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Theorem lemul1 10173
Description: Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)
Assertion
Ref Expression
lemul1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  ( A  x.  C )  <_  ( B  x.  C ) ) )

Proof of Theorem lemul1
StepHypRef Expression
1 ltmul1 10171 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )
2 recn 9364 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
3 recn 9364 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
4 recn 9364 . . . . . . 7  |-  ( C  e.  RR  ->  C  e.  CC )
54adantr 465 . . . . . 6  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  e.  CC )
6 gt0ne0 9796 . . . . . 6  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
75, 6jca 532 . . . . 5  |-  ( ( C  e.  RR  /\  0  <  C )  -> 
( C  e.  CC  /\  C  =/=  0 ) )
8 mulcan2 9966 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  x.  C )  =  ( B  x.  C )  <-> 
A  =  B ) )
92, 3, 7, 8syl3an 1260 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  x.  C )  =  ( B  x.  C )  <-> 
A  =  B ) )
109bicomd 201 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  =  B  <-> 
( A  x.  C
)  =  ( B  x.  C ) ) )
111, 10orbi12d 709 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  < 
B  \/  A  =  B )  <->  ( ( A  x.  C )  <  ( B  x.  C
)  \/  ( A  x.  C )  =  ( B  x.  C
) ) ) )
12 leloe 9453 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  ( A  <  B  \/  A  =  B )
) )
13123adant3 1008 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  ( A  <  B  \/  A  =  B )
) )
14 remulcl 9359 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  x.  C
)  e.  RR )
15143adant2 1007 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  x.  C )  e.  RR )
16 remulcl 9359 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  x.  C
)  e.  RR )
17163adant1 1006 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  x.  C )  e.  RR )
1815, 17leloed 9509 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
)  <_  ( B  x.  C )  <->  ( ( A  x.  C )  <  ( B  x.  C
)  \/  ( A  x.  C )  =  ( B  x.  C
) ) ) )
19183adant3r 1215 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  x.  C )  <_  ( B  x.  C )  <->  ( ( A  x.  C
)  <  ( B  x.  C )  \/  ( A  x.  C )  =  ( B  x.  C ) ) ) )
2011, 13, 193bitr4d 285 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  ( A  x.  C )  <_  ( B  x.  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2600   class class class wbr 4285  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274    x. cmul 9279    < clt 9410    <_ cle 9411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-op 3877  df-uni 4085  df-br 4286  df-opab 4344  df-mpt 4345  df-id 4628  df-po 4633  df-so 4634  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590
This theorem is referenced by:  lemul2  10174  lemul1a  10175  lediv23  10216  lemul1i  10247  ledivp1i  10250  lemul1d  11058  xlemul1a  11243  iccdil  11415  expgt1  11894  sqlecan  11964  facubnd  12068  sqrlem2  12725  sqrlem6  12729  eirrlem  13478  mbfi1fseqlem3  21164  mbfi1fseqlem4  21165  mbfi1fseqlem5  21166  itg2monolem3  21199  atans2  22295  log2tlbnd  22309  fsumfldivdiaglem  22498  chtublem  22519  bposlem2  22593  bposlem5  22596  selberglem2  22764  pntpbnd1a  22803  pntpbnd2  22805  ostth2lem3  22853  htthlem  24264  cnlnadjlem7  25422  bfplem1  28664  jm2.24nn  29245  jm3.1lem2  29310  stoweidlem14  29752  stoweidlem26  29764  stoweidlem34  29772
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