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Definition df-mp 8562
Description: Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 8697, and is intended to be used only by the construction. From Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
Assertion
Ref Expression
df-mp  |-  .P.  =  ( x  e.  P. ,  y  e.  P.  |->  { w  |  E. v  e.  x  E. u  e.  y  w  =  ( v  .Q  u ) } )
Distinct variable group:    x, y, w, v, u

Detailed syntax breakdown of Definition df-mp
StepHypRef Expression
1 cmp 8438 . 2  class  .P.
2 vx . . 3  set  x
3 vy . . 3  set  y
4 cnp 8435 . . 3  class  P.
5 vw . . . . . . . 8  set  w
65cv 1618 . . . . . . 7  class  w
7 vv . . . . . . . . 9  set  v
87cv 1618 . . . . . . . 8  class  v
9 vu . . . . . . . . 9  set  u
109cv 1618 . . . . . . . 8  class  u
11 cmq 8432 . . . . . . . 8  class  .Q
128, 10, 11co 5778 . . . . . . 7  class  ( v  .Q  u )
136, 12wceq 1619 . . . . . 6  wff  w  =  ( v  .Q  u
)
143cv 1618 . . . . . 6  class  y
1513, 9, 14wrex 2517 . . . . 5  wff  E. u  e.  y  w  =  ( v  .Q  u
)
162cv 1618 . . . . 5  class  x
1715, 7, 16wrex 2517 . . . 4  wff  E. v  e.  x  E. u  e.  y  w  =  ( v  .Q  u
)
1817, 5cab 2242 . . 3  class  { w  |  E. v  e.  x  E. u  e.  y  w  =  ( v  .Q  u ) }
192, 3, 4, 4, 18cmpt2 5780 . 2  class  ( x  e.  P. ,  y  e.  P.  |->  { w  |  E. v  e.  x  E. u  e.  y  w  =  ( v  .Q  u ) } )
201, 19wceq 1619 1  wff  .P.  =  ( x  e.  P. ,  y  e.  P.  |->  { w  |  E. v  e.  x  E. u  e.  y  w  =  ( v  .Q  u ) } )
Colors of variables: wff set class
This definition is referenced by:  mpv  8589  dmmp  8591  mulclprlem  8597  mulclpr  8598  mulasspr  8602  distrlem1pr  8603  distrlem4pr  8604  distrlem5pr  8605  1idpr  8607  reclem3pr  8627  reclem4pr  8628
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