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Definition df-plus 8158
Description: Define addition over complex numbers. (Contributed by NM, 28-May-1995.)
Assertion
Ref Expression
df-plus  |-  +  =  { <. <. 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 ) ,  ( v  +R  f ) >. )
) }
Distinct variable group:    x, y, z, w, v, u, f

Detailed syntax breakdown of Definition df-plus
StepHypRef Expression
1 caddc 8150 . 2  class  +
2 vx . . . . . . 7  set  x
32cv 1516 . . . . . 6  class  x
4 cc 8145 . . . . . 6  class  CC
53, 4wcel 1519 . . . . 5  wff  x  e.  CC
6 vy . . . . . . 7  set  y
76cv 1516 . . . . . 6  class  y
87, 4wcel 1519 . . . . 5  wff  y  e.  CC
95, 8wa 356 . . . 4  wff  ( x  e.  CC  /\  y  e.  CC )
10 vw . . . . . . . . . . . . 13  set  w
1110cv 1516 . . . . . . . . . . . 12  class  w
12 vv . . . . . . . . . . . . 13  set  v
1312cv 1516 . . . . . . . . . . . 12  class  v
1411, 13cop 3254 . . . . . . . . . . 11  class  <. w ,  v >.
153, 14wceq 1517 . . . . . . . . . 10  wff  x  = 
<. w ,  v >.
16 vu . . . . . . . . . . . . 13  set  u
1716cv 1516 . . . . . . . . . . . 12  class  u
18 vf . . . . . . . . . . . . 13  set  f
1918cv 1516 . . . . . . . . . . . 12  class  f
2017, 19cop 3254 . . . . . . . . . . 11  class  <. u ,  f >.
217, 20wceq 1517 . . . . . . . . . 10  wff  y  = 
<. u ,  f >.
2215, 21wa 356 . . . . . . . . 9  wff  ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )
23 vz . . . . . . . . . . 11  set  z
2423cv 1516 . . . . . . . . . 10  class  z
25 cplr 7903 . . . . . . . . . . . 12  class  +R
2611, 17, 25co 5353 . . . . . . . . . . 11  class  ( w  +R  u )
2713, 19, 25co 5353 . . . . . . . . . . 11  class  ( v  +R  f )
2826, 27cop 3254 . . . . . . . . . 10  class  <. (
w  +R  u ) ,  ( v  +R  f ) >.
2924, 28wceq 1517 . . . . . . . . 9  wff  z  = 
<. ( w  +R  u
) ,  ( v  +R  f ) >.
3022, 29wa 356 . . . . . . . 8  wff  ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( w  +R  u ) ,  ( v  +R  f )
>. )
3130, 18wex 1443 . . . . . . 7  wff  E. f
( ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( w  +R  u ) ,  ( v  +R  f ) >. )
3231, 16wex 1443 . . . . . 6  wff  E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( w  +R  u ) ,  ( v  +R  f ) >. )
3332, 12wex 1443 . . . . 5  wff  E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( w  +R  u ) ,  ( v  +R  f )
>. )
3433, 10wex 1443 . . . 4  wff  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( w  +R  u ) ,  ( v  +R  f ) >. )
359, 34wa 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 ) ,  ( v  +R  f )
>. ) )
3635, 2, 6, 23copab2 5354 . 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 ) ,  ( v  +R  f ) >. )
) }
371, 36wceq 1517 1  wff  +  =  { <. <. 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 ) ,  ( v  +R  f ) >. )
) }
Colors of variables: wff set class
This definition is referenced by:  addcnsr  8167  axaddf  8177
Copyright terms: Public domain