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Theorem ledivdiv 8909
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 8648 . . . 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 8854 . . 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 8891 . . . 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 8648 . . . 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 8854 . . 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 8891 . . . 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 8903 . . 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 1250 . 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 8048 . . . 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 8048 . . . 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 8648 . . . 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 8826 . . 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 8048 . . . 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 8048 . . . 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 8648 . . . 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 8826 . . 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 4042 . 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 2164   class class class wbr 4029  (class class class)co 5918   RRcr 7871   0cc0 7872   1c1 7873   < clt 8054   <_ cle 8055   / cdiv 8691
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-po 4327  df-iso 4328  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692
This theorem is referenced by:  ledivdivd  9788
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