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

Definition df-mpq 8413
Description: Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8351 . 2  class  .pQ
2 vx . . 3  set  x
3 vy . . 3  set  y
4 cnpi 8346 . . . 4  class  N.
54, 4cxp 4578 . . 3  class  ( N. 
X.  N. )
62cv 1618 . . . . . 6  class  x
7 c1st 5972 . . . . . 6  class  1st
86, 7cfv 4592 . . . . 5  class  ( 1st `  x )
93cv 1618 . . . . . 6  class  y
109, 7cfv 4592 . . . . 5  class  ( 1st `  y )
11 cmi 8348 . . . . 5  class  .N
128, 10, 11co 5710 . . . 4  class  ( ( 1st `  x )  .N  ( 1st `  y
) )
13 c2nd 5973 . . . . . 6  class  2nd
146, 13cfv 4592 . . . . 5  class  ( 2nd `  x )
159, 13cfv 4592 . . . . 5  class  ( 2nd `  y )
1614, 15, 11co 5710 . . . 4  class  ( ( 2nd `  x )  .N  ( 2nd `  y
) )
1712, 16cop 3547 . . 3  class  <. (
( 1st `  x
)  .N  ( 1st `  y ) ) ,  ( ( 2nd `  x
)  .N  ( 2nd `  y ) ) >.
182, 3, 5, 5, 17cmpt2 5712 . 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  8443  mulpqnq  8445  mulpqf  8450
  Copyright terms: Public domain W3C validator