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Theorem ltdivmul 9628
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 8822 . . . . . 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 9620 . . 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 8827 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
87adantr 451 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  B  e.  CC )
9 recn 8827 . . . . . 6  |-  ( C  e.  RR  ->  C  e.  CC )
109ad2antrl 708 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  CC )
11 gt0ne0 9239 . . . . . 6  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
1211adantl 452 . . . . 5  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  =/=  0 )
138, 10, 12divcan3d 9541 . . . 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 4035 . 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 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    x. cmul 8742    < clt 8867    / cdiv 9423
This theorem is referenced by:  ltdivmul2  9631  lt2mul2div  9632  ltrec  9637  supmul1  9719  avglt2  9950  rpnnen1lem1  10342  rpnnen1lem2  10343  rpnnen1lem3  10344  rpnnen1lem5  10346  ltdivmuld  10437  qbtwnre  10526  modid  10993  expnbnd  11230  mertenslem1  12340  tanhlt1  12440  eirrlem  12482  fldivp1  12945  pcfaclem  12946  4sqlem12  13003  icopnfcnv  18440  ovolscalem1  18872  mbfmulc2lem  19002  itg2monolem3  19107  dveflem  19326  dvlt0  19352  ftc1lem4  19386  radcnvlem1  19789  tangtx  19873  cosne0  19892  cosordlem  19893  efif1olem4  19907  logcnlem4  19992  logf1o2  19997  atantan  20219  atanbndlem  20221  birthdaylem3  20248  basellem3  20320  ppiub  20443  bposlem1  20523  bposlem2  20524  bposlem6  20528  bposlem8  20530  lgsquadlem1  20593  2sqlem8  20611  chebbnd1lem3  20620  chebbnd1  20621  ostth2lem2  20783  ex-fl  20834  mslb1  25607  nn0prpwlem  26238  stoweidlem13  27762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424
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