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Theorem lemul1 10202
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 10200 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )
2 recn 9393 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
3 recn 9393 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
4 recn 9393 . . . . . . 7  |-  ( C  e.  RR  ->  C  e.  CC )
54adantr 465 . . . . . 6  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  e.  CC )
6 gt0ne0 9825 . . . . . 6  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
75, 6jca 532 . . . . 5  |-  ( ( C  e.  RR  /\  0  <  C )  -> 
( C  e.  CC  /\  C  =/=  0 ) )
8 mulcan2 9995 . . . . 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 9482 . . 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 9388 . . . . 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 9388 . . . . 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 9538 . . 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 2620   class class class wbr 4313  (class class class)co 6112   CCcc 9301   RRcr 9302   0cc0 9303    x. cmul 9308    < clt 9439    <_ cle 9440
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 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-po 4662  df-so 4663  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619
This theorem is referenced by:  lemul2  10203  lemul1a  10204  lediv23  10245  lemul1i  10276  ledivp1i  10279  lemul1d  11087  xlemul1a  11272  iccdil  11444  expgt1  11923  sqlecan  11993  facubnd  12097  sqrlem2  12754  sqrlem6  12758  eirrlem  13507  mbfi1fseqlem3  21217  mbfi1fseqlem4  21218  mbfi1fseqlem5  21219  itg2monolem3  21252  atans2  22348  log2tlbnd  22362  fsumfldivdiaglem  22551  chtublem  22572  bposlem2  22646  bposlem5  22649  selberglem2  22817  pntpbnd1a  22856  pntpbnd2  22858  ostth2lem3  22906  htthlem  24341  cnlnadjlem7  25499  bfplem1  28747  jm2.24nn  29328  jm3.1lem2  29393  stoweidlem14  29835  stoweidlem26  29847  stoweidlem34  29855
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