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Theorem lediv1 10403
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 10402 . . . 4  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  <  A  <->  ( B  /  C )  <  ( A  /  C ) ) )
213com12 1200 . . 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 9659 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  -.  B  <  A ) )
543adant3 1016 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  -.  B  <  A ) )
6 gt0ne0 10013 . . . . . . 7  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
763adant1 1014 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
8 redivcl 10259 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  C  =/=  0 )  ->  ( A  /  C )  e.  RR )
97, 8syld3an3 1273 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  ( A  /  C )  e.  RR )
1093expb 1197 . . . 4  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  /  C )  e.  RR )
11103adant2 1015 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  /  C
)  e.  RR )
1263adant1 1014 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
13 redivcl 10259 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  C  =/=  0 )  ->  ( B  /  C )  e.  RR )
1412, 13syld3an3 1273 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  ( B  /  C )  e.  RR )
15143expb 1197 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( B  /  C )  e.  RR )
16153adant1 1014 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  /  C
)  e.  RR )
1711, 16lenltd 9726 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <_  ( B  /  C )  <->  -.  ( B  /  C )  < 
( A  /  C
) ) )
183, 5, 173bitr4d 285 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 184    /\ wa 369    /\ w3a 973    e. wcel 1767    =/= wne 2662   class class class wbr 4447  (class class class)co 6282   RRcr 9487   0cc0 9488    < clt 9624    <_ cle 9625    / cdiv 10202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203
This theorem is referenced by:  ge0div  10405  ledivmul  10414  ledivmulOLD  10415  lediv23  10433  lediv1d  11294  icccntr  11656  quoremz  11946  quoremnn0ALT  11948  sin01bnd  13777  cos01bnd  13778  sin02gt0  13784  hashdvds  14160  ovolscalem1  21659  dyadf  21735  dyadovol  21737  dyadmaxlem  21741  mbfi1fseqlem6  21862  cosordlem  22651  cxpcn3lem  22849  dvdsflf1o  23191  ppiub  23207  logfacrlim  23227  bposlem5  23291  lgseisenlem1  23352  vmadivsum  23395  mulog2sumlem2  23448  logdivbnd  23469  cdj1i  27028  cos2h  29623  heiborlem8  29917  reglogleb  30432  areaquad  30789  stoweidlem1  31301  stoweidlem11  31311  stoweidlem14  31314  taupilem1  36767
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