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Theorem ledivdiv 9892
Description: Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.)
Assertion
Ref Expression
ledivdiv  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( A  /  B )  <_ 
( C  /  D
)  <->  ( D  /  C )  <_  ( B  /  A ) ) )

Proof of Theorem ledivdiv
StepHypRef Expression
1 simpl 444 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  e.  RR )
2 gt0ne0 9486 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  =/=  0 )
31, 2jca 519 . . . . . 6  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( B  e.  RR  /\  B  =/=  0 ) )
4 redivcl 9726 . . . . . . 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 )
76adantlr 696 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  /  B
)  e.  RR )
8 divgt0 9871 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  ( A  /  B ) )
97, 8jca 519 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( ( A  /  B )  e.  RR  /\  0  <  ( A  /  B ) ) )
10 simpl 444 . . . . . . 7  |-  ( ( D  e.  RR  /\  0  <  D )  ->  D  e.  RR )
11 gt0ne0 9486 . . . . . . 7  |-  ( ( D  e.  RR  /\  0  <  D )  ->  D  =/=  0 )
1210, 11jca 519 . . . . . 6  |-  ( ( D  e.  RR  /\  0  <  D )  -> 
( D  e.  RR  /\  D  =/=  0 ) )
13 redivcl 9726 . . . . . . 7  |-  ( ( C  e.  RR  /\  D  e.  RR  /\  D  =/=  0 )  ->  ( C  /  D )  e.  RR )
14133expb 1154 . . . . . 6  |-  ( ( C  e.  RR  /\  ( D  e.  RR  /\  D  =/=  0 ) )  ->  ( C  /  D )  e.  RR )
1512, 14sylan2 461 . . . . 5  |-  ( ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) )  ->  ( C  /  D )  e.  RR )
1615adantlr 696 . . . 4  |-  ( ( ( C  e.  RR  /\  0  <  C )  /\  ( D  e.  RR  /\  0  < 
D ) )  -> 
( C  /  D
)  e.  RR )
17 divgt0 9871 . . . 4  |-  ( ( ( C  e.  RR  /\  0  <  C )  /\  ( D  e.  RR  /\  0  < 
D ) )  -> 
0  <  ( C  /  D ) )
1816, 17jca 519 . . 3  |-  ( ( ( C  e.  RR  /\  0  <  C )  /\  ( D  e.  RR  /\  0  < 
D ) )  -> 
( ( C  /  D )  e.  RR  /\  0  <  ( C  /  D ) ) )
19 lerec 9885 . . 3  |-  ( ( ( ( A  /  B )  e.  RR  /\  0  <  ( A  /  B ) )  /\  ( ( C  /  D )  e.  RR  /\  0  < 
( C  /  D
) ) )  -> 
( ( A  /  B )  <_  ( C  /  D )  <->  ( 1  /  ( C  /  D ) )  <_ 
( 1  /  ( A  /  B ) ) ) )
209, 18, 19syl2an 464 . 2  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( A  /  B )  <_ 
( C  /  D
)  <->  ( 1  / 
( C  /  D
) )  <_  (
1  /  ( A  /  B ) ) ) )
21 recn 9073 . . . . . 6  |-  ( C  e.  RR  ->  C  e.  CC )
2221adantr 452 . . . . 5  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  e.  CC )
23 gt0ne0 9486 . . . . 5  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
2422, 23jca 519 . . . 4  |-  ( ( C  e.  RR  /\  0  <  C )  -> 
( C  e.  CC  /\  C  =/=  0 ) )
25 recn 9073 . . . . . 6  |-  ( D  e.  RR  ->  D  e.  CC )
2625adantr 452 . . . . 5  |-  ( ( D  e.  RR  /\  0  <  D )  ->  D  e.  CC )
2726, 11jca 519 . . . 4  |-  ( ( D  e.  RR  /\  0  <  D )  -> 
( D  e.  CC  /\  D  =/=  0 ) )
28 recdiv 9713 . . . 4  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( 1  /  ( C  /  D ) )  =  ( D  /  C ) )
2924, 27, 28syl2an 464 . . 3  |-  ( ( ( C  e.  RR  /\  0  <  C )  /\  ( D  e.  RR  /\  0  < 
D ) )  -> 
( 1  /  ( C  /  D ) )  =  ( D  /  C ) )
30 recn 9073 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
3130adantr 452 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  CC )
32 gt0ne0 9486 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0 )
3331, 32jca 519 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  e.  CC  /\  A  =/=  0 ) )
34 recn 9073 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
3534adantr 452 . . . . 5  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  e.  CC )
3635, 2jca 519 . . . 4  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
37 recdiv 9713 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( 1  /  ( A  /  B ) )  =  ( B  /  A ) )
3833, 36, 37syl2an 464 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( 1  /  ( A  /  B ) )  =  ( B  /  A ) )
3929, 38breqan12rd 4221 . 2  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( 1  /  ( C  /  D ) )  <_ 
( 1  /  ( A  /  B ) )  <-> 
( D  /  C
)  <_  ( B  /  A ) ) )
4020, 39bitrd 245 1  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( A  /  B )  <_ 
( C  /  D
)  <->  ( D  /  C )  <_  ( B  /  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4205  (class class class)co 6074   CCcc 8981   RRcr 8982   0cc0 8983   1c1 8984    < clt 9113    <_ cle 9114    / cdiv 9670
This theorem is referenced by:  ledivdivd  10666
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 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-po 4496  df-so 4497  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-riota 6542  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671
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