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Theorem ledivdiv 9069
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 535 . . . 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 537 . . . 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 538 . . . . 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 8808 . . . 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 9014 . . 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 9051 . . . 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 539 . . . 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 541 . . . 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 542 . . . . 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 8808 . . . 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 9014 . . 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 9051 . . . 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 9063 . . 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 1274 . 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 8207 . . . 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 8207 . . . 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 540 . . . . 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 8808 . . . 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 8986 . . 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 8207 . . . 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 8207 . . . 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 536 . . . . 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 8808 . . . 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 8986 . . 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 4101 . 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 2202   class class class wbr 4088  (class class class)co 6017   RRcr 8030   0cc0 8031   1c1 8032   < clt 8213   <_ cle 8214   / cdiv 8851
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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149
This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852
This theorem is referenced by:  ledivdivd  9956
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