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Theorem lediv1 8393
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 8392 . . . 4  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  <  A  <->  ( B  /  C )  <  ( A  /  C ) ) )
213com12 1148 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  <  A  <->  ( B  /  C )  <  ( A  /  C ) ) )
32notbid 628 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( -.  B  < 
A  <->  -.  ( B  /  C )  <  ( A  /  C ) ) )
4 lenlt 7624 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  -.  B  <  A ) )
543adant3 964 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  -.  B  <  A ) )
6 gt0ap0 8165 . . . . . . 7  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C #  0 )
763adant1 962 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  C #  0 )
8 redivclap 8261 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  C #  0 )  ->  ( A  /  C )  e.  RR )
97, 8syld3an3 1220 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  ( A  /  C )  e.  RR )
1093expb 1145 . . . 4  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  /  C )  e.  RR )
11103adant2 963 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  /  C
)  e.  RR )
1263adant1 962 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  C #  0 )
13 redivclap 8261 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  C #  0 )  ->  ( B  /  C )  e.  RR )
1412, 13syld3an3 1220 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  ( B  /  C )  e.  RR )
15143expb 1145 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( B  /  C )  e.  RR )
16153adant1 962 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  /  C
)  e.  RR )
1711, 16lenltd 7664 . 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 925    e. wcel 1439   class class class wbr 3853  (class class class)co 5668   RRcr 7412   0cc0 7413    < clt 7585    <_ cle 7586   # cap 8121    / cdiv 8202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-1re 7502  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-mulrcl 7507  ax-addcom 7508  ax-mulcom 7509  ax-addass 7510  ax-mulass 7511  ax-distr 7512  ax-i2m1 7513  ax-0lt1 7514  ax-1rid 7515  ax-0id 7516  ax-rnegex 7517  ax-precex 7518  ax-cnre 7519  ax-pre-ltirr 7520  ax-pre-ltwlin 7521  ax-pre-lttrn 7522  ax-pre-apti 7523  ax-pre-ltadd 7524  ax-pre-mulgt0 7525  ax-pre-mulext 7526
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2624  df-sbc 2844  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-id 4131  df-po 4134  df-iso 4135  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-iota 4995  df-fun 5032  df-fv 5038  df-riota 5624  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591  df-sub 7718  df-neg 7719  df-reap 8115  df-ap 8122  df-div 8203
This theorem is referenced by:  ge0div  8395  ledivmul  8401  lediv23  8417  lediv1d  9283  icccntr  9480  sin01bnd  11111  cos01bnd  11112  sin02gt0  11117  hashdvds  11538
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