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Theorem ltdivmul 8831
Description: 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
Assertion
Ref Expression
ltdivmul  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <  B  <->  A  <  ( C  x.  B ) ) )

Proof of Theorem ltdivmul
StepHypRef Expression
1 remulcl 7938 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  x.  B
)  e.  RR )
21ancoms 268 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  x.  B
)  e.  RR )
32adantrr 479 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( C  x.  B )  e.  RR )
433adant1 1015 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( C  x.  B
)  e.  RR )
5 ltdiv1 8823 . . 3  |-  ( ( A  e.  RR  /\  ( C  x.  B
)  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <  ( C  x.  B
)  <->  ( A  /  C )  <  (
( C  x.  B
)  /  C ) ) )
64, 5syld3an2 1285 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  ( C  x.  B )  <->  ( A  /  C )  <  ( ( C  x.  B )  /  C ) ) )
7 recn 7943 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
87adantr 276 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  B  e.  CC )
9 recn 7943 . . . . . 6  |-  ( C  e.  RR  ->  C  e.  CC )
109ad2antrl 490 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  CC )
11 gt0ap0 8581 . . . . . 6  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C #  0 )
1211adantl 277 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C #  0
)
138, 10, 12divcanap3d 8750 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( C  x.  B )  /  C )  =  B )
14133adant1 1015 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( C  x.  B )  /  C
)  =  B )
1514breq2d 4015 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <  (
( C  x.  B
)  /  C )  <-> 
( A  /  C
)  <  B )
)
166, 15bitr2d 189 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <  B  <->  A  <  ( C  x.  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   class class class wbr 4003  (class class class)co 5874   CCcc 7808   RRcr 7809   0cc0 7810    x. cmul 7815    < clt 7990   # cap 8536    / cdiv 8627
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-po 4296  df-iso 4297  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628
This theorem is referenced by:  ltdivmul2  8833  lt2mul2div  8834  ltrec  8838  avglt2  9156  3halfnz  9348  ltdivmuld  9746  modqid  10346  expnbnd  10640  mertenslemi1  11538  eirraplem  11779  fldivp1  12340  pcfaclem  12341  dveflem  14118  coseq0negpitopi  14188  tangtx  14190  cosordlem  14201  cos02pilt1  14203  2sqlem8  14390  ex-fl  14397
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