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Theorem divrcnv 12632
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 abscl 12083 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
2 rerpdivcl 10639 . . . . 5  |-  ( ( ( abs `  A
)  e.  RR  /\  x  e.  RR+ )  -> 
( ( abs `  A
)  /  x )  e.  RR )
31, 2sylan 458 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( abs `  A
)  /  x )  e.  RR )
4 simpll 731 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  A  e.  CC )
5 rpcn 10620 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  n  e.  CC )
65ad2antrl 709 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  n  e.  CC )
7 rpne0 10627 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  n  =/=  0 )
87ad2antrl 709 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  n  =/=  0
)
94, 6, 8absdivd 12257 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  ( A  /  n ) )  =  ( ( abs `  A )  /  ( abs `  n ) ) )
10 rpre 10618 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  RR )
1110ad2antrl 709 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  n  e.  RR )
12 rpge0 10624 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  0  <_  n )
1312ad2antrl 709 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  0  <_  n
)
1411, 13absidd 12225 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  n
)  =  n )
1514oveq2d 6097 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( ( abs `  A )  /  ( abs `  n ) )  =  ( ( abs `  A )  /  n
) )
169, 15eqtrd 2468 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  ( A  /  n ) )  =  ( ( abs `  A )  /  n
) )
17 simprr 734 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( ( abs `  A )  /  x
)  <  n )
184abscld 12238 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  A
)  e.  RR )
19 rpre 10618 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  x  e.  RR )
2019ad2antlr 708 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  x  e.  RR )
21 rpgt0 10623 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  0  < 
x )
2221ad2antlr 708 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  0  <  x
)
23 rpgt0 10623 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  0  < 
n )
2423ad2antrl 709 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  0  <  n
)
25 ltdiv23 9901 . . . . . . . . 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 1193 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( ( ( abs `  A )  /  x )  < 
n  <->  ( ( abs `  A )  /  n
)  <  x )
)
2717, 26mpbid 202 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( ( abs `  A )  /  n
)  <  x )
2816, 27eqbrtrd 4232 . . . . . 6  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  ( A  /  n ) )  <  x )
2928expr 599 . . . . 5  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  n  e.  RR+ )  ->  ( ( ( abs `  A )  /  x
)  <  n  ->  ( abs `  ( A  /  n ) )  <  x ) )
3029ralrimiva 2789 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  A. n  e.  RR+  (
( ( abs `  A
)  /  x )  <  n  ->  ( abs `  ( A  /  n ) )  < 
x ) )
31 breq1 4215 . . . . . . 7  |-  ( y  =  ( ( abs `  A )  /  x
)  ->  ( y  <  n  <->  ( ( abs `  A )  /  x
)  <  n )
)
3231imbi1d 309 . . . . . 6  |-  ( y  =  ( ( abs `  A )  /  x
)  ->  ( (
y  <  n  ->  ( abs `  ( A  /  n ) )  <  x )  <->  ( (
( abs `  A
)  /  x )  <  n  ->  ( abs `  ( A  /  n ) )  < 
x ) ) )
3332ralbidv 2725 . . . . 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
) ) )
3433rspcev 3052 . . . 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 643 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR  A. n  e.  RR+  (
y  <  n  ->  ( abs `  ( A  /  n ) )  <  x ) )
3635ralrimiva 2789 . 2  |-  ( A  e.  CC  ->  A. x  e.  RR+  E. y  e.  RR  A. n  e.  RR+  ( y  <  n  ->  ( abs `  ( A  /  n ) )  <  x ) )
37 simpl 444 . . . . 5  |-  ( ( A  e.  CC  /\  n  e.  RR+ )  ->  A  e.  CC )
385adantl 453 . . . . 5  |-  ( ( A  e.  CC  /\  n  e.  RR+ )  ->  n  e.  CC )
397adantl 453 . . . . 5  |-  ( ( A  e.  CC  /\  n  e.  RR+ )  ->  n  =/=  0 )
4037, 38, 39divcld 9790 . . . 4  |-  ( ( A  e.  CC  /\  n  e.  RR+ )  -> 
( A  /  n
)  e.  CC )
4140ralrimiva 2789 . . 3  |-  ( A  e.  CC  ->  A. n  e.  RR+  ( A  /  n )  e.  CC )
42 rpssre 10622 . . . 4  |-  RR+  C_  RR
4342a1i 11 . . 3  |-  ( A  e.  CC  ->  RR+  C_  RR )
4441, 43rlim0lt 12303 . 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 224 1  |-  ( A  e.  CC  ->  (
n  e.  RR+  |->  ( A  /  n ) )  ~~> r  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706    C_ wss 3320   class class class wbr 4212    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990    < clt 9120    <_ cle 9121    / cdiv 9677   RR+crp 10612   abscabs 12039    ~~> r crli 12279
This theorem is referenced by:  divcnv  12633  cxp2limlem  20814  logfacrlim  21008  dchrmusumlema  21187  mudivsum  21224  selberg2lem  21244  pntrsumo1  21259
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-rlim 12283
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