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Theorem lemuldiv 10466
Description: 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
Assertion
Ref Expression
lemuldiv  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  x.  C )  <_  B  <->  A  <_  ( B  /  C ) ) )

Proof of Theorem lemuldiv
StepHypRef Expression
1 ltdivmul2 10462 . . . 4  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( B  /  C )  <  A  <->  B  <  ( A  x.  C ) ) )
213com12 1203 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( B  /  C )  <  A  <->  B  <  ( A  x.  C ) ) )
32notbid 294 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( -.  ( B  /  C )  < 
A  <->  -.  B  <  ( A  x.  C ) ) )
4 simp1 999 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  A  e.  RR )
5 gt0ne0 10060 . . . . . . 7  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
653adant1 1017 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
7 redivcl 10306 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  C  =/=  0 )  ->  ( B  /  C )  e.  RR )
86, 7syld3an3 1277 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR  /\  0  <  C )  ->  ( B  /  C )  e.  RR )
983expb 1200 . . . 4  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( B  /  C )  e.  RR )
1093adant1 1017 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  /  C
)  e.  RR )
114, 10lenltd 9765 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  ( B  /  C )  <->  -.  ( B  /  C )  < 
A ) )
12 remulcl 9609 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  x.  C
)  e.  RR )
13123adant2 1018 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  x.  C )  e.  RR )
14 simp2 1000 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
1513, 14lenltd 9765 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
)  <_  B  <->  -.  B  <  ( A  x.  C
) ) )
16153adant3r 1229 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  x.  C )  <_  B  <->  -.  B  <  ( A  x.  C ) ) )
173, 11, 163bitr4rd 288 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  x.  C )  <_  B  <->  A  <_  ( B  /  C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    e. wcel 1844    =/= wne 2600   class class class wbr 4397  (class class class)co 6280   RRcr 9523   0cc0 9524    x. cmul 9529    < clt 9660    <_ cle 9661    / cdiv 10249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-po 4746  df-so 4747  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250
This theorem is referenced by:  lemuldiv2  10467  lemuldivd  11351  hashdvds  14516  nmoleub2lem3  21892  mbfi1fseqlem4  22419  mbfi1fseqlem5  22420  radcnvlem1  23102  pige3  23204  fsumfldivdiaglem  23848  bposlem2  23943  bposlem3  23944  bposlem4  23945  bposlem7  23948  lgsquadlem1  24012  lgsquadlem2  24013  chebbnd1lem2  24038  chebbnd1lem3  24039  dchrisum0flblem1  24076  mulog2sumlem2  24103  pntibndlem3  24160  lemuldiv3d  36013  lemuldiv4d  36014  divge1b  38640
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