MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lemul1a Unicode version

Theorem lemul1a 9796
Description: Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by NM, 21-Feb-2005.)
Assertion
Ref Expression
lemul1a  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  /\  A  <_  B )  ->  ( A  x.  C )  <_  ( B  x.  C )
)

Proof of Theorem lemul1a
StepHypRef Expression
1 0re 9024 . . . . . . 7  |-  0  e.  RR
2 leloe 9094 . . . . . . 7  |-  ( ( 0  e.  RR  /\  C  e.  RR )  ->  ( 0  <_  C  <->  ( 0  <  C  \/  0  =  C )
) )
31, 2mpan 652 . . . . . 6  |-  ( C  e.  RR  ->  (
0  <_  C  <->  ( 0  <  C  \/  0  =  C ) ) )
43pm5.32i 619 . . . . 5  |-  ( ( C  e.  RR  /\  0  <_  C )  <->  ( C  e.  RR  /\  ( 0  <  C  \/  0  =  C ) ) )
5 lemul1 9794 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  ( A  x.  C )  <_  ( B  x.  C ) ) )
65biimpd 199 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  ->  ( A  x.  C
)  <_  ( B  x.  C ) ) )
763expia 1155 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( C  e.  RR  /\  0  < 
C )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C
) ) ) )
87com12 29 . . . . . 6  |-  ( ( C  e.  RR  /\  0  <  C )  -> 
( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C
) ) ) )
91leidi 9493 . . . . . . . . . 10  |-  0  <_  0
10 recn 9013 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  A  e.  CC )
1110mul01d 9197 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  x.  0 )  =  0 )
12 recn 9013 . . . . . . . . . . . 12  |-  ( B  e.  RR  ->  B  e.  CC )
1312mul01d 9197 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  ( B  x.  0 )  =  0 )
1411, 13breqan12d 4168 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  0 )  <_  ( B  x.  0 )  <->  0  <_  0 ) )
159, 14mpbiri 225 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  0 )  <_  ( B  x.  0 ) )
16 oveq2 6028 . . . . . . . . . 10  |-  ( 0  =  C  ->  ( A  x.  0 )  =  ( A  x.  C ) )
17 oveq2 6028 . . . . . . . . . 10  |-  ( 0  =  C  ->  ( B  x.  0 )  =  ( B  x.  C ) )
1816, 17breq12d 4166 . . . . . . . . 9  |-  ( 0  =  C  ->  (
( A  x.  0 )  <_  ( B  x.  0 )  <->  ( A  x.  C )  <_  ( B  x.  C )
) )
1915, 18syl5ib 211 . . . . . . . 8  |-  ( 0  =  C  ->  (
( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  C )  <_  ( B  x.  C )
) )
2019a1dd 44 . . . . . . 7  |-  ( 0  =  C  ->  (
( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C )
) ) )
2120adantl 453 . . . . . 6  |-  ( ( C  e.  RR  /\  0  =  C )  ->  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C
) ) ) )
228, 21jaodan 761 . . . . 5  |-  ( ( C  e.  RR  /\  ( 0  <  C  \/  0  =  C
) )  ->  (
( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C )
) ) )
234, 22sylbi 188 . . . 4  |-  ( ( C  e.  RR  /\  0  <_  C )  -> 
( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C
) ) ) )
2423com12 29 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( C  e.  RR  /\  0  <_  C )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C
) ) ) )
25243impia 1150 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( A  <_  B  ->  ( A  x.  C
)  <_  ( B  x.  C ) ) )
2625imp 419 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  /\  A  <_  B )  ->  ( A  x.  C )  <_  ( B  x.  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4153  (class class class)co 6020   RRcr 8922   0cc0 8923    x. cmul 8928    < clt 9053    <_ cle 9054
This theorem is referenced by:  lemul2a  9797  ltmul12a  9798  lemul12b  9799  lt2msq1  9825  lemul1ad  9882  faclbnd4lem1  11511  facavg  11519  mulcn2  12316  o1fsum  12519  eftlub  12637  bddmulibl  19597  cxpaddlelem  20502  dchrmusum2  21055  nmoub3i  22122  siilem1  22200  ubthlem3  22222  bcs2  22532  cnlnadjlem2  23419  leopnmid  23489  axcontlem7  25623  rrntotbnd  26236  jm2.17a  26716
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226
  Copyright terms: Public domain W3C validator