ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltdiv2 Unicode version
Theorem ltdiv2 8549
Description: Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.)
Assertion
Ref Expression
ltdiv2 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( C / B ) < ( C / A ) ) )
Proof of Theorem ltdiv2
StepHypRef Expression
1 simp2l 988 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B e. RR )
2 gt0ap0 8300 . . . . 5 |- ( ( B e. RR /\ 0 < B ) -> B # 0 )
323ad2ant2 984 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B # 0 )
41, 3rerecclapd 8500 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( 1 / B ) e. RR )
5 gt0ap0 8300 . . . . 5 |- ( ( A e. RR /\ 0 < A ) -> A # 0 )
6 rerecclap 8397 . . . . 5 |- ( ( A e. RR /\ A # 0 ) -> ( 1 / A ) e. RR )
75, 6syldan 278 . . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. RR )
873ad2ant1 983 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( 1 / A ) e. RR )
9 simp3 964 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( C e. RR /\ 0 < C ) )
10 ltmul2 8518 . . 3 |- ( ( ( 1 / B ) e. RR /\ ( 1 / A ) e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( 1 / B ) < ( 1 / A ) <-> ( C x. ( 1 / B ) ) < ( C x. ( 1 / A ) ) ) )
114, 8, 9, 10syl3anc 1197 . 2 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( 1 / B ) < ( 1 / A ) <-> ( C x. ( 1 / B ) ) < ( C x. ( 1 / A ) ) ) )
12 ltrec 8545 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) )
13123adant3 982 . 2 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) )
14 simp3l 990 . . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C e. RR )
1514recnd 7712 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C e. CC )
161recnd 7712 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B e. CC )
1715, 16, 3divrecapd 8460 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( C / B ) = ( C x. ( 1 / B ) ) )
18 simp1l 986 . . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A e. RR )
1918recnd 7712 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A e. CC )
2053ad2ant1 983 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A # 0 )
2115, 19, 20divrecapd 8460 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( C / A ) = ( C x. ( 1 / A ) ) )
2217, 21breq12d 3906 . 2 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( C / B ) < ( C / A ) <-> ( C x. ( 1 / B ) ) < ( C x. ( 1 / A ) ) ) )
2311, 13, 223bitr4d 219 1 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( C / B ) < ( C / A ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 943   e. wcel 1461   class class class wbr 3893  (class class class)co 5726   RRcr 7540   0cc0 7541   1c1 7542   x. cmul 7546   < clt 7718   # cap 8255   / cdiv 8339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-mulrcl 7638  ax-addcom 7639  ax-mulcom 7640  ax-addass 7641  ax-mulass 7642  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-1rid 7646  ax-0id 7647  ax-rnegex 7648  ax-precex 7649  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-apti 7654  ax-pre-ltadd 7655  ax-pre-mulgt0 7656  ax-pre-mulext 7657
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-id 4173  df-po 4176  df-iso 4177  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-reap 8249  df-ap 8256  df-div 8340
This theorem is referenced by:  ltdiv2d  9400  sin01gt0  11313
  Copyright terms: Public domain W3C validator