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Theorem lediv1 10295
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 10294 . . . 4  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  <  A  <->  ( B  /  C )  <  ( A  /  C ) ) )
213com12 1192 . . 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 9554 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  -.  B  <  A ) )
543adant3 1008 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  -.  B  <  A ) )
6 gt0ne0 9905 . . . . . . 7  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
763adant1 1006 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
8 redivcl 10151 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  C  =/=  0 )  ->  ( A  /  C )  e.  RR )
97, 8syld3an3 1264 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  ( A  /  C )  e.  RR )
1093expb 1189 . . . 4  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  /  C )  e.  RR )
11103adant2 1007 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  /  C
)  e.  RR )
1263adant1 1006 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
13 redivcl 10151 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  C  =/=  0 )  ->  ( B  /  C )  e.  RR )
1412, 13syld3an3 1264 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  ( B  /  C )  e.  RR )
15143expb 1189 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( B  /  C )  e.  RR )
16153adant1 1006 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  /  C
)  e.  RR )
1711, 16lenltd 9621 . 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 965    e. wcel 1758    =/= wne 2644   class class class wbr 4390  (class class class)co 6190   RRcr 9382   0cc0 9383    < clt 9519    <_ cle 9520    / cdiv 10094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-po 4739  df-so 4740  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095
This theorem is referenced by:  ge0div  10297  ledivmul  10306  ledivmulOLD  10307  lediv23  10325  lediv1d  11170  icccntr  11526  quoremz  11795  quoremnn0ALT  11797  sin01bnd  13571  cos01bnd  13572  sin02gt0  13578  hashdvds  13952  ovolscalem1  21112  dyadf  21187  dyadovol  21189  dyadmaxlem  21193  mbfi1fseqlem6  21314  cosordlem  22103  cxpcn3lem  22301  dvdsflf1o  22643  ppiub  22659  logfacrlim  22679  bposlem5  22743  lgseisenlem1  22804  vmadivsum  22847  mulog2sumlem2  22900  logdivbnd  22921  cdj1i  25972  cos2h  28561  heiborlem8  28855  reglogleb  29371  areaquad  29730  stoweidlem1  29934  stoweidlem11  29944  stoweidlem14  29947  taupilem1  35921
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