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Theorem ltdiv2 9045
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 1047 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B e. RR )
2 gt0ap0 8784 . . . . 5 |- ( ( B e. RR /\ 0 < B ) -> B # 0 )
323ad2ant2 1043 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B # 0 )
41, 3rerecclapd 8992 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( 1 / B ) e. RR )
5 gt0ap0 8784 . . . . 5 |- ( ( A e. RR /\ 0 < A ) -> A # 0 )
6 rerecclap 8888 . . . . 5 |- ( ( A e. RR /\ A # 0 ) -> ( 1 / A ) e. RR )
75, 6syldan 282 . . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. RR )
873ad2ant1 1042 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( 1 / A ) e. RR )
9 simp3 1023 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( C e. RR /\ 0 < C ) )
10 ltmul2 9014 . . 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 1271 . 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 9041 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) )
13123adant3 1041 . 2 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) )
14 simp3l 1049 . . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C e. RR )
1514recnd 8186 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C e. CC )
161recnd 8186 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B e. CC )
1715, 16, 3divrecapd 8951 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( C / B ) = ( C x. ( 1 / B ) ) )
18 simp1l 1045 . . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A e. RR )
1918recnd 8186 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A e. CC )
2053ad2ant1 1042 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A # 0 )
2115, 19, 20divrecapd 8951 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( C / A ) = ( C x. ( 1 / A ) ) )
2217, 21breq12d 4096 . 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 220 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 104   <-> wb 105   /\ w3a 1002   e. wcel 2200   class class class wbr 4083  (class class class)co 6007   RRcr 8009   0cc0 8010   1c1 8011   x. cmul 8015   < clt 8192   # cap 8739   / cdiv 8830
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128
This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831
This theorem is referenced by:  ltdiv2d  9928  sin01gt0  12288  coseq0negpitopi  15525  sincos6thpi  15531  cos02pilt1  15540  perfectlem2  15689
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