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Theorem divrcnv 12238
Description: The sequence of reciprocals of real numbers, multiplied by the factor  A, converges to zero. (Contributed by Mario Carneiro, 18-Sep-2014.)
Assertion
Ref Expression
divrcnv  |-  ( A  e.  CC  ->  (
n  e.  RR+  |->  ( A  /  n ) )  ~~> r  0 )
Distinct variable group:    A, n

Proof of Theorem divrcnv
StepHypRef Expression
1 abscl 11693 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
2 rerpdivcl 10313 . . . . 5  |-  ( ( ( abs `  A
)  e.  RR  /\  x  e.  RR+ )  -> 
( ( abs `  A
)  /  x )  e.  RR )
31, 2sylan 459 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( abs `  A
)  /  x )  e.  RR )
4 simpll 733 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  A  e.  CC )
5 rpcn 10294 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  n  e.  CC )
65ad2antrl 711 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  n  e.  CC )
7 rpne0 10301 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  n  =/=  0 )
87ad2antrl 711 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  n  =/=  0
)
94, 6, 8absdivd 11867 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  ( A  /  n ) )  =  ( ( abs `  A )  /  ( abs `  n ) ) )
10 rpre 10292 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  RR )
1110ad2antrl 711 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  n  e.  RR )
12 rpge0 10298 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  0  <_  n )
1312ad2antrl 711 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  0  <_  n
)
1411, 13absidd 11835 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  n
)  =  n )
1514oveq2d 5773 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( ( abs `  A )  /  ( abs `  n ) )  =  ( ( abs `  A )  /  n
) )
169, 15eqtrd 2288 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  ( A  /  n ) )  =  ( ( abs `  A )  /  n
) )
17 simprr 736 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( ( abs `  A )  /  x
)  <  n )
184abscld 11848 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  A
)  e.  RR )
19 rpre 10292 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  x  e.  RR )
2019ad2antlr 710 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  x  e.  RR )
21 rpgt0 10297 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  0  < 
x )
2221ad2antlr 710 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  0  <  x
)
23 rpgt0 10297 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  0  < 
n )
2423ad2antrl 711 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  0  <  n
)
25 ltdiv23 9580 . . . . . . . . 9  |-  ( ( ( abs `  A
)  e.  RR  /\  ( x  e.  RR  /\  0  <  x )  /\  ( n  e.  RR  /\  0  < 
n ) )  -> 
( ( ( abs `  A )  /  x
)  <  n  <->  ( ( abs `  A )  /  n )  <  x
) )
2618, 20, 22, 11, 24, 25syl122anc 1196 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( ( ( abs `  A )  /  x )  < 
n  <->  ( ( abs `  A )  /  n
)  <  x )
)
2717, 26mpbid 203 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( ( abs `  A )  /  n
)  <  x )
2816, 27eqbrtrd 3983 . . . . . 6  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  ( A  /  n ) )  <  x )
2928expr 601 . . . . 5  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  n  e.  RR+ )  ->  ( ( ( abs `  A )  /  x
)  <  n  ->  ( abs `  ( A  /  n ) )  <  x ) )
3029ralrimiva 2597 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  A. n  e.  RR+  (
( ( abs `  A
)  /  x )  <  n  ->  ( abs `  ( A  /  n ) )  < 
x ) )
31 breq1 3966 . . . . . . 7  |-  ( y  =  ( ( abs `  A )  /  x
)  ->  ( y  <  n  <->  ( ( abs `  A )  /  x
)  <  n )
)
3231imbi1d 310 . . . . . 6  |-  ( y  =  ( ( abs `  A )  /  x
)  ->  ( (
y  <  n  ->  ( abs `  ( A  /  n ) )  <  x )  <->  ( (
( abs `  A
)  /  x )  <  n  ->  ( abs `  ( A  /  n ) )  < 
x ) ) )
3332ralbidv 2534 . . . . 5  |-  ( y  =  ( ( abs `  A )  /  x
)  ->  ( A. n  e.  RR+  ( y  <  n  ->  ( abs `  ( A  /  n ) )  < 
x )  <->  A. n  e.  RR+  ( ( ( abs `  A )  /  x )  < 
n  ->  ( abs `  ( A  /  n
) )  <  x
) ) )
3433rcla4ev 2835 . . . 4  |-  ( ( ( ( abs `  A
)  /  x )  e.  RR  /\  A. n  e.  RR+  ( ( ( abs `  A
)  /  x )  <  n  ->  ( abs `  ( A  /  n ) )  < 
x ) )  ->  E. y  e.  RR  A. n  e.  RR+  (
y  <  n  ->  ( abs `  ( A  /  n ) )  <  x ) )
353, 30, 34syl2anc 645 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR  A. n  e.  RR+  (
y  <  n  ->  ( abs `  ( A  /  n ) )  <  x ) )
3635ralrimiva 2597 . 2  |-  ( A  e.  CC  ->  A. x  e.  RR+  E. y  e.  RR  A. n  e.  RR+  ( y  <  n  ->  ( abs `  ( A  /  n ) )  <  x ) )
37 simpl 445 . . . . 5  |-  ( ( A  e.  CC  /\  n  e.  RR+ )  ->  A  e.  CC )
385adantl 454 . . . . 5  |-  ( ( A  e.  CC  /\  n  e.  RR+ )  ->  n  e.  CC )
397adantl 454 . . . . 5  |-  ( ( A  e.  CC  /\  n  e.  RR+ )  ->  n  =/=  0 )
4037, 38, 39divcld 9469 . . . 4  |-  ( ( A  e.  CC  /\  n  e.  RR+ )  -> 
( A  /  n
)  e.  CC )
4140ralrimiva 2597 . . 3  |-  ( A  e.  CC  ->  A. n  e.  RR+  ( A  /  n )  e.  CC )
42 rpssre 10296 . . . 4  |-  RR+  C_  RR
4342a1i 12 . . 3  |-  ( A  e.  CC  ->  RR+  C_  RR )
4441, 43rlim0lt 11913 . 2  |-  ( A  e.  CC  ->  (
( n  e.  RR+  |->  ( A  /  n
) )  ~~> r  0  <->  A. x  e.  RR+  E. y  e.  RR  A. n  e.  RR+  ( y  <  n  ->  ( abs `  ( A  /  n ) )  <  x ) ) )
4536, 44mpbird 225 1  |-  ( A  e.  CC  ->  (
n  e.  RR+  |->  ( A  /  n ) )  ~~> r  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   E.wrex 2517    C_ wss 3094   class class class wbr 3963    e. cmpt 4017   ` cfv 4638  (class class class)co 5757   CCcc 8668   RRcr 8669   0cc0 8670    < clt 8800    <_ cle 8801    / cdiv 9356   RR+crp 10286   abscabs 11649    ~~> r crli 11889
This theorem is referenced by:  divcnv  12239  cxp2limlem  20197  logfacrlim  20390  dchrmusumlema  20569  mudivsum  20606  selberg2lem  20626  pntrsumo1  20641
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-pm 6708  df-en 6797  df-dom 6798  df-sdom 6799  df-sup 7127  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-n0 9898  df-z 9957  df-uz 10163  df-rp 10287  df-seq 10978  df-exp 11036  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-rlim 11893
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