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Theorem dfmq0qs 7442
Description: Multiplication on nonnegative fractions. This definition is similar to df-mq0 7441 but expands Q0. (Contributed by Jim Kingdon, 22-Nov-2019.)
Assertion
Ref Expression
dfmq0qs  |- ·Q0 
=  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  ( ( om  X.  N. ) /. ~Q0  )  /\  y  e.  ( ( om  X.  N. ) /. ~Q0  ) )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u
) ,  ( v  .o  f ) >. ] ~Q0  ) ) }
Distinct variable group:    x, y, z, w, v, u, f

Proof of Theorem dfmq0qs
StepHypRef Expression
1 df-mq0 7441 . 2  |- ·Q0 
=  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e. Q0  /\  y  e. Q0 )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u
) ,  ( v  .o  f ) >. ] ~Q0  ) ) }
2 df-nq0 7438 . . . . . 6  |- Q0  =  ( ( om 
X.  N. ) /. ~Q0  )
32eleq2i 2254 . . . . 5  |-  ( x  e. Q0  <->  x  e.  ( ( om 
X.  N. ) /. ~Q0  ) )
42eleq2i 2254 . . . . 5  |-  ( y  e. Q0  <->  y  e.  ( ( om 
X.  N. ) /. ~Q0  ) )
53, 4anbi12i 460 . . . 4  |-  ( ( x  e. Q0  /\  y  e. Q0 )  <->  ( x  e.  ( ( om  X.  N. ) /. ~Q0  )  /\  y  e.  ( ( om  X.  N. ) /. ~Q0  ) ) )
65anbi1i 458 . . 3  |-  ( ( ( x  e. Q0  /\  y  e. Q0 )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u
) ,  ( v  .o  f ) >. ] ~Q0  ) )  <->  ( ( x  e.  ( ( om 
X.  N. ) /. ~Q0  )  /\  y  e.  ( ( om  X.  N. ) /. ~Q0  ) )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u
) ,  ( v  .o  f ) >. ] ~Q0  ) ) )
76oprabbii 5943 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e. Q0  /\  y  e. Q0 )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u
) ,  ( v  .o  f ) >. ] ~Q0  ) ) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( ( om  X.  N. ) /. ~Q0  )  /\  y  e.  ( ( om  X.  N. ) /. ~Q0  ) )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u
) ,  ( v  .o  f ) >. ] ~Q0  ) ) }
81, 7eqtri 2208 1  |- ·Q0 
=  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  ( ( om  X.  N. ) /. ~Q0  )  /\  y  e.  ( ( om  X.  N. ) /. ~Q0  ) )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u
) ,  ( v  .o  f ) >. ] ~Q0  ) ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1363   E.wex 1502    e. wcel 2158   <.cop 3607   omcom 4601    X. cxp 4636  (class class class)co 5888   {coprab 5889    .o comu 6429   [cec 6547   /.cqs 6548   N.cnpi 7285   ~Q0 ceq0 7299  Q0cnq0 7300   ·Q0 cmq0 7303
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-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-oprab 5892  df-nq0 7438  df-mq0 7441
This theorem is referenced by:  mulnnnq0  7463
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