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Theorem divrcnv 12402
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 11853 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
2 rerpdivcl 10470 . . . . 5  |-  ( ( ( abs `  A
)  e.  RR  /\  x  e.  RR+ )  -> 
( ( abs `  A
)  /  x )  e.  RR )
31, 2sylan 457 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( abs `  A
)  /  x )  e.  RR )
4 simpll 730 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  A  e.  CC )
5 rpcn 10451 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  n  e.  CC )
65ad2antrl 708 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  n  e.  CC )
7 rpne0 10458 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  n  =/=  0 )
87ad2antrl 708 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  n  =/=  0
)
94, 6, 8absdivd 12027 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  ( A  /  n ) )  =  ( ( abs `  A )  /  ( abs `  n ) ) )
10 rpre 10449 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  RR )
1110ad2antrl 708 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  n  e.  RR )
12 rpge0 10455 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  0  <_  n )
1312ad2antrl 708 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  0  <_  n
)
1411, 13absidd 11995 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  n
)  =  n )
1514oveq2d 5958 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( ( abs `  A )  /  ( abs `  n ) )  =  ( ( abs `  A )  /  n
) )
169, 15eqtrd 2390 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  ( A  /  n ) )  =  ( ( abs `  A )  /  n
) )
17 simprr 733 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( ( abs `  A )  /  x
)  <  n )
184abscld 12008 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  A
)  e.  RR )
19 rpre 10449 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  x  e.  RR )
2019ad2antlr 707 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  x  e.  RR )
21 rpgt0 10454 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  0  < 
x )
2221ad2antlr 707 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  0  <  x
)
23 rpgt0 10454 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  0  < 
n )
2423ad2antrl 708 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  0  <  n
)
25 ltdiv23 9734 . . . . . . . . 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 1191 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( ( ( abs `  A )  /  x )  < 
n  <->  ( ( abs `  A )  /  n
)  <  x )
)
2717, 26mpbid 201 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( ( abs `  A )  /  n
)  <  x )
2816, 27eqbrtrd 4122 . . . . . 6  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  ( A  /  n ) )  <  x )
2928expr 598 . . . . 5  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  n  e.  RR+ )  ->  ( ( ( abs `  A )  /  x
)  <  n  ->  ( abs `  ( A  /  n ) )  <  x ) )
3029ralrimiva 2702 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  A. n  e.  RR+  (
( ( abs `  A
)  /  x )  <  n  ->  ( abs `  ( A  /  n ) )  < 
x ) )
31 breq1 4105 . . . . . . 7  |-  ( y  =  ( ( abs `  A )  /  x
)  ->  ( y  <  n  <->  ( ( abs `  A )  /  x
)  <  n )
)
3231imbi1d 308 . . . . . 6  |-  ( y  =  ( ( abs `  A )  /  x
)  ->  ( (
y  <  n  ->  ( abs `  ( A  /  n ) )  <  x )  <->  ( (
( abs `  A
)  /  x )  <  n  ->  ( abs `  ( A  /  n ) )  < 
x ) ) )
3332ralbidv 2639 . . . . 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 2960 . . . 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 642 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR  A. n  e.  RR+  (
y  <  n  ->  ( abs `  ( A  /  n ) )  <  x ) )
3635ralrimiva 2702 . 2  |-  ( A  e.  CC  ->  A. x  e.  RR+  E. y  e.  RR  A. n  e.  RR+  ( y  <  n  ->  ( abs `  ( A  /  n ) )  <  x ) )
37 simpl 443 . . . . 5  |-  ( ( A  e.  CC  /\  n  e.  RR+ )  ->  A  e.  CC )
385adantl 452 . . . . 5  |-  ( ( A  e.  CC  /\  n  e.  RR+ )  ->  n  e.  CC )
397adantl 452 . . . . 5  |-  ( ( A  e.  CC  /\  n  e.  RR+ )  ->  n  =/=  0 )
4037, 38, 39divcld 9623 . . . 4  |-  ( ( A  e.  CC  /\  n  e.  RR+ )  -> 
( A  /  n
)  e.  CC )
4140ralrimiva 2702 . . 3  |-  ( A  e.  CC  ->  A. n  e.  RR+  ( A  /  n )  e.  CC )
42 rpssre 10453 . . . 4  |-  RR+  C_  RR
4342a1i 10 . . 3  |-  ( A  e.  CC  ->  RR+  C_  RR )
4441, 43rlim0lt 12073 . 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 223 1  |-  ( A  e.  CC  ->  (
n  e.  RR+  |->  ( A  /  n ) )  ~~> r  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   E.wrex 2620    C_ wss 3228   class class class wbr 4102    e. cmpt 4156   ` cfv 5334  (class class class)co 5942   CCcc 8822   RRcr 8823   0cc0 8824    < clt 8954    <_ cle 8955    / cdiv 9510   RR+crp 10443   abscabs 11809    ~~> r crli 12049
This theorem is referenced by:  divcnv  12403  cxp2limlem  20375  logfacrlim  20569  dchrmusumlema  20748  mudivsum  20785  selberg2lem  20805  pntrsumo1  20820
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-pm 6860  df-en 6949  df-dom 6950  df-sdom 6951  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-rp 10444  df-seq 11136  df-exp 11195  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-rlim 12053
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