MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lt2mul2divd Unicode version

Theorem lt2mul2divd 10670
Description: The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
Hypotheses
Ref Expression
lt2mul2divd.1  |-  ( ph  ->  A  e.  RR )
lt2mul2divd.2  |-  ( ph  ->  B  e.  RR+ )
lt2mul2divd.3  |-  ( ph  ->  C  e.  RR )
lt2mul2divd.4  |-  ( ph  ->  D  e.  RR+ )
Assertion
Ref Expression
lt2mul2divd  |-  ( ph  ->  ( ( A  x.  B )  <  ( C  x.  D )  <->  ( A  /  D )  <  ( C  /  B ) ) )

Proof of Theorem lt2mul2divd
StepHypRef Expression
1 lt2mul2divd.1 . 2  |-  ( ph  ->  A  e.  RR )
2 lt2mul2divd.2 . . 3  |-  ( ph  ->  B  e.  RR+ )
32rpregt0d 10618 . 2  |-  ( ph  ->  ( B  e.  RR  /\  0  <  B ) )
4 lt2mul2divd.3 . 2  |-  ( ph  ->  C  e.  RR )
5 lt2mul2divd.4 . . 3  |-  ( ph  ->  D  e.  RR+ )
65rpregt0d 10618 . 2  |-  ( ph  ->  ( D  e.  RR  /\  0  <  D ) )
7 lt2mul2div 9850 . 2  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( A  x.  B )  < 
( C  x.  D
)  <->  ( A  /  D )  <  ( C  /  B ) ) )
81, 3, 4, 6, 7syl22anc 1185 1  |-  ( ph  ->  ( ( A  x.  B )  <  ( C  x.  D )  <->  ( A  /  D )  <  ( C  /  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721   class class class wbr 4180  (class class class)co 6048   RRcr 8953   0cc0 8954    x. cmul 8959    < clt 9084    / cdiv 9641   RR+crp 10576
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-rp 10577
  Copyright terms: Public domain W3C validator