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Definition df-mr 9485
Description: Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9547, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
df-mr  |-  .R  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
R.  /\  y  e.  R. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  f >. ]  ~R  )  /\  z  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  f )
) ,  ( ( w  .P.  f )  +P.  ( v  .P.  u ) ) >. ]  ~R  ) ) }
Distinct variable group:    x, y, z, w, v, u, f

Detailed syntax breakdown of Definition df-mr
StepHypRef Expression
1 cmr 9297 . 2  class  .R
2 vx . . . . . . 7  setvar  x
32cv 1437 . . . . . 6  class  x
4 cnr 9292 . . . . . 6  class  R.
53, 4wcel 1869 . . . . 5  wff  x  e. 
R.
6 vy . . . . . . 7  setvar  y
76cv 1437 . . . . . 6  class  y
87, 4wcel 1869 . . . . 5  wff  y  e. 
R.
95, 8wa 371 . . . 4  wff  ( x  e.  R.  /\  y  e.  R. )
10 vw . . . . . . . . . . . . . 14  setvar  w
1110cv 1437 . . . . . . . . . . . . 13  class  w
12 vv . . . . . . . . . . . . . 14  setvar  v
1312cv 1437 . . . . . . . . . . . . 13  class  v
1411, 13cop 4003 . . . . . . . . . . . 12  class  <. w ,  v >.
15 cer 9291 . . . . . . . . . . . 12  class  ~R
1614, 15cec 7367 . . . . . . . . . . 11  class  [ <. w ,  v >. ]  ~R
173, 16wceq 1438 . . . . . . . . . 10  wff  x  =  [ <. w ,  v
>. ]  ~R
18 vu . . . . . . . . . . . . . 14  setvar  u
1918cv 1437 . . . . . . . . . . . . 13  class  u
20 vf . . . . . . . . . . . . . 14  setvar  f
2120cv 1437 . . . . . . . . . . . . 13  class  f
2219, 21cop 4003 . . . . . . . . . . . 12  class  <. u ,  f >.
2322, 15cec 7367 . . . . . . . . . . 11  class  [ <. u ,  f >. ]  ~R
247, 23wceq 1438 . . . . . . . . . 10  wff  y  =  [ <. u ,  f
>. ]  ~R
2517, 24wa 371 . . . . . . . . 9  wff  ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  f >. ]  ~R  )
26 vz . . . . . . . . . . 11  setvar  z
2726cv 1437 . . . . . . . . . 10  class  z
28 cmp 9289 . . . . . . . . . . . . . 14  class  .P.
2911, 19, 28co 6303 . . . . . . . . . . . . 13  class  ( w  .P.  u )
3013, 21, 28co 6303 . . . . . . . . . . . . 13  class  ( v  .P.  f )
31 cpp 9288 . . . . . . . . . . . . 13  class  +P.
3229, 30, 31co 6303 . . . . . . . . . . . 12  class  ( ( w  .P.  u )  +P.  ( v  .P.  f ) )
3311, 21, 28co 6303 . . . . . . . . . . . . 13  class  ( w  .P.  f )
3413, 19, 28co 6303 . . . . . . . . . . . . 13  class  ( v  .P.  u )
3533, 34, 31co 6303 . . . . . . . . . . . 12  class  ( ( w  .P.  f )  +P.  ( v  .P.  u ) )
3632, 35cop 4003 . . . . . . . . . . 11  class  <. (
( w  .P.  u
)  +P.  ( v  .P.  f ) ) ,  ( ( w  .P.  f )  +P.  (
v  .P.  u )
) >.
3736, 15cec 7367 . . . . . . . . . 10  class  [ <. ( ( w  .P.  u
)  +P.  ( v  .P.  f ) ) ,  ( ( w  .P.  f )  +P.  (
v  .P.  u )
) >. ]  ~R
3827, 37wceq 1438 . . . . . . . . 9  wff  z  =  [ <. ( ( w  .P.  u )  +P.  ( v  .P.  f
) ) ,  ( ( w  .P.  f
)  +P.  ( v  .P.  u ) ) >. ]  ~R
3925, 38wa 371 . . . . . . . 8  wff  ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  f >. ]  ~R  )  /\  z  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  f )
) ,  ( ( w  .P.  f )  +P.  ( v  .P.  u ) ) >. ]  ~R  )
4039, 20wex 1660 . . . . . . 7  wff  E. f
( ( x  =  [ <. w ,  v
>. ]  ~R  /\  y  =  [ <. u ,  f
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  f ) ) ,  ( ( w  .P.  f )  +P.  (
v  .P.  u )
) >. ]  ~R  )
4140, 18wex 1660 . . . . . 6  wff  E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  f >. ]  ~R  )  /\  z  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  f )
) ,  ( ( w  .P.  f )  +P.  ( v  .P.  u ) ) >. ]  ~R  )
4241, 12wex 1660 . . . . 5  wff  E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  f
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  f ) ) ,  ( ( w  .P.  f )  +P.  (
v  .P.  u )
) >. ]  ~R  )
4342, 10wex 1660 . . . 4  wff  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  f >. ]  ~R  )  /\  z  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  f )
) ,  ( ( w  .P.  f )  +P.  ( v  .P.  u ) ) >. ]  ~R  )
449, 43wa 371 . . 3  wff  ( ( x  e.  R.  /\  y  e.  R. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  f
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  f ) ) ,  ( ( w  .P.  f )  +P.  (
v  .P.  u )
) >. ]  ~R  )
)
4544, 2, 6, 26coprab 6304 . 2  class  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  R.  /\  y  e.  R. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  f
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  f ) ) ,  ( ( w  .P.  f )  +P.  (
v  .P.  u )
) >. ]  ~R  )
) }
461, 45wceq 1438 1  wff  .R  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
R.  /\  y  e.  R. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  f >. ]  ~R  )  /\  z  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  f )
) ,  ( ( w  .P.  f )  +P.  ( v  .P.  u ) ) >. ]  ~R  ) ) }
Colors of variables: wff setvar class
This definition is referenced by:  mulsrpr  9502  dmmulsr  9512
  Copyright terms: Public domain W3C validator