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Theorem divrcnv 13643
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 13090 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
2 rerpdivcl 11256 . . . . 5  |-  ( ( ( abs `  A
)  e.  RR  /\  x  e.  RR+ )  -> 
( ( abs `  A
)  /  x )  e.  RR )
31, 2sylan 471 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( abs `  A
)  /  x )  e.  RR )
4 simpll 753 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  A  e.  CC )
5 rpcn 11237 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  n  e.  CC )
65ad2antrl 727 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  n  e.  CC )
7 rpne0 11244 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  n  =/=  0 )
87ad2antrl 727 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  n  =/=  0
)
94, 6, 8absdivd 13265 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  ( A  /  n ) )  =  ( ( abs `  A )  /  ( abs `  n ) ) )
10 rpre 11235 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  RR )
1110ad2antrl 727 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  n  e.  RR )
12 rpge0 11241 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  0  <_  n )
1312ad2antrl 727 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  0  <_  n
)
1411, 13absidd 13233 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  n
)  =  n )
1514oveq2d 6297 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( ( abs `  A )  /  ( abs `  n ) )  =  ( ( abs `  A )  /  n
) )
169, 15eqtrd 2484 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  ( A  /  n ) )  =  ( ( abs `  A )  /  n
) )
17 simprr 757 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( ( abs `  A )  /  x
)  <  n )
184abscld 13246 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  A
)  e.  RR )
19 rpre 11235 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  x  e.  RR )
2019ad2antlr 726 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  x  e.  RR )
21 rpgt0 11240 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  0  < 
x )
2221ad2antlr 726 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  0  <  x
)
23 rpgt0 11240 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  0  < 
n )
2423ad2antrl 727 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  0  <  n
)
25 ltdiv23 10442 . . . . . . . . 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 1238 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( ( ( abs `  A )  /  x )  < 
n  <->  ( ( abs `  A )  /  n
)  <  x )
)
2717, 26mpbid 210 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( ( abs `  A )  /  n
)  <  x )
2816, 27eqbrtrd 4457 . . . . . 6  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  ( n  e.  RR+  /\  ( ( abs `  A
)  /  x )  <  n ) )  ->  ( abs `  ( A  /  n ) )  <  x )
2928expr 615 . . . . 5  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  n  e.  RR+ )  ->  ( ( ( abs `  A )  /  x
)  <  n  ->  ( abs `  ( A  /  n ) )  <  x ) )
3029ralrimiva 2857 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  A. n  e.  RR+  (
( ( abs `  A
)  /  x )  <  n  ->  ( abs `  ( A  /  n ) )  < 
x ) )
31 breq1 4440 . . . . . . 7  |-  ( y  =  ( ( abs `  A )  /  x
)  ->  ( y  <  n  <->  ( ( abs `  A )  /  x
)  <  n )
)
3231imbi1d 317 . . . . . 6  |-  ( y  =  ( ( abs `  A )  /  x
)  ->  ( (
y  <  n  ->  ( abs `  ( A  /  n ) )  <  x )  <->  ( (
( abs `  A
)  /  x )  <  n  ->  ( abs `  ( A  /  n ) )  < 
x ) ) )
3332ralbidv 2882 . . . . 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 3196 . . . 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 661 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR  A. n  e.  RR+  (
y  <  n  ->  ( abs `  ( A  /  n ) )  <  x ) )
3635ralrimiva 2857 . 2  |-  ( A  e.  CC  ->  A. x  e.  RR+  E. y  e.  RR  A. n  e.  RR+  ( y  <  n  ->  ( abs `  ( A  /  n ) )  <  x ) )
37 simpl 457 . . . . 5  |-  ( ( A  e.  CC  /\  n  e.  RR+ )  ->  A  e.  CC )
385adantl 466 . . . . 5  |-  ( ( A  e.  CC  /\  n  e.  RR+ )  ->  n  e.  CC )
397adantl 466 . . . . 5  |-  ( ( A  e.  CC  /\  n  e.  RR+ )  ->  n  =/=  0 )
4037, 38, 39divcld 10326 . . . 4  |-  ( ( A  e.  CC  /\  n  e.  RR+ )  -> 
( A  /  n
)  e.  CC )
4140ralrimiva 2857 . . 3  |-  ( A  e.  CC  ->  A. n  e.  RR+  ( A  /  n )  e.  CC )
42 rpssre 11239 . . . 4  |-  RR+  C_  RR
4342a1i 11 . . 3  |-  ( A  e.  CC  ->  RR+  C_  RR )
4441, 43rlim0lt 13311 . 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 232 1  |-  ( A  e.  CC  ->  (
n  e.  RR+  |->  ( A  /  n ) )  ~~> r  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   E.wrex 2794    C_ wss 3461   class class class wbr 4437    |-> cmpt 4495   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495    < clt 9631    <_ cle 9632    / cdiv 10212   RR+crp 11229   abscabs 13046    ~~> r crli 13287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-n0 10802  df-z 10871  df-uz 11091  df-rp 11230  df-seq 12087  df-exp 12146  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-rlim 13291
This theorem is referenced by:  divcnv  13644  cxp2limlem  23177  logfacrlim  23371  dchrmusumlema  23550  mudivsum  23587  selberg2lem  23607  pntrsumo1  23622
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