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Theorem ltdiv23 9893
Description: Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)
Assertion
Ref Expression
ltdiv23  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( A  /  B )  <  C  <->  ( A  /  C )  <  B ) )

Proof of Theorem ltdiv23
StepHypRef Expression
1 simpl 444 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  e.  RR )
2 gt0ne0 9485 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  =/=  0 )
31, 2jca 519 . . . . . 6  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( B  e.  RR  /\  B  =/=  0 ) )
4 redivcl 9725 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A  /  B )  e.  RR )
543expb 1154 . . . . . 6  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  ( A  /  B )  e.  RR )
63, 5sylan2 461 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( A  /  B )  e.  RR )
763adant3 977 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  ( A  /  B )  e.  RR )
8 simp3 959 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  C  e.  RR )
9 simp2 958 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  ( B  e.  RR  /\  0  < 
B ) )
10 ltmul1 9852 . . . 4  |-  ( ( ( A  /  B
)  e.  RR  /\  C  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( A  /  B )  <  C  <->  ( ( A  /  B
)  x.  B )  <  ( C  x.  B ) ) )
117, 8, 9, 10syl3anc 1184 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  ( ( A  /  B )  < 
C  <->  ( ( A  /  B )  x.  B )  <  ( C  x.  B )
) )
12113adant3r 1181 . 2  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( A  /  B )  <  C  <->  ( ( A  /  B
)  x.  B )  <  ( C  x.  B ) ) )
13 recn 9072 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
1413adantr 452 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  A  e.  CC )
15 recn 9072 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
1615ad2antrl 709 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  B  e.  CC )
172adantl 453 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  B  =/=  0 )
1814, 16, 17divcan1d 9783 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  x.  B )  =  A )
19183adant3 977 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( A  /  B )  x.  B
)  =  A )
2019breq1d 4214 . 2  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( ( A  /  B )  x.  B )  <  ( C  x.  B )  <->  A  <  ( C  x.  B ) ) )
21 remulcl 9067 . . . . . . . 8  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  x.  B
)  e.  RR )
2221ancoms 440 . . . . . . 7  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  x.  B
)  e.  RR )
2322adantrr 698 . . . . . 6  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( C  x.  B )  e.  RR )
24233adant1 975 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( C  x.  B
)  e.  RR )
25 ltdiv1 9866 . . . . 5  |-  ( ( A  e.  RR  /\  ( C  x.  B
)  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <  ( C  x.  B
)  <->  ( A  /  C )  <  (
( C  x.  B
)  /  C ) ) )
2624, 25syld3an2 1231 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  ( C  x.  B )  <->  ( A  /  C )  <  ( ( C  x.  B )  /  C ) ) )
27 recn 9072 . . . . . . . . 9  |-  ( C  e.  RR  ->  C  e.  CC )
2827adantr 452 . . . . . . . 8  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  e.  CC )
29 gt0ne0 9485 . . . . . . . 8  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
3028, 29jca 519 . . . . . . 7  |-  ( ( C  e.  RR  /\  0  <  C )  -> 
( C  e.  CC  /\  C  =/=  0 ) )
31 divcan3 9694 . . . . . . . 8  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  (
( C  x.  B
)  /  C )  =  B )
32313expb 1154 . . . . . . 7  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( C  x.  B )  /  C )  =  B )
3315, 30, 32syl2an 464 . . . . . 6  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( C  x.  B )  /  C )  =  B )
34333adant1 975 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( C  x.  B )  /  C
)  =  B )
3534breq2d 4216 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <  (
( C  x.  B
)  /  C )  <-> 
( A  /  C
)  <  B )
)
3626, 35bitrd 245 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  ( C  x.  B )  <->  ( A  /  C )  <  B ) )
37363adant2r 1179 . 2  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( A  <  ( C  x.  B )  <->  ( A  /  C )  <  B ) )
3812, 20, 373bitrd 271 1  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( A  /  B )  <  C  <->  ( A  /  C )  <  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982    x. cmul 8987    < clt 9112    / cdiv 9669
This theorem is referenced by:  ltdiv23i  9927  ltdiv23d  10696  divrcnv  12624  prmind2  13082  lebnumii  18983  bposlem2  21061  pntibndlem1  21275  stoweidlem7  27723
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670
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