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Theorem ltdiv23 10427
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 457 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  e.  RR )
2 gt0ne0 10008 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  =/=  0 )
31, 2jca 532 . . . . . 6  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( B  e.  RR  /\  B  =/=  0 ) )
4 redivcl 10254 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A  /  B )  e.  RR )
543expb 1192 . . . . . 6  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  ( A  /  B )  e.  RR )
63, 5sylan2 474 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( A  /  B )  e.  RR )
763adant3 1011 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  ( A  /  B )  e.  RR )
8 simp3 993 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  C  e.  RR )
9 simp2 992 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  ( B  e.  RR  /\  0  < 
B ) )
10 ltmul1 10383 . . . 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 1223 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  ( ( A  /  B )  < 
C  <->  ( ( A  /  B )  x.  B )  <  ( C  x.  B )
) )
12113adant3r 1220 . 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 9573 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
1413adantr 465 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  A  e.  CC )
15 recn 9573 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
1615ad2antrl 727 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  B  e.  CC )
172adantl 466 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  B  =/=  0 )
1814, 16, 17divcan1d 10312 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  x.  B )  =  A )
19183adant3 1011 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( A  /  B )  x.  B
)  =  A )
2019breq1d 4452 . 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 9568 . . . . . . . 8  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  x.  B
)  e.  RR )
2221ancoms 453 . . . . . . 7  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  x.  B
)  e.  RR )
2322adantrr 716 . . . . . 6  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( C  x.  B )  e.  RR )
24233adant1 1009 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( C  x.  B
)  e.  RR )
25 ltdiv1 10397 . . . . 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 1270 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  ( C  x.  B )  <->  ( A  /  C )  <  ( ( C  x.  B )  /  C ) ) )
27 recn 9573 . . . . . . . . 9  |-  ( C  e.  RR  ->  C  e.  CC )
2827adantr 465 . . . . . . . 8  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  e.  CC )
29 gt0ne0 10008 . . . . . . . 8  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
3028, 29jca 532 . . . . . . 7  |-  ( ( C  e.  RR  /\  0  <  C )  -> 
( C  e.  CC  /\  C  =/=  0 ) )
31 divcan3 10222 . . . . . . . 8  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  (
( C  x.  B
)  /  C )  =  B )
32313expb 1192 . . . . . . 7  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( C  x.  B )  /  C )  =  B )
3315, 30, 32syl2an 477 . . . . . 6  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( C  x.  B )  /  C )  =  B )
34333adant1 1009 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( C  x.  B )  /  C
)  =  B )
3534breq2d 4454 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <  (
( C  x.  B
)  /  C )  <-> 
( A  /  C
)  <  B )
)
3626, 35bitrd 253 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  ( C  x.  B )  <->  ( A  /  C )  <  B ) )
37363adant2r 1218 . 2  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( A  <  ( C  x.  B )  <->  ( A  /  C )  <  B ) )
3812, 20, 373bitrd 279 1  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( A  /  B )  <  C  <->  ( A  /  C )  <  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657   class class class wbr 4442  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483    x. cmul 9488    < clt 9619    / cdiv 10197
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 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
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 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198
This theorem is referenced by:  ltdiv23i  10461  ltdiv23d  11303  divrcnv  13618  prmind2  14078  lebnumii  21196  bposlem2  23283  pntibndlem1  23497  stoweidlem7  31264
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