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Theorem divrcnv 12274
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 11729 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
2 rerpdivcl 10349 . . . . 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 10330 . . . . . . . . . 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 10337 . . . . . . . . . 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 11903 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  ( A  /  n ) )  =  ( ( abs `  A )  /  ( abs `  n ) ) )
10 rpre 10328 . . . . . . . . . . 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 10334 . . . . . . . . . . 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 11871 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  n
)  =  n )
1514oveq2d 5808 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( ( abs `  A )  /  ( abs `  n ) )  =  ( ( abs `  A )  /  n
) )
169, 15eqtrd 2290 . . . . . . 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 11884 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  A
)  e.  RR )
19 rpre 10328 . . . . . . . . . 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 10333 . . . . . . . . . 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 10333 . . . . . . . . . 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 9615 . . . . . . . . 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 4017 . . . . . 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 2601 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  A. n  e.  RR+  (
( ( abs `  A
)  /  x )  <  n  ->  ( abs `  ( A  /  n ) )  < 
x ) )
31 breq1 4000 . . . . . . 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 2538 . . . . 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 2859 . . . 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 2601 . 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 9504 . . . 4  |-  ( ( A  e.  CC  /\  n  e.  RR+ )  -> 
( A  /  n
)  e.  CC )
4140ralrimiva 2601 . . 3  |-  ( A  e.  CC  ->  A. n  e.  RR+  ( A  /  n )  e.  CC )
42 rpssre 10332 . . . 4  |-  RR+  C_  RR
4342a1i 12 . . 3  |-  ( A  e.  CC  ->  RR+  C_  RR )
4441, 43rlim0lt 11949 . 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 2421   A.wral 2518   E.wrex 2519    C_ wss 3127   class class class wbr 3997    e. cmpt 4051   ` cfv 4673  (class class class)co 5792   CCcc 8703   RRcr 8704   0cc0 8705    < clt 8835    <_ cle 8836    / cdiv 9391   RR+crp 10322   abscabs 11685    ~~> r crli 11925
This theorem is referenced by:  divcnv  12275  cxp2limlem  20233  logfacrlim  20426  dchrmusumlema  20605  mudivsum  20642  selberg2lem  20662  pntrsumo1  20677
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 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-pm 6743  df-en 6832  df-dom 6833  df-sdom 6834  df-sup 7162  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9934  df-z 9993  df-uz 10199  df-rp 10323  df-seq 11014  df-exp 11072  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-rlim 11929
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