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Theorem lediv1 10450
Description: Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)
Assertion
Ref Expression
lediv1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  ( A  /  C )  <_  ( B  /  C ) ) )

Proof of Theorem lediv1
StepHypRef Expression
1 ltdiv1 10449 . . . 4  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  <  A  <->  ( B  /  C )  <  ( A  /  C ) ) )
213com12 1203 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  <  A  <->  ( B  /  C )  <  ( A  /  C ) ) )
32notbid 294 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( -.  B  < 
A  <->  -.  ( B  /  C )  <  ( A  /  C ) ) )
4 lenlt 9696 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  -.  B  <  A ) )
543adant3 1019 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  -.  B  <  A ) )
6 gt0ne0 10060 . . . . . . 7  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
763adant1 1017 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
8 redivcl 10306 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  C  =/=  0 )  ->  ( A  /  C )  e.  RR )
97, 8syld3an3 1277 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  ( A  /  C )  e.  RR )
1093expb 1200 . . . 4  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  /  C )  e.  RR )
11103adant2 1018 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  /  C
)  e.  RR )
1263adant1 1017 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
13 redivcl 10306 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  C  =/=  0 )  ->  ( B  /  C )  e.  RR )
1412, 13syld3an3 1277 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  ( B  /  C )  e.  RR )
15143expb 1200 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( B  /  C )  e.  RR )
16153adant1 1017 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  /  C
)  e.  RR )
1711, 16lenltd 9765 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <_  ( B  /  C )  <->  -.  ( B  /  C )  < 
( A  /  C
) ) )
183, 5, 173bitr4d 287 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  ( A  /  C )  <_  ( B  /  C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    e. wcel 1844    =/= wne 2600   class class class wbr 4397  (class class class)co 6280   RRcr 9523   0cc0 9524    < clt 9660    <_ cle 9661    / cdiv 10249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-po 4746  df-so 4747  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250
This theorem is referenced by:  ge0div  10452  ledivmul  10461  lediv23  10479  lediv1d  11348  icccntr  11716  quoremz  12022  quoremnn0ALT  12024  sin01bnd  14131  cos01bnd  14132  sin02gt0  14138  hashdvds  14516  ovolscalem1  22218  dyadf  22294  dyadovol  22296  dyadmaxlem  22300  mbfi1fseqlem6  22421  cosordlem  23212  cxpcn3lem  23419  dvdsflf1o  23846  ppiub  23862  logfacrlim  23882  bposlem5  23946  lgseisenlem1  24007  vmadivsum  24050  mulog2sumlem2  24103  logdivbnd  24124  cdj1i  27778  taupilem1  31260  cos2h  31431  heiborlem8  31609  reglogleb  35202  areaquad  35561  stoweidlem1  37164  stoweidlem11  37174  stoweidlem14  37177  flnn0div2ge  38673
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