MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-plpq Unicode version

Definition df-plpq 8548
Description: Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8759, 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 8554) works with the equivalence classes of these ordered pairs determined by the equivalence relation  ~Q (df-enq 8551). (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 8486 . 2  class  +pQ
2 vx . . 3  set  x
3 vy . . 3  set  y
4 cnpi 8482 . . . 4  class  N.
54, 4cxp 4703 . . 3  class  ( N. 
X.  N. )
62cv 1631 . . . . . . 7  class  x
7 c1st 6136 . . . . . . 7  class  1st
86, 7cfv 5271 . . . . . 6  class  ( 1st `  x )
93cv 1631 . . . . . . 7  class  y
10 c2nd 6137 . . . . . . 7  class  2nd
119, 10cfv 5271 . . . . . 6  class  ( 2nd `  y )
12 cmi 8484 . . . . . 6  class  .N
138, 11, 12co 5874 . . . . 5  class  ( ( 1st `  x )  .N  ( 2nd `  y
) )
149, 7cfv 5271 . . . . . 6  class  ( 1st `  y )
156, 10cfv 5271 . . . . . 6  class  ( 2nd `  x )
1614, 15, 12co 5874 . . . . 5  class  ( ( 1st `  y )  .N  ( 2nd `  x
) )
17 cpli 8483 . . . . 5  class  +N
1813, 16, 17co 5874 . . . 4  class  ( ( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) )
1915, 11, 12co 5874 . . . 4  class  ( ( 2nd `  x )  .N  ( 2nd `  y
) )
2018, 19cop 3656 . . 3  class  <. (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>.
212, 3, 5, 5, 20cmpt2 5876 . 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 1632 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  8576  addpqnq  8578  addpqf  8584
  Copyright terms: Public domain W3C validator