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Theorem ledivdiv 8860
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 533 . . . 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 535 . . . 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 536 . . . . 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 8599 . . . 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 8805 . . 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 8842 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  ( A  /  B ) )
76adantr 276 . . 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 537 . . . 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 539 . . . 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 540 . . . . 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 8599 . . . 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 8805 . . 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 8842 . . . 4  |-  ( ( ( C  e.  RR  /\  0  <  C )  /\  ( D  e.  RR  /\  0  < 
D ) )  -> 
0  <  ( C  /  D ) )
1413adantl 277 . . 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 8854 . . 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 1249 . 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 7999 . . . 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 7999 . . . 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 538 . . . . 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 8599 . . . 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 8777 . . 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 7999 . . . 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 7999 . . . 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 534 . . . . 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 8599 . . . 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 8777 . . 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 4028 . 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 188 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 104    <-> wb 105    e. wcel 2158   class class class wbr 4015  (class class class)co 5888   RRcr 7823   0cc0 7824   1c1 7825    < clt 8005    <_ cle 8006    / cdiv 8642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643
This theorem is referenced by:  ledivdivd  9735
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