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Theorem ltdiv2 10909
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 ltrec 10905 . . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) )
213adant3 1081 . 2 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) )
3 gt0ne0 10493 . . . . . 6 |- ( ( B e. RR /\ 0 < B ) -> B =/= 0 )
4 rereccl 10743 . . . . . 6 |- ( ( B e. RR /\ B =/= 0 ) -> ( 1 / B ) e. RR )
53, 4syldan 487 . . . . 5 |- ( ( B e. RR /\ 0 < B ) -> ( 1 / B ) e. RR )
6 gt0ne0 10493 . . . . . . 7 |- ( ( A e. RR /\ 0 < A ) -> A =/= 0 )
7 rereccl 10743 . . . . . . 7 |- ( ( A e. RR /\ A =/= 0 ) -> ( 1 / A ) e. RR )
86, 7syldan 487 . . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. RR )
9 ltmul2 10874 . . . . . 6 |- ( ( ( 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 ) ) ) )
108, 9syl3an2 1360 . . . . 5 |- ( ( ( 1 / B ) e. RR /\ ( A e. RR /\ 0 < A ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( 1 / B ) < ( 1 / A ) <-> ( C x. ( 1 / B ) ) < ( C x. ( 1 / A ) ) ) )
115, 10syl3an1 1359 . . . 4 |- ( ( ( B e. RR /\ 0 < B ) /\ ( A e. RR /\ 0 < A ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( 1 / B ) < ( 1 / A ) <-> ( C x. ( 1 / B ) ) < ( C x. ( 1 / A ) ) ) )
12 recn 10026 . . . . . . 7 |- ( C e. RR -> C e. CC )
1312adantr 481 . . . . . 6 |- ( ( C e. RR /\ 0 < C ) -> C e. CC )
14 recn 10026 . . . . . . . 8 |- ( B e. RR -> B e. CC )
1514adantr 481 . . . . . . 7 |- ( ( B e. RR /\ 0 < B ) -> B e. CC )
1615, 3jca 554 . . . . . 6 |- ( ( B e. RR /\ 0 < B ) -> ( B e. CC /\ B =/= 0 ) )
17 recn 10026 . . . . . . . 8 |- ( A e. RR -> A e. CC )
1817adantr 481 . . . . . . 7 |- ( ( A e. RR /\ 0 < A ) -> A e. CC )
1918, 6jca 554 . . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> ( A e. CC /\ A =/= 0 ) )
20 divrec 10701 . . . . . . . . 9 |- ( ( C e. CC /\ B e. CC /\ B =/= 0 ) -> ( C / B ) = ( C x. ( 1 / B ) ) )
21203expb 1266 . . . . . . . 8 |- ( ( C e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( C / B ) = ( C x. ( 1 / B ) ) )
22213adant3 1081 . . . . . . 7 |- ( ( C e. CC /\ ( B e. CC /\ B =/= 0 ) /\ ( A e. CC /\ A =/= 0 ) ) -> ( C / B ) = ( C x. ( 1 / B ) ) )
23 divrec 10701 . . . . . . . . 9 |- ( ( C e. CC /\ A e. CC /\ A =/= 0 ) -> ( C / A ) = ( C x. ( 1 / A ) ) )
24233expb 1266 . . . . . . . 8 |- ( ( C e. CC /\ ( A e. CC /\ A =/= 0 ) ) -> ( C / A ) = ( C x. ( 1 / A ) ) )
25243adant2 1080 . . . . . . 7 |- ( ( C e. CC /\ ( B e. CC /\ B =/= 0 ) /\ ( A e. CC /\ A =/= 0 ) ) -> ( C / A ) = ( C x. ( 1 / A ) ) )
2622, 25breq12d 4666 . . . . . 6 |- ( ( C e. CC /\ ( B e. CC /\ B =/= 0 ) /\ ( A e. CC /\ A =/= 0 ) ) -> ( ( C / B ) < ( C / A ) <-> ( C x. ( 1 / B ) ) < ( C x. ( 1 / A ) ) ) )
2713, 16, 19, 26syl3an 1368 . . . . 5 |- ( ( ( C e. RR /\ 0 < C ) /\ ( B e. RR /\ 0 < B ) /\ ( A e. RR /\ 0 < A ) ) -> ( ( C / B ) < ( C / A ) <-> ( C x. ( 1 / B ) ) < ( C x. ( 1 / A ) ) ) )
28273coml 1272 . . . 4 |- ( ( ( B e. RR /\ 0 < B ) /\ ( A e. RR /\ 0 < A ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( C / B ) < ( C / A ) <-> ( C x. ( 1 / B ) ) < ( C x. ( 1 / A ) ) ) )
2911, 28bitr4d 271 . . 3 |- ( ( ( B e. RR /\ 0 < B ) /\ ( A e. RR /\ 0 < A ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( 1 / B ) < ( 1 / A ) <-> ( C / B ) < ( C / A ) ) )
30293com12 1269 . 2 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( 1 / B ) < ( 1 / A ) <-> ( C / B ) < ( C / A ) ) )
312, 30bitrd 268 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 setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   / cdiv 10684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685
This theorem is referenced by:  ltdiv2d  11895  sin01gt0  14920  sincos6thpi  24267  tanord1  24283  basellem1  24807  perfectlem2  24955  bposlem6  25014  dchrisum0flblem2  25198  pntpbnd1a  25274  pntlemr  25291  hgt750lem  30729  stoweidlem42  40259
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