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Theorem ltmul1 9539
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.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
ltmul1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )

Proof of Theorem ltmul1
StepHypRef Expression
1 ltmul1a 9538 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  A  < 
B )  ->  ( A  x.  C )  <  ( B  x.  C
) )
21ex 425 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  ->  ( A  x.  C
)  <  ( B  x.  C ) ) )
3 oveq1 5764 . . . . . 6  |-  ( A  =  B  ->  ( A  x.  C )  =  ( B  x.  C ) )
43a1i 12 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  =  B  ->  ( A  x.  C )  =  ( B  x.  C ) ) )
5 ltmul1a 9538 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  A  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  B  < 
A )  ->  ( B  x.  C )  <  ( A  x.  C
) )
65ex 425 . . . . . 6  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  <  A  ->  ( B  x.  C
)  <  ( A  x.  C ) ) )
763com12 1160 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  <  A  ->  ( B  x.  C
)  <  ( A  x.  C ) ) )
84, 7orim12d 814 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  =  B  \/  B  < 
A )  ->  (
( A  x.  C
)  =  ( B  x.  C )  \/  ( B  x.  C
)  <  ( A  x.  C ) ) ) )
98con3d 127 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( -.  ( ( A  x.  C )  =  ( B  x.  C )  \/  ( B  x.  C )  <  ( A  x.  C
) )  ->  -.  ( A  =  B  \/  B  <  A ) ) )
10 simp1 960 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  A  e.  RR )
11 simp3l 988 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  RR )
1210, 11remulcld 8796 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  x.  C
)  e.  RR )
13 simp2 961 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  B  e.  RR )
1413, 11remulcld 8796 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  x.  C
)  e.  RR )
1512, 14lttrid 8890 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  x.  C )  <  ( B  x.  C )  <->  -.  ( ( A  x.  C )  =  ( B  x.  C )  \/  ( B  x.  C )  <  ( A  x.  C )
) ) )
1610, 13lttrid 8890 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  -.  ( A  =  B  \/  B  <  A
) ) )
179, 15, 163imtr4d 261 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  x.  C )  <  ( B  x.  C )  ->  A  <  B ) )
182, 17impbid 185 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:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3963  (class class class)co 5757   RRcr 8669   0cc0 8670    x. cmul 8675    < clt 8800
This theorem is referenced by:  ltmul2  9540  lemul1  9541  ltdiv1  9553  ltdiv23  9580  recp1lt1  9587  ltmul1i  9608  ltdivp1i  9616  ltmul1d  10359  expmulnbnd  11164  discr1  11168  mertenslem1  12267  qnumgt0  12748  4sqlem12  12930  pgpfaclem2  15244  mbfi1fseqlem4  19000  itg2monolem1  19032  dgrcolem2  19582  tangtx  19800  ftalem1  20237  basellem4  20248  lgsquadlem1  20520  lgsquadlem2  20521  pntpbnd1  20662  ostth2lem1  20694  nn0prpwlem  25570  pellexlem2  26247  stoweidlem11  27060  stoweidlem13  27062  stoweidlem26  27075  stoweidlem34  27083  stoweidlem59  27108
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-po 4251  df-so 4252  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-iota 6190  df-riota 6237  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-ltxr 8805  df-sub 8972  df-neg 8973
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