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Theorem lt2mul2div 8630
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 520 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  C  e.  RR )
21recnd 7787 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  C  e.  CC )
3 simprrl 528 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  D  e.  RR )
43recnd 7787 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  D  e.  CC )
52, 4mulcomd 7780 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( C  x.  D )  =  ( D  x.  C ) )
65oveq1d 5782 . . . 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 524 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  B  e.  RR )
87recnd 7787 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  B  e.  CC )
9 simplrr 525 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  0  <  B
)
107, 9gt0ap0d 8384 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  B #  0 )
114, 2, 8, 10divassapd 8579 . . . 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 2170 . . 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 3936 . 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 518 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  A  e.  RR )
151, 3remulcld 7789 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( C  x.  D )  e.  RR )
16 simplr 519 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( B  e.  RR  /\  0  < 
B ) )
17 ltmuldiv 8625 . . 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 1216 . 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 8587 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( C  /  B )  e.  RR )
20 simprr 521 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( D  e.  RR  /\  0  < 
D ) )
21 ltdivmul 8627 . . 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 1216 . 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 219 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 103    <-> wb 104    e. wcel 1480   class class class wbr 3924  (class class class)co 5767   RRcr 7612   0cc0 7613    x. cmul 7618    < clt 7793    / cdiv 8425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426
This theorem is referenced by:  lt2mul2divd  9545
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