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Theorem lemul2a 8845
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 8844 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  /\  A  <_  B )  ->  ( A  x.  C )  <_  ( B  x.  C )
)
2 recn 7973 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
3 recn 7973 . . . . . 6  |-  ( C  e.  RR  ->  C  e.  CC )
4 mulcom 7969 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  =  ( C  x.  A ) )
52, 3, 4syl2an 289 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  x.  C
)  =  ( C  x.  A ) )
65adantrr 479 . . . 4  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  ->  ( A  x.  C )  =  ( C  x.  A ) )
763adant2 1018 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( A  x.  C
)  =  ( C  x.  A ) )
87adantr 276 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  /\  A  <_  B )  ->  ( A  x.  C )  =  ( C  x.  A ) )
9 recn 7973 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
10 mulcom 7969 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
119, 3, 10syl2an 289 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
1211adantrr 479 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  ->  ( B  x.  C )  =  ( C  x.  B ) )
13123adant1 1017 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( B  x.  C
)  =  ( C  x.  B ) )
1413adantr 276 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  /\  A  <_  B )  ->  ( B  x.  C )  =  ( C  x.  B ) )
151, 8, 143brtr3d 4049 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 104    /\ w3a 980    = wceq 1364    e. wcel 2160   class class class wbr 4018  (class class class)co 5895   CCcc 7838   RRcr 7839   0cc0 7840    x. cmul 7845    <_ cle 8022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568
This theorem is referenced by:  lemul12b  8847  ledivp1  8889  lemul2ad  8926  facavg  10757  mulcn2  11351  cvgratnnlemnexp  11563  cvgratnnlemmn  11564  mertenslemi1  11574
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