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Theorem ltdivmul 8833
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 7939 . . . . . 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 8825 . . 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 7944 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
87adantr 276 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  B  e.  CC )
9 recn 7944 . . . . . 6  |-  ( C  e.  RR  ->  C  e.  CC )
109ad2antrl 490 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  CC )
11 gt0ap0 8583 . . . . . 6  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C #  0 )
1211adantl 277 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C #  0
)
138, 10, 12divcanap3d 8752 . . . 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 4016 . 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 4004  (class class class)co 5875   CCcc 7809   RRcr 7810   0cc0 7811    x. cmul 7816    < clt 7992   # cap 8538    / cdiv 8629
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 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929
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 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-po 4297  df-iso 4298  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630
This theorem is referenced by:  ltdivmul2  8835  lt2mul2div  8836  ltrec  8840  avglt2  9158  3halfnz  9350  ltdivmuld  9748  modqid  10349  expnbnd  10644  mertenslemi1  11543  eirraplem  11784  fldivp1  12346  pcfaclem  12347  dveflem  14190  coseq0negpitopi  14260  tangtx  14262  cosordlem  14273  cos02pilt1  14275  2sqlem8  14473  ex-fl  14480
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