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Theorem ltdivmul 9644
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 8838 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  x.  B
)  e.  RR )
21ancoms 439 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  x.  B
)  e.  RR )
32adantrr 697 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( C  x.  B )  e.  RR )
433adant1 973 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( C  x.  B
)  e.  RR )
5 ltdiv1 9636 . . 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 1229 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  ( C  x.  B )  <->  ( A  /  C )  <  ( ( C  x.  B )  /  C ) ) )
7 recn 8843 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
87adantr 451 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  B  e.  CC )
9 recn 8843 . . . . . 6  |-  ( C  e.  RR  ->  C  e.  CC )
109ad2antrl 708 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  CC )
11 gt0ne0 9255 . . . . . 6  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
1211adantl 452 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  =/=  0 )
138, 10, 12divcan3d 9557 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( C  x.  B )  /  C )  =  B )
14133adant1 973 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( C  x.  B )  /  C
)  =  B )
1514breq2d 4051 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <  (
( C  x.  B
)  /  C )  <-> 
( A  /  C
)  <  B )
)
166, 15bitr2d 245 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    x. cmul 8758    < clt 8883    / cdiv 9439
This theorem is referenced by:  ltdivmul2  9647  lt2mul2div  9648  ltrec  9653  supmul1  9735  avglt2  9966  rpnnen1lem1  10358  rpnnen1lem2  10359  rpnnen1lem3  10360  rpnnen1lem5  10362  ltdivmuld  10453  qbtwnre  10542  modid  11009  expnbnd  11246  mertenslem1  12356  tanhlt1  12456  eirrlem  12498  fldivp1  12961  pcfaclem  12962  4sqlem12  13019  icopnfcnv  18456  ovolscalem1  18888  mbfmulc2lem  19018  itg2monolem3  19123  dveflem  19342  dvlt0  19368  ftc1lem4  19402  radcnvlem1  19805  tangtx  19889  cosne0  19908  cosordlem  19909  efif1olem4  19923  logcnlem4  20008  logf1o2  20013  atantan  20235  atanbndlem  20237  birthdaylem3  20264  basellem3  20336  ppiub  20459  bposlem1  20539  bposlem2  20540  bposlem6  20544  bposlem8  20546  lgsquadlem1  20609  2sqlem8  20627  chebbnd1lem3  20636  chebbnd1  20637  ostth2lem2  20799  ex-fl  20850  ftc1cnnclem  25024  mslb1  25710  nn0prpwlem  26341  stoweidlem13  27865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440
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