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Theorem ltmul2 9786
Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.)
Assertion
Ref Expression
ltmul2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( C  x.  A )  <  ( C  x.  B ) ) )

Proof of Theorem ltmul2
StepHypRef Expression
1 ltmul1 9785 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )
2 recn 9006 . . . 4  |-  ( C  e.  RR  ->  C  e.  CC )
3 recn 9006 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
4 mulcom 9002 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  =  ( C  x.  A ) )
53, 4sylan 458 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  CC )  ->  ( A  x.  C
)  =  ( C  x.  A ) )
653adant2 976 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  CC )  ->  ( A  x.  C )  =  ( C  x.  A ) )
7 recn 9006 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  CC )
8 mulcom 9002 . . . . . . 7  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
97, 8sylan 458 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  CC )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
1093adant1 975 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  CC )  ->  ( B  x.  C )  =  ( C  x.  B ) )
116, 10breq12d 4159 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  CC )  ->  (
( A  x.  C
)  <  ( B  x.  C )  <->  ( C  x.  A )  <  ( C  x.  B )
) )
122, 11syl3an3 1219 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
)  <  ( B  x.  C )  <->  ( C  x.  A )  <  ( C  x.  B )
) )
13123adant3r 1181 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  x.  C )  <  ( B  x.  C )  <->  ( C  x.  A )  <  ( C  x.  B ) ) )
141, 13bitrd 245 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    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4146  (class class class)co 6013   CCcc 8914   RRcr 8915   0cc0 8916    x. cmul 8921    < clt 9046
This theorem is referenced by:  ltmul12a  9791  mulgt1  9794  ltmulgt11  9795  lt2msq1  9818  ltdiv2  9820  ltmul2i  9857  ltmul2d  10611  ef01bndlem  12705  cos01gt0  12712  sin4lt0  12716  iserodd  13129  pockthg  13194  prmreclem1  13204  prmreclem5  13208  blcvx  18693  dvcvx  19764  itgulm  20184  tangtx  20273  chtub  20856  bposlem1  20928  bposlem2  20929  bposlem7  20934  lgsdilem  20966  lgsquadlem1  20998  lgsquadlem2  20999  chebbnd1lem3  21025  chto1ub  21030  pntlemb  21151  irrapxlem1  26569  irrapxlem2  26570  irrapxlem5  26573  pellexlem2  26577  rmspecsqrnq  26653  stoweidlem11  27421  stoweidlem26  27436
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 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-ltxr 9051  df-sub 9218  df-neg 9219
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