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Theorem lemul1 10395
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 10393 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )
2 recn 9580 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
3 recn 9580 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
4 recn 9580 . . . . . . 7  |-  ( C  e.  RR  ->  C  e.  CC )
54adantr 465 . . . . . 6  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  e.  CC )
6 gt0ne0 10018 . . . . . 6  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
75, 6jca 532 . . . . 5  |-  ( ( C  e.  RR  /\  0  <  C )  -> 
( C  e.  CC  /\  C  =/=  0 ) )
8 mulcan2 10188 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  x.  C )  =  ( B  x.  C )  <-> 
A  =  B ) )
92, 3, 7, 8syl3an 1269 . . . 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 9669 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  ( A  <  B  \/  A  =  B )
) )
13123adant3 1015 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  ( A  <  B  \/  A  =  B )
) )
14 remulcl 9575 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  x.  C
)  e.  RR )
15143adant2 1014 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  x.  C )  e.  RR )
16 remulcl 9575 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  x.  C
)  e.  RR )
17163adant1 1013 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  x.  C )  e.  RR )
1815, 17leloed 9726 . . 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 1224 . 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 972    = wceq 1381    e. wcel 1802    =/= wne 2636   class class class wbr 4433  (class class class)co 6277   CCcc 9488   RRcr 9489   0cc0 9490    x. cmul 9495    < clt 9626    <_ cle 9627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-po 4786  df-so 4787  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808
This theorem is referenced by:  lemul2  10396  lemul1a  10397  lediv23  10438  lemul1i  10469  ledivp1i  10472  lemul1d  11299  xlemul1a  11484  iccdil  11662  expgt1  12178  sqlecan  12248  facubnd  12352  sqrlem2  13051  sqrlem6  13055  eirrlem  13809  mbfi1fseqlem3  21990  mbfi1fseqlem4  21991  mbfi1fseqlem5  21992  itg2monolem3  22025  atans2  23127  log2tlbnd  23141  fsumfldivdiaglem  23330  chtublem  23351  bposlem2  23425  bposlem5  23428  selberglem2  23596  pntpbnd1a  23635  pntpbnd2  23637  ostth2lem3  23685  htthlem  25699  cnlnadjlem7  26857  bfplem1  30286  jm2.24nn  30865  jm3.1lem2  30928  stoweidlem14  31681  stoweidlem26  31693  stoweidlem34  31701
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