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Theorem lemul1a 10183
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 9386 . . . . . . 7  |-  0  e.  RR
2 leloe 9461 . . . . . . 7  |-  ( ( 0  e.  RR  /\  C  e.  RR )  ->  ( 0  <_  C  <->  ( 0  <  C  \/  0  =  C )
) )
31, 2mpan 670 . . . . . 6  |-  ( C  e.  RR  ->  (
0  <_  C  <->  ( 0  <  C  \/  0  =  C ) ) )
43pm5.32i 637 . . . . 5  |-  ( ( C  e.  RR  /\  0  <_  C )  <->  ( C  e.  RR  /\  ( 0  <  C  \/  0  =  C ) ) )
5 lemul1 10181 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  ( A  x.  C )  <_  ( B  x.  C ) ) )
65biimpd 207 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  ->  ( A  x.  C
)  <_  ( B  x.  C ) ) )
763expia 1189 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( C  e.  RR  /\  0  < 
C )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C
) ) ) )
87com12 31 . . . . . 6  |-  ( ( C  e.  RR  /\  0  <  C )  -> 
( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C
) ) ) )
91leidi 9874 . . . . . . . . . 10  |-  0  <_  0
10 recn 9372 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  A  e.  CC )
1110mul01d 9568 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  x.  0 )  =  0 )
12 recn 9372 . . . . . . . . . . . 12  |-  ( B  e.  RR  ->  B  e.  CC )
1312mul01d 9568 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  ( B  x.  0 )  =  0 )
1411, 13breqan12d 4307 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  0 )  <_  ( B  x.  0 )  <->  0  <_  0 ) )
159, 14mpbiri 233 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  0 )  <_  ( B  x.  0 ) )
16 oveq2 6099 . . . . . . . . . 10  |-  ( 0  =  C  ->  ( A  x.  0 )  =  ( A  x.  C ) )
17 oveq2 6099 . . . . . . . . . 10  |-  ( 0  =  C  ->  ( B  x.  0 )  =  ( B  x.  C ) )
1816, 17breq12d 4305 . . . . . . . . 9  |-  ( 0  =  C  ->  (
( A  x.  0 )  <_  ( B  x.  0 )  <->  ( A  x.  C )  <_  ( B  x.  C )
) )
1915, 18syl5ib 219 . . . . . . . 8  |-  ( 0  =  C  ->  (
( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  C )  <_  ( B  x.  C )
) )
2019a1dd 46 . . . . . . 7  |-  ( 0  =  C  ->  (
( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C )
) ) )
2120adantl 466 . . . . . 6  |-  ( ( C  e.  RR  /\  0  =  C )  ->  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C
) ) ) )
228, 21jaodan 783 . . . . 5  |-  ( ( C  e.  RR  /\  ( 0  <  C  \/  0  =  C
) )  ->  (
( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C )
) ) )
234, 22sylbi 195 . . . 4  |-  ( ( C  e.  RR  /\  0  <_  C )  -> 
( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C
) ) ) )
2423com12 31 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( C  e.  RR  /\  0  <_  C )  ->  ( A  <_  B  ->  ( A  x.  C )  <_  ( B  x.  C
) ) ) )
25243impia 1184 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( A  <_  B  ->  ( A  x.  C
)  <_  ( B  x.  C ) ) )
2625imp 429 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 965    = wceq 1369    e. wcel 1756   class class class wbr 4292  (class class class)co 6091   RRcr 9281   0cc0 9282    x. cmul 9287    < clt 9418    <_ cle 9419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598
This theorem is referenced by:  lemul2a  10184  ltmul12a  10185  lemul12b  10186  lt2msq1  10215  lemul1ad  10272  faclbnd4lem1  12069  facavg  12077  mulcn2  13073  o1fsum  13276  eftlub  13393  bddmulibl  21316  cxpaddlelem  22189  dchrmusum2  22743  axcontlem7  23216  nmoub3i  24173  siilem1  24251  ubthlem3  24273  bcs2  24584  cnlnadjlem2  25472  leopnmid  25542  eulerpartlemgc  26745  rrntotbnd  28735  jm2.17a  29303
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