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Theorem ledivdiv 9661
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 443 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  e.  RR )
2 gt0ne0 9255 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  =/=  0 )
31, 2jca 518 . . . . . 6  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( B  e.  RR  /\  B  =/=  0 ) )
4 redivcl 9495 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A  /  B )  e.  RR )
543expb 1152 . . . . . 6  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  ( A  /  B )  e.  RR )
63, 5sylan2 460 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( A  /  B )  e.  RR )
76adantlr 695 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  /  B
)  e.  RR )
8 divgt0 9640 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  ( A  /  B ) )
97, 8jca 518 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( ( A  /  B )  e.  RR  /\  0  <  ( A  /  B ) ) )
10 simpl 443 . . . . . . 7  |-  ( ( D  e.  RR  /\  0  <  D )  ->  D  e.  RR )
11 gt0ne0 9255 . . . . . . 7  |-  ( ( D  e.  RR  /\  0  <  D )  ->  D  =/=  0 )
1210, 11jca 518 . . . . . 6  |-  ( ( D  e.  RR  /\  0  <  D )  -> 
( D  e.  RR  /\  D  =/=  0 ) )
13 redivcl 9495 . . . . . . 7  |-  ( ( C  e.  RR  /\  D  e.  RR  /\  D  =/=  0 )  ->  ( C  /  D )  e.  RR )
14133expb 1152 . . . . . 6  |-  ( ( C  e.  RR  /\  ( D  e.  RR  /\  D  =/=  0 ) )  ->  ( C  /  D )  e.  RR )
1512, 14sylan2 460 . . . . 5  |-  ( ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) )  ->  ( C  /  D )  e.  RR )
1615adantlr 695 . . . 4  |-  ( ( ( C  e.  RR  /\  0  <  C )  /\  ( D  e.  RR  /\  0  < 
D ) )  -> 
( C  /  D
)  e.  RR )
17 divgt0 9640 . . . 4  |-  ( ( ( C  e.  RR  /\  0  <  C )  /\  ( D  e.  RR  /\  0  < 
D ) )  -> 
0  <  ( C  /  D ) )
1816, 17jca 518 . . 3  |-  ( ( ( C  e.  RR  /\  0  <  C )  /\  ( D  e.  RR  /\  0  < 
D ) )  -> 
( ( C  /  D )  e.  RR  /\  0  <  ( C  /  D ) ) )
19 lerec 9654 . . 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 463 . 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 8843 . . . . . 6  |-  ( C  e.  RR  ->  C  e.  CC )
2221adantr 451 . . . . 5  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  e.  CC )
23 gt0ne0 9255 . . . . 5  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
2422, 23jca 518 . . . 4  |-  ( ( C  e.  RR  /\  0  <  C )  -> 
( C  e.  CC  /\  C  =/=  0 ) )
25 recn 8843 . . . . . 6  |-  ( D  e.  RR  ->  D  e.  CC )
2625adantr 451 . . . . 5  |-  ( ( D  e.  RR  /\  0  <  D )  ->  D  e.  CC )
2726, 11jca 518 . . . 4  |-  ( ( D  e.  RR  /\  0  <  D )  -> 
( D  e.  CC  /\  D  =/=  0 ) )
28 recdiv 9482 . . . 4  |-  ( ( ( C  e.  CC  /\  C  =/=  0 )  /\  ( D  e.  CC  /\  D  =/=  0 ) )  -> 
( 1  /  ( C  /  D ) )  =  ( D  /  C ) )
2924, 27, 28syl2an 463 . . 3  |-  ( ( ( C  e.  RR  /\  0  <  C )  /\  ( D  e.  RR  /\  0  < 
D ) )  -> 
( 1  /  ( C  /  D ) )  =  ( D  /  C ) )
30 recn 8843 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
3130adantr 451 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  CC )
32 gt0ne0 9255 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0 )
3331, 32jca 518 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  e.  CC  /\  A  =/=  0 ) )
34 recn 8843 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
3534adantr 451 . . . . 5  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  e.  CC )
3635, 2jca 518 . . . 4  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
37 recdiv 9482 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( 1  /  ( A  /  B ) )  =  ( B  /  A ) )
3833, 36, 37syl2an 463 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( 1  /  ( A  /  B ) )  =  ( B  /  A ) )
3929, 38breqan12rd 4055 . 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 244 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    < clt 8883    <_ cle 8884    / cdiv 9439
This theorem is referenced by:  ledivdivd  10431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440
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