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Theorem lediv1 8620
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 8619 . . . 4  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  <  A  <->  ( B  /  C )  <  ( A  /  C ) ) )
213com12 1185 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  <  A  <->  ( B  /  C )  <  ( A  /  C ) ) )
32notbid 656 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( -.  B  < 
A  <->  -.  ( B  /  C )  <  ( A  /  C ) ) )
4 lenlt 7833 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  -.  B  <  A ) )
543adant3 1001 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  -.  B  <  A ) )
6 gt0ap0 8381 . . . . . . 7  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C #  0 )
763adant1 999 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  C #  0 )
8 redivclap 8484 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  C #  0 )  ->  ( A  /  C )  e.  RR )
97, 8syld3an3 1261 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  ( A  /  C )  e.  RR )
1093expb 1182 . . . 4  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  /  C )  e.  RR )
11103adant2 1000 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  /  C
)  e.  RR )
1263adant1 999 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  C #  0 )
13 redivclap 8484 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  C #  0 )  ->  ( B  /  C )  e.  RR )
1412, 13syld3an3 1261 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  ( B  /  C )  e.  RR )
15143expb 1182 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( B  /  C )  e.  RR )
16153adant1 999 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  /  C
)  e.  RR )
1711, 16lenltd 7873 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <_  ( B  /  C )  <->  -.  ( B  /  C )  < 
( A  /  C
) ) )
183, 5, 173bitr4d 219 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  ( A  /  C )  <_  ( B  /  C ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    e. wcel 1480   class class class wbr 3924  (class class class)co 5767   RRcr 7612   0cc0 7613    < clt 7793    <_ cle 7794   # cap 8336    / cdiv 8425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426
This theorem is referenced by:  ge0div  8622  ledivmul  8628  lediv23  8644  lediv1d  9523  icccntr  9776  sin01bnd  11453  cos01bnd  11454  sin02gt0  11459  hashdvds  11886  cosordlem  12919
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