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Definition df-mul 8165
Description: Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
df-mul  |-  x.  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
) }
Distinct variable group:    x, y, z, w, v, u, f

Detailed syntax breakdown of Definition df-mul
StepHypRef Expression
1 cmul 8158 . 2  class  x.
2 vx . . . . . . 7  set  x
32cv 1517 . . . . . 6  class  x
4 cc 8151 . . . . . 6  class  CC
53, 4wcel 1520 . . . . 5  wff  x  e.  CC
6 vy . . . . . . 7  set  y
76cv 1517 . . . . . 6  class  y
87, 4wcel 1520 . . . . 5  wff  y  e.  CC
95, 8wa 356 . . . 4  wff  ( x  e.  CC  /\  y  e.  CC )
10 vw . . . . . . . . . . . . 13  set  w
1110cv 1517 . . . . . . . . . . . 12  class  w
12 vv . . . . . . . . . . . . 13  set  v
1312cv 1517 . . . . . . . . . . . 12  class  v
1411, 13cop 3256 . . . . . . . . . . 11  class  <. w ,  v >.
153, 14wceq 1518 . . . . . . . . . 10  wff  x  = 
<. w ,  v >.
16 vu . . . . . . . . . . . . 13  set  u
1716cv 1517 . . . . . . . . . . . 12  class  u
18 vf . . . . . . . . . . . . 13  set  f
1918cv 1517 . . . . . . . . . . . 12  class  f
2017, 19cop 3256 . . . . . . . . . . 11  class  <. u ,  f >.
217, 20wceq 1518 . . . . . . . . . 10  wff  y  = 
<. u ,  f >.
2215, 21wa 356 . . . . . . . . 9  wff  ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )
23 vz . . . . . . . . . . 11  set  z
2423cv 1517 . . . . . . . . . 10  class  z
25 cmr 7910 . . . . . . . . . . . . 13  class  .R
2611, 17, 25co 5356 . . . . . . . . . . . 12  class  ( w  .R  u )
27 cm1r 7908 . . . . . . . . . . . . 13  class  -1R
2813, 19, 25co 5356 . . . . . . . . . . . . 13  class  ( v  .R  f )
2927, 28, 25co 5356 . . . . . . . . . . . 12  class  ( -1R 
.R  ( v  .R  f ) )
30 cplr 7909 . . . . . . . . . . . 12  class  +R
3126, 29, 30co 5356 . . . . . . . . . . 11  class  ( ( w  .R  u )  +R  ( -1R  .R  ( v  .R  f
) ) )
3213, 17, 25co 5356 . . . . . . . . . . . 12  class  ( v  .R  u )
3311, 19, 25co 5356 . . . . . . . . . . . 12  class  ( w  .R  f )
3432, 33, 30co 5356 . . . . . . . . . . 11  class  ( ( v  .R  u )  +R  ( w  .R  f ) )
3531, 34cop 3256 . . . . . . . . . 10  class  <. (
( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >.
3624, 35wceq 1518 . . . . . . . . 9  wff  z  = 
<. ( ( w  .R  u )  +R  ( -1R  .R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >.
3722, 36wa 356 . . . . . . . 8  wff  ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( ( w  .R  u )  +R  ( -1R  .R  (
v  .R  f )
) ) ,  ( ( v  .R  u
)  +R  ( w  .R  f ) )
>. )
3837, 18wex 1444 . . . . . . 7  wff  E. f
( ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
3938, 16wex 1444 . . . . . 6  wff  E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
4039, 12wex 1444 . . . . 5  wff  E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( ( w  .R  u )  +R  ( -1R  .R  (
v  .R  f )
) ) ,  ( ( v  .R  u
)  +R  ( w  .R  f ) )
>. )
4140, 10wex 1444 . . . 4  wff  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
429, 41wa 356 . . 3  wff  ( ( x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( ( w  .R  u )  +R  ( -1R  .R  (
v  .R  f )
) ) ,  ( ( v  .R  u
)  +R  ( w  .R  f ) )
>. ) )
4342, 2, 6, 23copab2 5357 . 2  class  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f
( ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
) }
441, 43wceq 1518 1  wff  x.  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .R  u
)  +R  ( -1R 
.R  ( v  .R  f ) ) ) ,  ( ( v  .R  u )  +R  ( w  .R  f
) ) >. )
) }
Colors of variables: wff set class
This definition is referenced by:  mulcnsr  8174  axmulf  8184
Copyright terms: Public domain