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Theorem lemul1a 10406
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 9606 . . . . . . 7  |-  0  e.  RR
2 leloe 9681 . . . . . . 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 10404 . . . . . . . . 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 1198 . . . . . . 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 10097 . . . . . . . . . 10  |-  0  <_  0
10 recn 9592 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  A  e.  CC )
1110mul01d 9788 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  x.  0 )  =  0 )
12 recn 9592 . . . . . . . . . . . 12  |-  ( B  e.  RR  ->  B  e.  CC )
1312mul01d 9788 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  ( B  x.  0 )  =  0 )
1411, 13breqan12d 4467 . . . . . . . . . 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 6302 . . . . . . . . . 10  |-  ( 0  =  C  ->  ( A  x.  0 )  =  ( A  x.  C ) )
17 oveq2 6302 . . . . . . . . . 10  |-  ( 0  =  C  ->  ( B  x.  0 )  =  ( B  x.  C ) )
1816, 17breq12d 4465 . . . . . . . . 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 1193 . 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 973    = wceq 1379    e. wcel 1767   class class class wbr 4452  (class class class)co 6294   RRcr 9501   0cc0 9502    x. cmul 9507    < clt 9638    <_ cle 9639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-po 4805  df-so 4806  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818
This theorem is referenced by:  lemul2a  10407  ltmul12a  10408  lemul12b  10409  lt2msq1  10438  lemul1ad  10495  faclbnd4lem1  12349  facavg  12357  mulcn2  13393  o1fsum  13602  eftlub  13717  bddmulibl  22090  cxpaddlelem  22968  dchrmusum2  23522  axcontlem7  24064  nmoub3i  25479  siilem1  25557  ubthlem3  25579  bcs2  25890  cnlnadjlem2  26778  leopnmid  26848  eulerpartlemgc  28094  rrntotbnd  30227  jm2.17a  30794
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