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Theorem ltdiv23 8657
Description: Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)
Assertion
Ref Expression
ltdiv23 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) < C <-> ( A / C ) < B ) )
Proof of Theorem ltdiv23
StepHypRef Expression
1 simp1 981 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A e. RR )
2 simp2l 1007 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B e. RR )
3 simp2r 1008 . . . . 5 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> 0 < B )
42, 3gt0ap0d 8398 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B # 0 )
51, 2, 4redivclapd 8601 . . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A / B ) e. RR )
6 simp3l 1009 . . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C e. RR )
7 simp2 982 . . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( B e. RR /\ 0 < B ) )
8 ltmul1 8361 . . 3 |- ( ( ( A / B ) e. RR /\ C e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) < C <-> ( ( A / B ) x. B ) < ( C x. B ) ) )
95, 6, 7, 8syl3anc 1216 . 2 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) < C <-> ( ( A / B ) x. B ) < ( C x. B ) ) )
101recnd 7801 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A e. CC )
112recnd 7801 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B e. CC )
1210, 11, 4divcanap1d 8558 . . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) x. B ) = A )
1312breq1d 3939 . 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 ) ) )
146, 2remulcld 7803 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( C x. B ) e. RR )
15 ltdiv1 8633 . . . 4 |- ( ( A e. RR /\ ( C x. B ) e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < ( ( C x. B ) / C ) ) )
1614, 15syld3an2 1263 . . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < ( ( C x. B ) / C ) ) )
176recnd 7801 . . . . 5 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C e. CC )
18 simp3r 1010 . . . . . 6 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> 0 < C )
196, 18gt0ap0d 8398 . . . . 5 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C # 0 )
2011, 17, 19divcanap3d 8562 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. B ) / C ) = B )
2120breq2d 3941 . . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / C ) < ( ( C x. B ) / C ) <-> ( A / C ) < B ) )
2216, 21bitrd 187 . 2 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < B ) )
239, 13, 223bitrd 213 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 103   <-> wb 104   /\ w3a 962   e. wcel 1480   class class class wbr 3929  (class class class)co 5774   RRcr 7626   0cc0 7627   x. cmul 7632   < clt 7807   / cdiv 8439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440
This theorem is referenced by:  ltdiv23i  8691  divlt1lt  9518  ltdiv23d  9551  prmind2  11808
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