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Theorem ltdivmul 10209
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 9372 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  x.  B
)  e.  RR )
21ancoms 453 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  x.  B
)  e.  RR )
32adantrr 716 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( C  x.  B )  e.  RR )
433adant1 1006 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( C  x.  B
)  e.  RR )
5 ltdiv1 10198 . . 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 1265 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  ( C  x.  B )  <->  ( A  /  C )  <  ( ( C  x.  B )  /  C ) ) )
7 recn 9377 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
87adantr 465 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  B  e.  CC )
9 recn 9377 . . . . . 6  |-  ( C  e.  RR  ->  C  e.  CC )
109ad2antrl 727 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  CC )
11 gt0ne0 9809 . . . . . 6  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
1211adantl 466 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  =/=  0 )
138, 10, 12divcan3d 10117 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( C  x.  B )  /  C )  =  B )
14133adant1 1006 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( C  x.  B )  /  C
)  =  B )
1514breq2d 4309 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <  (
( C  x.  B
)  /  C )  <-> 
( A  /  C
)  <  B )
)
166, 15bitr2d 254 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <  B  <->  A  <  ( C  x.  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   class class class wbr 4297  (class class class)co 6096   CCcc 9285   RRcr 9286   0cc0 9287    x. cmul 9292    < clt 9423    / cdiv 9998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999
This theorem is referenced by:  ltdivmul2  10212  lt2mul2div  10213  ltrec  10218  supmul1  10300  avglt2  10568  rpnnen1lem1  10984  rpnnen1lem2  10985  rpnnen1lem3  10986  rpnnen1lem5  10988  ltdivmuld  11079  qbtwnre  11174  modid  11737  expnbnd  11998  mertenslem1  13349  tanhlt1  13449  eirrlem  13491  fldivp1  13964  pcfaclem  13965  4sqlem12  14022  icopnfcnv  20519  ovolscalem1  21001  mbfmulc2lem  21130  itg2monolem3  21235  dveflem  21456  dvlt0  21482  ftc1lem4  21516  radcnvlem1  21883  tangtx  21972  cosne0  21991  cosordlem  21992  efif1olem4  22006  logcnlem4  22095  logf1o2  22100  atantan  22323  atanbndlem  22325  birthdaylem3  22352  basellem3  22425  ppiub  22548  bposlem1  22628  bposlem2  22629  bposlem6  22633  bposlem8  22635  lgsquadlem1  22698  2sqlem8  22716  chebbnd1lem3  22725  chebbnd1  22726  ostth2lem2  22888  ex-fl  23659  ftc1cnnclem  28470  nn0prpwlem  28522  stoweidlem13  29813
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