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Theorem ledivmul 9888
Description: 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
Assertion
Ref Expression
ledivmul  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <_  B  <->  A  <_  ( C  x.  B ) ) )

Proof of Theorem ledivmul
StepHypRef Expression
1 remulcl 9080 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  x.  B
)  e.  RR )
21ancoms 441 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  x.  B
)  e.  RR )
32adantrr 699 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( C  x.  B )  e.  RR )
433adant1 976 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( C  x.  B
)  e.  RR )
5 lediv1 9880 . . 3  |-  ( ( A  e.  RR  /\  ( C  x.  B
)  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <_  ( C  x.  B
)  <->  ( A  /  C )  <_  (
( C  x.  B
)  /  C ) ) )
64, 5syld3an2 1232 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  ( C  x.  B )  <->  ( A  /  C )  <_  ( ( C  x.  B )  /  C ) ) )
7 recn 9085 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
87adantr 453 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  B  e.  CC )
9 recn 9085 . . . . . 6  |-  ( C  e.  RR  ->  C  e.  CC )
109ad2antrl 710 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  CC )
11 gt0ne0 9498 . . . . . 6  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
1211adantl 454 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  =/=  0 )
138, 10, 12divcan3d 9800 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( C  x.  B )  /  C )  =  B )
14133adant1 976 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( C  x.  B )  /  C
)  =  B )
1514breq2d 4227 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <_  (
( C  x.  B
)  /  C )  <-> 
( A  /  C
)  <_  B )
)
166, 15bitr2d 247 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <_  B  <->  A  <_  ( C  x.  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215  (class class class)co 6084   CCcc 8993   RRcr 8994   0cc0 8995    x. cmul 9000    < clt 9125    <_ cle 9126    / cdiv 9682
This theorem is referenced by:  ledivmul2  9892  ledivmul2OLD  9893  rpnnen1lem3  10607  ledivmuld  10702  discr1  11520  faclbnd2  11587  sqrlem7  12059  o1fsum  12597  eftlub  12715  eflegeo  12727  4sqlem16  13333  iihalf2  18963  lebnumii  18996  ovolscalem1  19414  itg2mulclem  19641  abelthlem7  20359  pilem2  20373  sinhalfpilem  20379  sincosq1lem  20410  cxpaddle  20641  leibpi  20787  log2ublem1  20791  jensenlem2  20831  harmonicbnd4  20854  fsumfldivdiaglem  20979  bcmono  21066  lgsquadlem1  21143  rplogsumlem1  21183  rplogsumlem2  21184  dchrisum0lem2a  21216  mulogsumlem  21230  pntlemr  21301  unitdivcld  24304  cvmliftlem2  24978  snmlff  25021  divelunit  25190  sin2h  26250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683
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