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Theorem lemul1a 9610
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 8838 . . . . . . 7  |-  0  e.  RR
2 leloe 8908 . . . . . . 7  |-  ( ( 0  e.  RR  /\  C  e.  RR )  ->  ( 0  <_  C  <->  ( 0  <  C  \/  0  =  C )
) )
31, 2mpan 651 . . . . . 6  |-  ( C  e.  RR  ->  (
0  <_  C  <->  ( 0  <  C  \/  0  =  C ) ) )
43pm5.32i 618 . . . . 5  |-  ( ( C  e.  RR  /\  0  <_  C )  <->  ( C  e.  RR  /\  ( 0  <  C  \/  0  =  C ) ) )
5 lemul1 9608 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  ( A  x.  C )  <_  ( B  x.  C ) ) )
65biimpd 198 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  ->  ( A  x.  C
)  <_  ( B  x.  C ) ) )
763expia 1153 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( C  e.  RR  /\  0  < 
C )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C
) ) ) )
87com12 27 . . . . . 6  |-  ( ( C  e.  RR  /\  0  <  C )  -> 
( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C
) ) ) )
91leidi 9307 . . . . . . . . . 10  |-  0  <_  0
10 recn 8827 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  A  e.  CC )
1110mul01d 9011 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  x.  0 )  =  0 )
12 recn 8827 . . . . . . . . . . . 12  |-  ( B  e.  RR  ->  B  e.  CC )
1312mul01d 9011 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  ( B  x.  0 )  =  0 )
1411, 13breqan12d 4038 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  0 )  <_  ( B  x.  0 )  <->  0  <_  0 ) )
159, 14mpbiri 224 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  0 )  <_  ( B  x.  0 ) )
16 oveq2 5866 . . . . . . . . . 10  |-  ( 0  =  C  ->  ( A  x.  0 )  =  ( A  x.  C ) )
17 oveq2 5866 . . . . . . . . . 10  |-  ( 0  =  C  ->  ( B  x.  0 )  =  ( B  x.  C ) )
1816, 17breq12d 4036 . . . . . . . . 9  |-  ( 0  =  C  ->  (
( A  x.  0 )  <_  ( B  x.  0 )  <->  ( A  x.  C )  <_  ( B  x.  C )
) )
1915, 18syl5ib 210 . . . . . . . 8  |-  ( 0  =  C  ->  (
( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  C )  <_  ( B  x.  C )
) )
2019a1dd 42 . . . . . . 7  |-  ( 0  =  C  ->  (
( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C )
) ) )
2120adantl 452 . . . . . 6  |-  ( ( C  e.  RR  /\  0  =  C )  ->  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C
) ) ) )
228, 21jaodan 760 . . . . 5  |-  ( ( C  e.  RR  /\  ( 0  <  C  \/  0  =  C
) )  ->  (
( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C )
) ) )
234, 22sylbi 187 . . . 4  |-  ( ( C  e.  RR  /\  0  <_  C )  -> 
( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C
) ) ) )
2423com12 27 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( C  e.  RR  /\  0  <_  C )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C
) ) ) )
25243impia 1148 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( A  <_  B  ->  ( A  x.  C
)  <_  ( B  x.  C ) ) )
2625imp 418 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023  (class class class)co 5858   RRcr 8736   0cc0 8737    x. cmul 8742    < clt 8867    <_ cle 8868
This theorem is referenced by:  lemul2a  9611  ltmul12a  9612  lemul12b  9613  lt2msq1  9639  lemul1ad  9696  faclbnd4lem1  11306  facavg  11314  mulcn2  12069  o1fsum  12271  eftlub  12389  bddmulibl  19193  cxpaddlelem  20091  dchrmusum2  20643  nmoub3i  21351  siilem1  21429  ubthlem3  21451  bcs2  21761  cnlnadjlem2  22648  leopnmid  22718  axcontlem7  24598  rrntotbnd  26560  jm2.17a  27047  fmul01lt1lem2  27715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040
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