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Definition df-plpq 8500
Description: Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8711, and is intended to be used only by the construction. This "pre-addition" operation works directly with ordered pairs of integers. The actual positive fraction addition  +Q (df-plq 8506) works with the equivalence classes of these ordered pairs determined by the equivalence relation  ~Q (df-enq 8503). (Analogous remarks apply to the other "pre-" operations in the complex number construction that follows.) From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 28-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
df-plpq  |-  +pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>. )
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-plpq
StepHypRef Expression
1 cplpq 8438 . 2  class  +pQ
2 vx . . 3  set  x
3 vy . . 3  set  y
4 cnpi 8434 . . . 4  class  N.
54, 4cxp 4659 . . 3  class  ( N. 
X.  N. )
62cv 1618 . . . . . . 7  class  x
7 c1st 6054 . . . . . . 7  class  1st
86, 7cfv 4673 . . . . . 6  class  ( 1st `  x )
93cv 1618 . . . . . . 7  class  y
10 c2nd 6055 . . . . . . 7  class  2nd
119, 10cfv 4673 . . . . . 6  class  ( 2nd `  y )
12 cmi 8436 . . . . . 6  class  .N
138, 11, 12co 5792 . . . . 5  class  ( ( 1st `  x )  .N  ( 2nd `  y
) )
149, 7cfv 4673 . . . . . 6  class  ( 1st `  y )
156, 10cfv 4673 . . . . . 6  class  ( 2nd `  x )
1614, 15, 12co 5792 . . . . 5  class  ( ( 1st `  y )  .N  ( 2nd `  x
) )
17 cpli 8435 . . . . 5  class  +N
1813, 16, 17co 5792 . . . 4  class  ( ( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) )
1915, 11, 12co 5792 . . . 4  class  ( ( 2nd `  x )  .N  ( 2nd `  y
) )
2018, 19cop 3617 . . 3  class  <. (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>.
212, 3, 5, 5, 20cmpt2 5794 . 2  class  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N.  X.  N. )  |->  <. ( ( ( 1st `  x )  .N  ( 2nd `  y
) )  +N  (
( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>. )
221, 21wceq 1619 1  wff  +pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>. )
Colors of variables: wff set class
This definition is referenced by:  addpipq2  8528  addpqnq  8530  addpqf  8536
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