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Definition df-mq0 6583
Description: Define multiplication on non-negative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)
Assertion
Ref Expression
df-mq0  |- ·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  ) ) }
Distinct variable group:    x, y, z, w, v, u, f

Detailed syntax breakdown of Definition df-mq0
StepHypRef Expression
1 cmq0 6445 . 2  class ·Q0
2 vx . . . . . . 7  setvar  x
32cv 1258 . . . . . 6  class  x
4 cnq0 6442 . . . . . 6  class Q0
53, 4wcel 1409 . . . . 5  wff  x  e. Q0
6 vy . . . . . . 7  setvar  y
76cv 1258 . . . . . 6  class  y
87, 4wcel 1409 . . . . 5  wff  y  e. Q0
95, 8wa 101 . . . 4  wff  ( x  e. Q0  /\  y  e. Q0 )
10 vw . . . . . . . . . . . . . 14  setvar  w
1110cv 1258 . . . . . . . . . . . . 13  class  w
12 vv . . . . . . . . . . . . . 14  setvar  v
1312cv 1258 . . . . . . . . . . . . 13  class  v
1411, 13cop 3405 . . . . . . . . . . . 12  class  <. w ,  v >.
15 ceq0 6441 . . . . . . . . . . . 12  class ~Q0
1614, 15cec 6134 . . . . . . . . . . 11  class  [ <. w ,  v >. ] ~Q0
173, 16wceq 1259 . . . . . . . . . 10  wff  x  =  [ <. w ,  v
>. ] ~Q0
18 vu . . . . . . . . . . . . . 14  setvar  u
1918cv 1258 . . . . . . . . . . . . 13  class  u
20 vf . . . . . . . . . . . . . 14  setvar  f
2120cv 1258 . . . . . . . . . . . . 13  class  f
2219, 21cop 3405 . . . . . . . . . . . 12  class  <. u ,  f >.
2322, 15cec 6134 . . . . . . . . . . 11  class  [ <. u ,  f >. ] ~Q0
247, 23wceq 1259 . . . . . . . . . 10  wff  y  =  [ <. u ,  f
>. ] ~Q0
2517, 24wa 101 . . . . . . . . 9  wff  ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )
26 vz . . . . . . . . . . 11  setvar  z
2726cv 1258 . . . . . . . . . 10  class  z
28 comu 6029 . . . . . . . . . . . . 13  class  .o
2911, 19, 28co 5539 . . . . . . . . . . . 12  class  ( w  .o  u )
3013, 21, 28co 5539 . . . . . . . . . . . 12  class  ( v  .o  f )
3129, 30cop 3405 . . . . . . . . . . 11  class  <. (
w  .o  u ) ,  ( v  .o  f ) >.
3231, 15cec 6134 . . . . . . . . . 10  class  [ <. ( w  .o  u ) ,  ( v  .o  f ) >. ] ~Q0
3327, 32wceq 1259 . . . . . . . . 9  wff  z  =  [ <. ( w  .o  u ) ,  ( v  .o  f )
>. ] ~Q0
3425, 33wa 101 . . . . . . . 8  wff  ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u
) ,  ( v  .o  f ) >. ] ~Q0  )
3534, 20wex 1397 . . . . . . 7  wff  E. f
( ( x  =  [ <. w ,  v
>. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u
) ,  ( v  .o  f ) >. ] ~Q0  )
3635, 18wex 1397 . . . . . 6  wff  E. u E. f ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u
) ,  ( v  .o  f ) >. ] ~Q0  )
3736, 12wex 1397 . . . . 5  wff  E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u ) ,  ( v  .o  f )
>. ] ~Q0  )
3837, 10wex 1397 . . . 4  wff  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u
) ,  ( v  .o  f ) >. ] ~Q0  )
399, 38wa 101 . . 3  wff  ( ( 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  ) )
4039, 2, 6, 26coprab 5540 . 2  class  { <. <.
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  ) ) }
411, 40wceq 1259 1  wff ·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  ) ) }
Colors of variables: wff set class
This definition is referenced by:  dfmq0qs  6584
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