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Theorem lt2mul2div 8338
Description: 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)
Assertion
Ref Expression
lt2mul2div  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( A  x.  B )  < 
( C  x.  D
)  <->  ( A  /  D )  <  ( C  /  B ) ) )

Proof of Theorem lt2mul2div
StepHypRef Expression
1 simprl 498 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  C  e.  RR )
21recnd 7514 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  C  e.  CC )
3 simprrl 506 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  D  e.  RR )
43recnd 7514 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  D  e.  CC )
52, 4mulcomd 7507 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( C  x.  D )  =  ( D  x.  C ) )
65oveq1d 5667 . . . 4  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( C  x.  D )  /  B )  =  ( ( D  x.  C
)  /  B ) )
7 simplrl 502 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  B  e.  RR )
87recnd 7514 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  B  e.  CC )
9 simplrr 503 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  0  <  B
)
107, 9gt0ap0d 8103 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  B #  0 )
114, 2, 8, 10divassapd 8291 . . . 4  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( D  x.  C )  /  B )  =  ( D  x.  ( C  /  B ) ) )
126, 11eqtrd 2120 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( C  x.  D )  /  B )  =  ( D  x.  ( C  /  B ) ) )
1312breq2d 3857 . 2  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( A  < 
( ( C  x.  D )  /  B
)  <->  A  <  ( D  x.  ( C  /  B ) ) ) )
14 simpll 496 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  A  e.  RR )
151, 3remulcld 7516 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( C  x.  D )  e.  RR )
16 simplr 497 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( B  e.  RR  /\  0  < 
B ) )
17 ltmuldiv 8333 . . 3  |-  ( ( A  e.  RR  /\  ( C  x.  D
)  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  x.  B )  <  ( C  x.  D
)  <->  A  <  ( ( C  x.  D )  /  B ) ) )
1814, 15, 16, 17syl3anc 1174 . 2  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( A  x.  B )  < 
( C  x.  D
)  <->  A  <  ( ( C  x.  D )  /  B ) ) )
191, 7, 10redivclapd 8299 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( C  /  B )  e.  RR )
20 simprr 499 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( D  e.  RR  /\  0  < 
D ) )
21 ltdivmul 8335 . . 3  |-  ( ( A  e.  RR  /\  ( C  /  B
)  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) )  ->  ( ( A  /  D )  < 
( C  /  B
)  <->  A  <  ( D  x.  ( C  /  B ) ) ) )
2214, 19, 20, 21syl3anc 1174 . 2  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( A  /  D )  < 
( C  /  B
)  <->  A  <  ( D  x.  ( C  /  B ) ) ) )
2313, 18, 223bitr4d 218 1  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( A  x.  B )  < 
( C  x.  D
)  <->  ( A  /  D )  <  ( C  /  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1438   class class class wbr 3845  (class class class)co 5652   RRcr 7347   0cc0 7348    x. cmul 7353    < clt 7520    / cdiv 8137
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 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461
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-rmo 2367  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-po 4123  df-iso 4124  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 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138
This theorem is referenced by:  lt2mul2divd  9234
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