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Definition df-plqqs 6419
Description: Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 24-Aug-1995.)
Assertion
Ref Expression
df-plqqs  |-  +Q  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
Q.  /\  y  e.  Q. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  +pQ  <. u ,  f
>. ) ]  ~Q  )
) }
Distinct variable group:    x, y, z, w, v, u, f

Detailed syntax breakdown of Definition df-plqqs
StepHypRef Expression
1 cplq 6352 . 2  class  +Q
2 vx . . . . . . 7  setvar  x
32cv 1242 . . . . . 6  class  x
4 cnq 6350 . . . . . 6  class  Q.
53, 4wcel 1393 . . . . 5  wff  x  e. 
Q.
6 vy . . . . . . 7  setvar  y
76cv 1242 . . . . . 6  class  y
87, 4wcel 1393 . . . . 5  wff  y  e. 
Q.
95, 8wa 97 . . . 4  wff  ( x  e.  Q.  /\  y  e.  Q. )
10 vw . . . . . . . . . . . . . 14  setvar  w
1110cv 1242 . . . . . . . . . . . . 13  class  w
12 vv . . . . . . . . . . . . . 14  setvar  v
1312cv 1242 . . . . . . . . . . . . 13  class  v
1411, 13cop 3375 . . . . . . . . . . . 12  class  <. w ,  v >.
15 ceq 6349 . . . . . . . . . . . 12  class  ~Q
1614, 15cec 6082 . . . . . . . . . . 11  class  [ <. w ,  v >. ]  ~Q
173, 16wceq 1243 . . . . . . . . . 10  wff  x  =  [ <. w ,  v
>. ]  ~Q
18 vu . . . . . . . . . . . . . 14  setvar  u
1918cv 1242 . . . . . . . . . . . . 13  class  u
20 vf . . . . . . . . . . . . . 14  setvar  f
2120cv 1242 . . . . . . . . . . . . 13  class  f
2219, 21cop 3375 . . . . . . . . . . . 12  class  <. u ,  f >.
2322, 15cec 6082 . . . . . . . . . . 11  class  [ <. u ,  f >. ]  ~Q
247, 23wceq 1243 . . . . . . . . . 10  wff  y  =  [ <. u ,  f
>. ]  ~Q
2517, 24wa 97 . . . . . . . . 9  wff  ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )
26 vz . . . . . . . . . . 11  setvar  z
2726cv 1242 . . . . . . . . . 10  class  z
28 cplpq 6346 . . . . . . . . . . . 12  class  +pQ
2914, 22, 28co 5490 . . . . . . . . . . 11  class  ( <.
w ,  v >.  +pQ  <. u ,  f
>. )
3029, 15cec 6082 . . . . . . . . . 10  class  [ (
<. w ,  v >.  +pQ  <. u ,  f
>. ) ]  ~Q
3127, 30wceq 1243 . . . . . . . . 9  wff  z  =  [ ( <. w ,  v >.  +pQ  <. u ,  f >. ) ]  ~Q
3225, 31wa 97 . . . . . . . 8  wff  ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  +pQ  <. u ,  f
>. ) ]  ~Q  )
3332, 20wex 1381 . . . . . . 7  wff  E. f
( ( x  =  [ <. w ,  v
>. ]  ~Q  /\  y  =  [ <. u ,  f
>. ]  ~Q  )  /\  z  =  [ ( <. w ,  v >.  +pQ  <. u ,  f
>. ) ]  ~Q  )
3433, 18wex 1381 . . . . . 6  wff  E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  +pQ  <. u ,  f
>. ) ]  ~Q  )
3534, 12wex 1381 . . . . 5  wff  E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f
>. ]  ~Q  )  /\  z  =  [ ( <. w ,  v >.  +pQ  <. u ,  f
>. ) ]  ~Q  )
3635, 10wex 1381 . . . 4  wff  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  +pQ  <. u ,  f
>. ) ]  ~Q  )
379, 36wa 97 . . 3  wff  ( ( x  e.  Q.  /\  y  e.  Q. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f
>. ]  ~Q  )  /\  z  =  [ ( <. w ,  v >.  +pQ  <. u ,  f
>. ) ]  ~Q  )
)
3837, 2, 6, 26coprab 5491 . 2  class  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  Q.  /\  y  e.  Q. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f
>. ]  ~Q  )  /\  z  =  [ ( <. w ,  v >.  +pQ  <. u ,  f
>. ) ]  ~Q  )
) }
391, 38wceq 1243 1  wff  +Q  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
Q.  /\  y  e.  Q. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ]  ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [
( <. w ,  v
>.  +pQ  <. u ,  f
>. ) ]  ~Q  )
) }
Colors of variables: wff set class
This definition is referenced by:  addpipqqs  6440  dmaddpq  6449
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