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Theorem ltdiv1 8619
Description: Division of both sides of 'less than' by a positive number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
ltdiv1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  /  C )  <  ( B  /  C ) ) )

Proof of Theorem ltdiv1
StepHypRef Expression
1 simp1 981 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  A  e.  RR )
2 simp2 982 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  B  e.  RR )
3 simp3l 1009 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  RR )
4 simp3r 1010 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
0  <  C )
53, 4gt0ap0d 8384 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C #  0 )
63, 5rerecclapd 8586 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( 1  /  C
)  e.  RR )
7 recgt0 8601 . . . 4  |-  ( ( C  e.  RR  /\  0  <  C )  -> 
0  <  ( 1  /  C ) )
873ad2ant3 1004 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
0  <  ( 1  /  C ) )
9 ltmul1 8347 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  (
( 1  /  C
)  e.  RR  /\  0  <  ( 1  /  C ) ) )  ->  ( A  < 
B  <->  ( A  x.  ( 1  /  C
) )  <  ( B  x.  ( 1  /  C ) ) ) )
101, 2, 6, 8, 9syl112anc 1220 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  x.  ( 1  /  C ) )  <  ( B  x.  ( 1  /  C
) ) ) )
111recnd 7787 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  A  e.  CC )
123recnd 7787 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  CC )
1311, 12, 5divrecapd 8546 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  /  C
)  =  ( A  x.  ( 1  /  C ) ) )
142recnd 7787 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  B  e.  CC )
1514, 12, 5divrecapd 8546 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  /  C
)  =  ( B  x.  ( 1  /  C ) ) )
1613, 15breq12d 3937 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <  ( B  /  C )  <->  ( A  x.  ( 1  /  C
) )  <  ( B  x.  ( 1  /  C ) ) ) )
1710, 16bitr4d 190 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  /  C )  <  ( B  /  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    e. wcel 1480   class class class wbr 3924  (class class class)co 5767   RRcr 7612   0cc0 7613   1c1 7614    x. cmul 7618    < clt 7793    / cdiv 8425
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 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731
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 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426
This theorem is referenced by:  lediv1  8620  gt0div  8621  ltmuldiv  8625  ltdivmul  8627  ltdiv23  8643  ltdiv1i  8672  ltdiv1d  9522  flltdivnn0lt  10070  flqdiv  10087  hashdvds  11886  hashgcdlem  11892  sinq12gt0  12900
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