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Theorem ltdiv2 9109
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 1050 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B e. RR )
2 gt0ap0 8848 . . . . 5 |- ( ( B e. RR /\ 0 < B ) -> B # 0 )
323ad2ant2 1046 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B # 0 )
41, 3rerecclapd 9056 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( 1 / B ) e. RR )
5 gt0ap0 8848 . . . . 5 |- ( ( A e. RR /\ 0 < A ) -> A # 0 )
6 rerecclap 8952 . . . . 5 |- ( ( A e. RR /\ A # 0 ) -> ( 1 / A ) e. RR )
75, 6syldan 282 . . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. RR )
873ad2ant1 1045 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( 1 / A ) e. RR )
9 simp3 1026 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( C e. RR /\ 0 < C ) )
10 ltmul2 9078 . . 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 1274 . 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 9105 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) )
13123adant3 1044 . 2 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) )
14 simp3l 1052 . . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C e. RR )
1514recnd 8250 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C e. CC )
161recnd 8250 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B e. CC )
1715, 16, 3divrecapd 9015 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( C / B ) = ( C x. ( 1 / B ) ) )
18 simp1l 1048 . . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A e. RR )
1918recnd 8250 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A e. CC )
2053ad2ant1 1045 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A # 0 )
2115, 19, 20divrecapd 9015 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( C / A ) = ( C x. ( 1 / A ) ) )
2217, 21breq12d 4106 . 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 1005   e. wcel 2202   class class class wbr 4093  (class class class)co 6028   RRcr 8074   0cc0 8075   1c1 8076   x. cmul 8080   < clt 8256   # cap 8803   / cdiv 8894
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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895
This theorem is referenced by:  ltdiv2d  9999  sin01gt0  12386  coseq0negpitopi  15630  sincos6thpi  15636  cos02pilt1  15645  perfectlem2  15797
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