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Theorem lmnn 18705
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 10393 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
32adantl 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR+ )
43rpred 10406 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR )
53rpge0d 10410 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  0  <_  ( 1  /  x ) )
6 flge0nn0 10964 . . . . . 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 10019 . . . . 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 10279 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
1514uztrn2 10261 . . . . . . . . 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 5679 . . . . . . . 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 17912 . . . . . . 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 9807 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  / 
k )  e.  RR )
2322rexrd 8897 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( 1  / 
k )  e.  RR* )
24 rpxr 10377 . . . . . . 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 9777 . . . . . . . . 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 9777 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  k  e.  RR )
33 flltp1 10948 . . . . . . . . 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 10256 . . . . . . . . 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 8992 . . . . . . 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 10383 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <  x ) )
40 nnrp 10379 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR+ )
4140rpregt0d 10412 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
42 ltrec1 9659 . . . . . . . . 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 10506 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )  ->  ( ( F `
 k ) D P )  <  x
)
4746ralrimiva 2639 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  A. k  e.  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) ( ( F `  k
) D P )  <  x )
48 fveq2 5541 . . . . . 6  |-  ( j  =  ( ( |_
`  ( 1  /  x ) )  +  1 )  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  ( ( |_ `  ( 1  /  x ) )  +  1 ) ) )
4948raleqdv 2755 . . . . 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 2897 . . . 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 2639 . 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 10069 . . . 4  |-  1  e.  ZZ
5554a1i 10 . . 3  |-  ( ph  ->  1  e.  ZZ )
56 eqidd 2297 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  =  ( F `  k
) )
5753, 10, 14, 55, 56, 12lmmbrf 18704 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756   RR*cxr 8882    < clt 8883    <_ cle 8884    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   |_cfl 10940   * Metcxmt 16385   MetOpencmopn 16388   ~~> tclm 16972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-fl 10941  df-topgen 13360  df-xmet 16389  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-lm 16975
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