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Theorem lemul2a 9867
Description: Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
Assertion
Ref Expression
lemul2a  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  /\  A  <_  B )  ->  ( C  x.  A )  <_  ( C  x.  B )
)

Proof of Theorem lemul2a
StepHypRef Expression
1 lemul1a 9866 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  /\  A  <_  B )  ->  ( A  x.  C )  <_  ( B  x.  C )
)
2 recn 9082 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
3 recn 9082 . . . . . 6  |-  ( C  e.  RR  ->  C  e.  CC )
4 mulcom 9078 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  =  ( C  x.  A ) )
52, 3, 4syl2an 465 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  x.  C
)  =  ( C  x.  A ) )
65adantrr 699 . . . 4  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  ->  ( A  x.  C )  =  ( C  x.  A ) )
763adant2 977 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( A  x.  C
)  =  ( C  x.  A ) )
87adantr 453 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  /\  A  <_  B )  ->  ( A  x.  C )  =  ( C  x.  A ) )
9 recn 9082 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
10 mulcom 9078 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
119, 3, 10syl2an 465 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
1211adantrr 699 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  ->  ( B  x.  C )  =  ( C  x.  B ) )
13123adant1 976 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( B  x.  C
)  =  ( C  x.  B ) )
1413adantr 453 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  /\  A  <_  B )  ->  ( B  x.  C )  =  ( C  x.  B ) )
151, 8, 143brtr3d 4243 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  /\  A  <_  B )  ->  ( C  x.  A )  <_  ( C  x.  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992    x. cmul 8997    <_ cle 9123
This theorem is referenced by:  lemul12b  9869  ledivp1  9914  ledivp1i  9938  ltdivp1i  9939  lemul2ad  9953  supmul1  9975  facavg  11594  mulcn2  12391  cvgrat  12662  mertenslem1  12663  prmreclem3  13288  nmoco  18773  blcvx  18831  fsumharmonic  20852  bposlem1  21070  dchrvmasumiflem1  21197  nmoub3i  22276  htthlem  22422  nmopub2tALT  23414  nmfnleub2  23431  nmophmi  23536  nmopadjlem  23594  nmopcoadji  23606  lediv2aALT  25119  stoweidlem24  27751
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
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-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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296
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