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Definition df-mp 9257
Description: Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 9392, 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 9133 . 2  class  .P.
2 vx . . 3  setvar  x
3 vy . . 3  setvar  y
4 cnp 9130 . . 3  class  P.
5 vw . . . . . . . 8  setvar  w
65cv 1369 . . . . . . 7  class  w
7 vv . . . . . . . . 9  setvar  v
87cv 1369 . . . . . . . 8  class  v
9 vu . . . . . . . . 9  setvar  u
109cv 1369 . . . . . . . 8  class  u
11 cmq 9127 . . . . . . . 8  class  .Q
128, 10, 11co 6193 . . . . . . 7  class  ( v  .Q  u )
136, 12wceq 1370 . . . . . 6  wff  w  =  ( v  .Q  u
)
143cv 1369 . . . . . 6  class  y
1513, 9, 14wrex 2796 . . . . 5  wff  E. u  e.  y  w  =  ( v  .Q  u
)
162cv 1369 . . . . 5  class  x
1715, 7, 16wrex 2796 . . . 4  wff  E. v  e.  x  E. u  e.  y  w  =  ( v  .Q  u
)
1817, 5cab 2436 . . 3  class  { w  |  E. v  e.  x  E. u  e.  y  w  =  ( v  .Q  u ) }
192, 3, 4, 4, 18cmpt2 6195 . 2  class  ( x  e.  P. ,  y  e.  P.  |->  { w  |  E. v  e.  x  E. u  e.  y  w  =  ( v  .Q  u ) } )
201, 19wceq 1370 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 setvar class
This definition is referenced by:  mpv  9284  dmmp  9286  mulclprlem  9292  mulclpr  9293  mulasspr  9297  distrlem1pr  9298  distrlem4pr  9299  distrlem5pr  9300  1idpr  9302  reclem3pr  9322  reclem4pr  9323
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