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Theorem ledivdiv 8616
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 simplll 507 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  A  e.  RR )
2 simplrl 509 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  B  e.  RR )
3 simplrr 510 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  0  <  B
)
42, 3gt0ap0d 8359 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  B #  0 )
51, 2, 4redivclapd 8562 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( A  /  B )  e.  RR )
6 divgt0 8598 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  ( A  /  B ) )
76adantr 274 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  0  <  ( A  /  B ) )
8 simprll 511 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  C  e.  RR )
9 simprrl 513 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  D  e.  RR )
10 simprrr 514 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  0  <  D
)
119, 10gt0ap0d 8359 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  D #  0 )
128, 9, 11redivclapd 8562 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( C  /  D )  e.  RR )
13 divgt0 8598 . . . 4  |-  ( ( ( C  e.  RR  /\  0  <  C )  /\  ( D  e.  RR  /\  0  < 
D ) )  -> 
0  <  ( C  /  D ) )
1413adantl 275 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  0  <  ( C  /  D ) )
15 lerec 8610 . . 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 ) ) ) )
165, 7, 12, 14, 15syl22anc 1202 . 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 ) ) ) )
178recnd 7762 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  C  e.  CC )
189recnd 7762 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  D  e.  CC )
19 simprlr 512 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  0  <  C
)
208, 19gt0ap0d 8359 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  C #  0 )
2117, 18, 20, 11recdivapd 8535 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( 1  / 
( C  /  D
) )  =  ( D  /  C ) )
221recnd 7762 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  A  e.  CC )
232recnd 7762 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  B  e.  CC )
24 simpllr 508 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  0  <  A
)
251, 24gt0ap0d 8359 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  A #  0 )
2622, 23, 25, 4recdivapd 8535 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  < 
C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( 1  / 
( A  /  B
) )  =  ( B  /  A ) )
2721, 26breq12d 3912 . 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 ) ) )
2816, 27bitrd 187 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    /\ wa 103    <-> wb 104    e. wcel 1465   class class class wbr 3899  (class class class)co 5742   RRcr 7587   0cc0 7588   1c1 7589    < clt 7768    <_ cle 7769    / cdiv 8400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401
This theorem is referenced by:  ledivdivd  9477
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