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Theorem lmnn 19208
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 10627 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
32adantl 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR+ )
43rpred 10640 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR )
53rpge0d 10644 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  0  <_  ( 1  /  x ) )
6 flge0nn0 11217 . . . . . 6  |-  ( ( ( 1  /  x
)  e.  RR  /\  0  <_  ( 1  /  x ) )  -> 
( |_ `  (
1  /  x ) )  e.  NN0 )
74, 5, 6syl2anc 643 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( |_ `  ( 1  /  x
) )  e.  NN0 )
8 nn0p1nn 10251 . . . . 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 707 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  D  e.  ( * Met `  X
) )
12 lmnn.5 . . . . . . . . 9  |-  ( ph  ->  F : NN --> X )
1312ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  F : NN --> X )
14 nnuz 10513 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
1514uztrn2 10495 . . . . . . . . 9  |-  ( ( ( ( |_ `  ( 1  /  x
) )  +  1 )  e.  NN  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x
) )  +  1 ) ) )  -> 
k  e.  NN )
169, 15sylan 458 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  k  e.  NN )
1713, 16ffvelrnd 5863 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( F `  k )  e.  X
)
181ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  P  e.  X
)
19 xmetcl 18353 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  ( F `  k )  e.  X  /\  P  e.  X
)  ->  ( ( F `  k ) D P )  e.  RR* )
2011, 17, 18, 19syl3anc 1184 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( F `
 k ) D P )  e.  RR* )
2116nnrecred 10037 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  / 
k )  e.  RR )
2221rexrd 9126 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  / 
k )  e.  RR* )
23 rpxr 10611 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e. 
RR* )
2423ad2antlr 708 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  x  e.  RR* )
25 lmnn.6 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( F `  k ) D P )  < 
( 1  /  k
) )
2625adantlr 696 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  NN )  ->  (
( F `  k
) D P )  <  ( 1  / 
k ) )
2716, 26syldan 457 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( F `
 k ) D P )  <  (
1  /  k ) )
284adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  /  x )  e.  RR )
299nnred 10007 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( |_ `  ( 1  /  x ) )  +  1 )  e.  RR )
3029adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( |_
`  ( 1  /  x ) )  +  1 )  e.  RR )
3116nnred 10007 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  k  e.  RR )
32 flltp1 11201 . . . . . . . . 9  |-  ( ( 1  /  x )  e.  RR  ->  (
1  /  x )  <  ( ( |_
`  ( 1  /  x ) )  +  1 ) )
3328, 32syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  /  x )  <  (
( |_ `  (
1  /  x ) )  +  1 ) )
34 eluzle 10490 . . . . . . . . 9  |-  ( k  e.  ( ZZ>= `  (
( |_ `  (
1  /  x ) )  +  1 ) )  ->  ( ( |_ `  ( 1  /  x ) )  +  1 )  <_  k
)
3534adantl 453 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( |_
`  ( 1  /  x ) )  +  1 )  <_  k
)
3628, 30, 31, 33, 35ltletrd 9222 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  /  x )  <  k
)
37 simplr 732 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  x  e.  RR+ )
38 rpregt0 10617 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <  x ) )
39 nnrp 10613 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR+ )
4039rpregt0d 10646 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
41 ltrec1 9889 . . . . . . . . 9  |-  ( ( ( x  e.  RR  /\  0  <  x )  /\  ( k  e.  RR  /\  0  < 
k ) )  -> 
( ( 1  /  x )  <  k  <->  ( 1  /  k )  <  x ) )
4238, 40, 41syl2an 464 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  k  e.  NN )  ->  (
( 1  /  x
)  <  k  <->  ( 1  /  k )  < 
x ) )
4337, 16, 42syl2anc 643 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( 1  /  x )  < 
k  <->  ( 1  / 
k )  <  x
) )
4436, 43mpbid 202 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  / 
k )  <  x
)
4520, 22, 24, 27, 44xrlttrd 10741 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( F `
 k ) D P )  <  x
)
4645ralrimiva 2781 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  A. k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) ( ( F `  k
) D P )  <  x )
47 fveq2 5720 . . . . . 6  |-  ( j  =  ( ( |_
`  ( 1  /  x ) )  +  1 )  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )
4847raleqdv 2902 . . . . 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 ) )
4948rspcev 3044 . . . 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 )
509, 46, 49syl2anc 643 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( ( F `  k ) D P )  <  x )
5150ralrimiva 2781 . 2  |-  ( ph  ->  A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( ( F `  k ) D P )  < 
x )
52 lmnn.2 . . 3  |-  J  =  ( MetOpen `  D )
53 1z 10303 . . . 4  |-  1  e.  ZZ
5453a1i 11 . . 3  |-  ( ph  ->  1  e.  ZZ )
55 eqidd 2436 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  =  ( F `  k
) )
5652, 10, 14, 54, 55, 12lmmbrf 19207 . 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 ) ) )
571, 51, 56mpbir2and 889 1  |-  ( ph  ->  F ( ~~> t `  J ) P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   class class class wbr 4204   -->wf 5442   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985   RR*cxr 9111    < clt 9112    <_ cle 9113    / cdiv 9669   NNcn 9992   NN0cn0 10213   ZZcz 10274   ZZ>=cuz 10480   RR+crp 10604   |_cfl 11193   * Metcxmt 16678   MetOpencmopn 16683   ~~> tclm 17282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-fl 11194  df-topgen 13659  df-psmet 16686  df-xmet 16687  df-bl 16689  df-mopn 16690  df-top 16955  df-bases 16957  df-topon 16958  df-lm 17285
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