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Definition df-mpq 8501
Description: Define pre-multiplication 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. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 28-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
df-mpq  |-  .pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
)
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-mpq
StepHypRef Expression
1 cmpq 8439 . 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 . . . . . 6  class  x
7 c1st 6054 . . . . . 6  class  1st
86, 7cfv 4673 . . . . 5  class  ( 1st `  x )
93cv 1618 . . . . . 6  class  y
109, 7cfv 4673 . . . . 5  class  ( 1st `  y )
11 cmi 8436 . . . . 5  class  .N
128, 10, 11co 5792 . . . 4  class  ( ( 1st `  x )  .N  ( 1st `  y
) )
13 c2nd 6055 . . . . . 6  class  2nd
146, 13cfv 4673 . . . . 5  class  ( 2nd `  x )
159, 13cfv 4673 . . . . 5  class  ( 2nd `  y )
1614, 15, 11co 5792 . . . 4  class  ( ( 2nd `  x )  .N  ( 2nd `  y
) )
1712, 16cop 3617 . . 3  class  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
182, 3, 5, 5, 17cmpt2 5794 . 2  class  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N.  X.  N. )  |->  <. ( ( 1st `  x )  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>. )
191, 18wceq 1619 1  wff  .pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
)
Colors of variables: wff set class
This definition is referenced by:  mulpipq2  8531  mulpqnq  8533  mulpqf  8538
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