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Theorem lmnn 20890
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 11115 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
32adantl 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR+ )
43rpred 11128 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR )
53rpge0d 11132 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  0  <_  ( 1  /  x ) )
6 flge0nn0 11767 . . . . . 6  |-  ( ( ( 1  /  x
)  e.  RR  /\  0  <_  ( 1  /  x ) )  -> 
( |_ `  (
1  /  x ) )  e.  NN0 )
74, 5, 6syl2anc 661 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( |_ `  ( 1  /  x
) )  e.  NN0 )
8 nn0p1nn 10720 . . . . 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 725 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  D  e.  ( *Met `  X
) )
12 lmnn.5 . . . . . . . . 9  |-  ( ph  ->  F : NN --> X )
1312ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  F : NN --> X )
14 eluznn 11026 . . . . . . . . 9  |-  ( ( ( ( |_ `  ( 1  /  x
) )  +  1 )  e.  NN  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x
) )  +  1 ) ) )  -> 
k  e.  NN )
159, 14sylan 471 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  k  e.  NN )
1613, 15ffvelrnd 5943 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( F `  k )  e.  X
)
171ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  P  e.  X
)
18 xmetcl 20022 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  ( F `  k )  e.  X  /\  P  e.  X
)  ->  ( ( F `  k ) D P )  e.  RR* )
1911, 16, 17, 18syl3anc 1219 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( F `
 k ) D P )  e.  RR* )
2015nnrecred 10468 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  / 
k )  e.  RR )
2120rexrd 9534 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  / 
k )  e.  RR* )
22 rpxr 11099 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e. 
RR* )
2322ad2antlr 726 . . . . . 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 714 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  NN )  ->  (
( F `  k
) D P )  <  ( 1  / 
k ) )
2615, 25syldan 470 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( F `
 k ) D P )  <  (
1  /  k ) )
274adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  /  x )  e.  RR )
289nnred 10438 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( |_ `  ( 1  /  x ) )  +  1 )  e.  RR )
2928adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( |_
`  ( 1  /  x ) )  +  1 )  e.  RR )
3015nnred 10438 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  k  e.  RR )
31 flltp1 11751 . . . . . . . . 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 10974 . . . . . . . . 9  |-  ( k  e.  ( ZZ>= `  (
( |_ `  (
1  /  x ) )  +  1 ) )  ->  ( ( |_ `  ( 1  /  x ) )  +  1 )  <_  k
)
3433adantl 466 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( |_
`  ( 1  /  x ) )  +  1 )  <_  k
)
3527, 29, 30, 32, 34ltletrd 9632 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  /  x )  <  k
)
36 simplr 754 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  x  e.  RR+ )
37 rpregt0 11105 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <  x ) )
38 nnrp 11101 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR+ )
3938rpregt0d 11134 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
40 ltrec1 10320 . . . . . . . . 9  |-  ( ( ( x  e.  RR  /\  0  <  x )  /\  ( k  e.  RR  /\  0  < 
k ) )  -> 
( ( 1  /  x )  <  k  <->  ( 1  /  k )  <  x ) )
4137, 39, 40syl2an 477 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  k  e.  NN )  ->  (
( 1  /  x
)  <  k  <->  ( 1  /  k )  < 
x ) )
4236, 15, 41syl2anc 661 . . . . . . 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 11234 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( F `
 k ) D P )  <  x
)
4544ralrimiva 2822 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  A. k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) ( ( F `  k
) D P )  <  x )
46 fveq2 5789 . . . . . 6  |-  ( j  =  ( ( |_
`  ( 1  /  x ) )  +  1 )  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )
4746raleqdv 3019 . . . . 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 3169 . . . 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 661 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( ( F `  k ) D P )  <  x )
5049ralrimiva 2822 . 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 10997 . . 3  |-  NN  =  ( ZZ>= `  1 )
53 1zzd 10778 . . 3  |-  ( ph  ->  1  e.  ZZ )
54 eqidd 2452 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  =  ( F `  k
) )
5551, 10, 52, 53, 54, 12lmmbrf 20889 . 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 913 1  |-  ( ph  ->  F ( ~~> t `  J ) P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   class class class wbr 4390   -->wf 5512   ` cfv 5516  (class class class)co 6190   RRcr 9382   0cc0 9383   1c1 9384    + caddc 9386   RR*cxr 9518    < clt 9519    <_ cle 9520    / cdiv 10094   NNcn 10423   NN0cn0 10680   ZZ>=cuz 10962   RR+crp 11092   |_cfl 11741   *Metcxmt 17910   MetOpencmopn 17915   ~~> tclm 18946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-er 7201  df-map 7316  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-n0 10681  df-z 10748  df-uz 10963  df-q 11055  df-rp 11093  df-xneg 11190  df-xadd 11191  df-xmul 11192  df-fl 11743  df-topgen 14484  df-psmet 17918  df-xmet 17919  df-bl 17921  df-mopn 17922  df-top 18619  df-bases 18621  df-topon 18622  df-lm 18949
This theorem is referenced by: (None)
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