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Theorem lediv23 10322
Description: Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)
Assertion
Ref Expression
lediv23  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( A  /  B )  <_  C  <->  ( A  /  C )  <_  B ) )

Proof of Theorem lediv23
StepHypRef Expression
1 simpl 457 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  e.  RR )
2 gt0ne0 9902 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  =/=  0 )
31, 2jca 532 . . . . . 6  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( B  e.  RR  /\  B  =/=  0 ) )
4 redivcl 10148 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A  /  B )  e.  RR )
543expb 1189 . . . . . 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 1008 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  ( A  /  B )  e.  RR )
8 simp3 990 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  C  e.  RR )
9 simp2 989 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  ( B  e.  RR  /\  0  < 
B ) )
10 lemul1 10279 . . . 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 1219 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  ( ( A  /  B )  <_  C 
<->  ( ( A  /  B )  x.  B
)  <_  ( C  x.  B ) ) )
12113adant3r 1216 . 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 9470 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
1413adantr 465 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  A  e.  CC )
15 recn 9470 . . . . . 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 10206 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  x.  B )  =  A )
19183adant3 1008 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( A  /  B )  x.  B
)  =  A )
2019breq1d 4397 . 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 9465 . . . . . . . 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 1006 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( C  x.  B
)  e.  RR )
25 lediv1 10292 . . . . 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 1266 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  ( C  x.  B )  <->  ( A  /  C )  <_  ( ( C  x.  B )  /  C ) ) )
27 recn 9470 . . . . . . . . 9  |-  ( C  e.  RR  ->  C  e.  CC )
2827adantr 465 . . . . . . . 8  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  e.  CC )
29 gt0ne0 9902 . . . . . . . 8  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
3028, 29jca 532 . . . . . . 7  |-  ( ( C  e.  RR  /\  0  <  C )  -> 
( C  e.  CC  /\  C  =/=  0 ) )
31 divcan3 10116 . . . . . . . 8  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  (
( C  x.  B
)  /  C )  =  B )
32313expb 1189 . . . . . . 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 1006 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( C  x.  B )  /  C
)  =  B )
3534breq2d 4399 . . . 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 1214 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2642   class class class wbr 4387  (class class class)co 6187   CCcc 9378   RRcr 9379   0cc0 9380    x. cmul 9385    < clt 9516    <_ cle 9517    / cdiv 10091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-po 4736  df-so 4737  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-div 10092
This theorem is referenced by:  lediv23d  11182  pntlemj  22965  minvecolem4  24413  stoweidlem36  29966
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