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Theorem ltmul2 8317
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 8069 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )
2 recn 7475 . . . 4  |-  ( C  e.  RR  ->  C  e.  CC )
3 recn 7475 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
4 mulcom 7471 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  =  ( C  x.  A ) )
53, 4sylan 277 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  CC )  ->  ( A  x.  C
)  =  ( C  x.  A ) )
653adant2 962 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  CC )  ->  ( A  x.  C )  =  ( C  x.  A ) )
7 recn 7475 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  CC )
8 mulcom 7471 . . . . . . 7  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
97, 8sylan 277 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  CC )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
1093adant1 961 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  CC )  ->  ( B  x.  C )  =  ( C  x.  B ) )
116, 10breq12d 3858 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  CC )  ->  (
( A  x.  C
)  <  ( B  x.  C )  <->  ( C  x.  A )  <  ( C  x.  B )
) )
122, 11syl3an3 1209 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
)  <  ( B  x.  C )  <->  ( C  x.  A )  <  ( C  x.  B )
) )
13123adant3r 1171 . 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 186 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    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   class class class wbr 3845  (class class class)co 5652   CCcc 7348   RRcr 7349   0cc0 7350    x. cmul 7355    < clt 7522
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltadd 7461  ax-pre-mulgt0 7462
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7524  df-mnf 7525  df-ltxr 7527  df-sub 7655  df-neg 7656
This theorem is referenced by:  ltmul12a  8321  mulgt1  8324  ltmulgt11  8325  lt2msq1  8346  ltdiv2  8348  ltmul2i  8384  ltmul2d  9216  ef01bndlem  11047  cos01gt0  11053  sin4lt0  11057
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