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Theorem lt2mul2div 10410
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 9571 . . . . . . . . 9  |-  ( C  e.  RR  ->  C  e.  CC )
2 recn 9571 . . . . . . . . 9  |-  ( D  e.  RR  ->  D  e.  CC )
3 mulcom 9567 . . . . . . . . 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 6290 . . . . . . 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 9571 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  e.  CC )
109adantr 465 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  e.  CC )
11 gt0ne0 10006 . . . . . . . . 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 10214 . . . . . . 7  |-  ( ( D  e.  CC  /\  C  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( D  x.  C )  /  B
)  =  ( D  x.  ( C  /  B ) ) )
157, 8, 13, 14syl3anc 1223 . . . . . 6  |-  ( ( ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( D  x.  C )  /  B
)  =  ( D  x.  ( C  /  B ) ) )
166, 15eqtrd 2501 . . . . 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 4452 . 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 9566 . . . . 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 10404 . . 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 1223 . 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 10252 . . . . . . 7  |-  ( ( C  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( C  /  B )  e.  RR )
30293expb 1192 . . . . . 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 10406 . . 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 1223 . 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 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481    x. cmul 9486    < clt 9617    / cdiv 10195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196
This theorem is referenced by:  lt2mul2divd  11303  efcllem  13664  icopnfhmeo  21171  nmoleub2lem3  21326  dvcvx  22149  log2ub  23001  chebbnd1lem3  23377  subfaclim  28258
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