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Theorem ledivdiv 9645
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 9239 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  =/=  0 )
31, 2jca 518 . . . . . 6  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( B  e.  RR  /\  B  =/=  0 ) )
4 redivcl 9479 . . . . . . 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 9624 . . . 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 9239 . . . . . . 7  |-  ( ( D  e.  RR  /\  0  <  D )  ->  D  =/=  0 )
1210, 11jca 518 . . . . . 6  |-  ( ( D  e.  RR  /\  0  <  D )  -> 
( D  e.  RR  /\  D  =/=  0 ) )
13 redivcl 9479 . . . . . . 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 9624 . . . 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 9638 . . 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 8827 . . . . . 6  |-  ( C  e.  RR  ->  C  e.  CC )
2221adantr 451 . . . . 5  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  e.  CC )
23 gt0ne0 9239 . . . . 5  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
2422, 23jca 518 . . . 4  |-  ( ( C  e.  RR  /\  0  <  C )  -> 
( C  e.  CC  /\  C  =/=  0 ) )
25 recn 8827 . . . . . 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 9466 . . . 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 8827 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
3130adantr 451 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  CC )
32 gt0ne0 9239 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0 )
3331, 32jca 518 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  e.  CC  /\  A  =/=  0 ) )
34 recn 8827 . . . . . 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 9466 . . . 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 4039 . 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 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    < clt 8867    <_ cle 8868    / cdiv 9423
This theorem is referenced by:  ledivdivd  10415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424
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