x ) )  +  1 )  <_  k
)
3527, 29, 30, 32, 34ltletrd 9527 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  /  x )  <  k
)
36 simplr 749 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  x  e.  RR+ )
37 rpregt0 11000 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <  x ) )
38 nnrp 10996 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR+ )
3938rpregt0d 11029 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
40 ltrec1 10215 . . . . . . . . 9  |-  ( ( ( x  e.  RR  /\  0  <  x )  /\  ( k  e.  RR  /\  0  < 
k ) )  -> 
( ( 1  /  x )  <  k  <->  ( 1  /  k )  <  x ) )
4137, 39, 40syl2an 474 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  k  e.  NN )  ->  (
( 1  /  x
)  <  k  <->  ( 1  /  k )  < 
x ) )
4236, 15, 41syl2anc 656 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( 1  /  x )  < 
k  <->  ( 1  / 
k )  <  x
) )
4335, 42mpbid 210 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  / 
k )  <  x
)
4419, 21, 23, 26, 43xrlttrd 11129 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( F `
 k ) D P )  <  x
)
4544ralrimiva 2797 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  A. k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) ( ( F `  k
) D P )  <  x )
46 fveq2 5688 . . . . . 6  |-  ( j  =  ( ( |_
`  ( 1  /  x ) )  +  1 )  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )
4746raleqdv 2921 . . . . 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 ) )
4847rspcev 3070 . . . 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 )
499, 45, 48syl2anc 656 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( ( F `  k ) D P )  <  x )
5049ralrimiva 2797 . 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 10892 . . 3  |-  NN  =  ( ZZ>= `  1 )
53 1zzd 10673 . . 3  |-  ( ph  ->  1  e.  ZZ )
54 eqidd 2442 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  =  ( F `  k
) )
5551, 10, 52, 53, 54, 12lmmbrf 20732 . 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 ) ) )
561, 50, 55mpbir2and 908 1  |-  ( ph  ->  F ( ~~> t `  J ) P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   E.wrex 2714   class class class wbr 4289   -->wf 5411   ` cfv 5415  (class class class)co 6090   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281   RR*cxr 9413    < clt 9414    <_ cle 9415    / cdiv 9989   NNcn 10318   NN0cn0 10575   ZZ>=cuz 10857   RR+crp 10987   |_cfl 11636   *Metcxmt 17760   MetOpencmopn 17765   ~~> tclm 18789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-fl 11638  df-topgen 14378  df-psmet 17768  df-xmet 17769  df-bl 17771  df-mopn 17772  df-top 18462  df-bases 18464  df-topon 18465  df-lm 18792
This theorem is referenced by: (None)
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