ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  lediv23 Unicode version
Theorem lediv23 7971
Description: Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)
Assertion
Ref Expression
lediv23 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) <_ C <-> ( A / C ) <_ B ) )
Proof of Theorem lediv23
StepHypRef Expression
1 simp1 938 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A e. RR )
2 simp2l 964 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B e. RR )
3 simp2r 965 . . . . 5 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> 0 < B )
42, 3gt0ap0d 7728 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B # 0 )
51, 2, 4redivclapd 7920 . . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A / B ) e. RR )
6 simp3l 966 . . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C e. RR )
7 lemul1 7693 . . 3 |- ( ( ( A / B ) e. RR /\ C e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) <_ C <-> ( ( A / B ) x. B ) <_ ( C x. B ) ) )
85, 6, 2, 3, 7syl112anc 1173 . 2 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) <_ C <-> ( ( A / B ) x. B ) <_ ( C x. B ) ) )
91recnd 7147 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A e. CC )
102recnd 7147 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B e. CC )
119, 10, 4divcanap1d 7878 . . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) x. B ) = A )
1211breq1d 3795 . 2 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( ( A / B ) x. B ) <_ ( C x. B ) <-> A <_ ( C x. B ) ) )
136, 2remulcld 7149 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( C x. B ) e. RR )
14 lediv1 7947 . . . 4 |- ( ( A e. RR /\ ( C x. B ) e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ ( C x. B ) <-> ( A / C ) <_ ( ( C x. B ) / C ) ) )
1513, 14syld3an2 1216 . . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ ( C x. B ) <-> ( A / C ) <_ ( ( C x. B ) / C ) ) )
166recnd 7147 . . . . 5 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C e. CC )
17 simp3r 967 . . . . . 6 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> 0 < C )
186, 17gt0ap0d 7728 . . . . 5 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C # 0 )
1910, 16, 18divcanap3d 7882 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. B ) / C ) = B )
2019breq2d 3797 . . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / C ) <_ ( ( C x. B ) / C ) <-> ( A / C ) <_ B ) )
2115, 20bitrd 186 . 2 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ ( C x. B ) <-> ( A / C ) <_ B ) )
228, 12, 213bitrd 212 1 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) <_ C <-> ( A / C ) <_ B ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   e. wcel 1433   class class class wbr 3785  (class class class)co 5532   RRcr 6980   0cc0 6981   x. cmul 6986   < clt 7153   <_ cle 7154   / cdiv 7760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094
This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761
This theorem is referenced by:  divle1le  8802  ledivge1le  8803  lediv23d  8835
  Copyright terms: Public domain W3C validator