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Theorem lt2mul2div 10422
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 recn 9580 . . . . . . . . 9  |-  ( C  e.  RR  ->  C  e.  CC )
2 recn 9580 . . . . . . . . 9  |-  ( D  e.  RR  ->  D  e.  CC )
3 mulcom 9576 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  x.  D
)  =  ( D  x.  C ) )
41, 2, 3syl2an 477 . . . . . . . 8  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  x.  D
)  =  ( D  x.  C ) )
54oveq1d 6292 . . . . . . 7  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( ( C  x.  D )  /  B
)  =  ( ( D  x.  C )  /  B ) )
65adantl 466 . . . . . 6  |-  ( ( ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( C  x.  D )  /  B
)  =  ( ( D  x.  C )  /  B ) )
72ad2antll 728 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  CC )
81ad2antrl 727 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  CC )
9 recn 9580 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  e.  CC )
109adantr 465 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  e.  CC )
11 gt0ne0 10018 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  =/=  0 )
1210, 11jca 532 . . . . . . . 8  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
1312adantr 465 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
14 divass 10226 . . . . . . 7  |-  ( ( D  e.  CC  /\  C  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( D  x.  C )  /  B
)  =  ( D  x.  ( C  /  B ) ) )
157, 8, 13, 14syl3anc 1227 . . . . . 6  |-  ( ( ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( D  x.  C )  /  B
)  =  ( D  x.  ( C  /  B ) ) )
166, 15eqtrd 2482 . . . . 5  |-  ( ( ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( C  x.  D )  /  B
)  =  ( D  x.  ( C  /  B ) ) )
1716adantrrr 724 . . . 4  |-  ( ( ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( C  x.  D )  /  B )  =  ( D  x.  ( C  /  B ) ) )
1817adantll 713 . . 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 ) ) )
1918breq2d 4445 . 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 ) ) ) )
20 simpll 753 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  A  e.  RR )
21 remulcl 9575 . . . . 5  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  x.  D
)  e.  RR )
2221adantrr 716 . . . 4  |-  ( ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) )  ->  ( C  x.  D )  e.  RR )
2322adantl 466 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( C  x.  D )  e.  RR )
24 simplr 754 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( B  e.  RR  /\  0  < 
B ) )
25 ltmuldiv 10416 . . 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 ) ) )
2620, 23, 24, 25syl3anc 1227 . 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 ) ) )
27 simpl 457 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  e.  RR )
2827, 11jca 532 . . . . . 6  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( B  e.  RR  /\  B  =/=  0 ) )
29 redivcl 10264 . . . . . . 7  |-  ( ( C  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( C  /  B )  e.  RR )
30293expb 1196 . . . . . 6  |-  ( ( C  e.  RR  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  ( C  /  B )  e.  RR )
3128, 30sylan2 474 . . . . 5  |-  ( ( C  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( C  /  B )  e.  RR )
3231ancoms 453 . . . 4  |-  ( ( ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  ( C  /  B )  e.  RR )
3332ad2ant2lr 747 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( C  /  B )  e.  RR )
34 simprr 756 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( D  e.  RR  /\  0  < 
D ) )
35 ltdivmul 10418 . . 3  |-  ( ( A  e.  RR  /\  ( C  /  B
)  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) )  ->  ( ( A  /  D )  < 
( C  /  B
)  <->  A  <  ( D  x.  ( C  /  B ) ) ) )
3620, 33, 34, 35syl3anc 1227 . 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 ) ) ) )
3719, 26, 363bitr4d 285 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   class class class wbr 4433  (class class class)co 6277   CCcc 9488   RRcr 9489   0cc0 9490    x. cmul 9495    < clt 9626    / cdiv 10207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-po 4786  df-so 4787  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208
This theorem is referenced by:  lt2mul2divd  11318  efcllem  13686  icopnfhmeo  21309  nmoleub2lem3  21464  dvcvx  22287  log2ub  23145  chebbnd1lem3  23521  subfaclim  28498
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