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

Definition df-plp 8623
Description: Define addition on positive reals. 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. From Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
Assertion
Ref Expression
df-plp  |-  +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-plp
StepHypRef Expression
1 cpp 8499 . 2  class  +P.
2 vx . . 3  set  x
3 vy . . 3  set  y
4 cnp 8497 . . 3  class  P.
5 vw . . . . . . . 8  set  w
65cv 1631 . . . . . . 7  class  w
7 vv . . . . . . . . 9  set  v
87cv 1631 . . . . . . . 8  class  v
9 vu . . . . . . . . 9  set  u
109cv 1631 . . . . . . . 8  class  u
11 cplq 8493 . . . . . . . 8  class  +Q
128, 10, 11co 5874 . . . . . . 7  class  ( v  +Q  u )
136, 12wceq 1632 . . . . . 6  wff  w  =  ( v  +Q  u
)
143cv 1631 . . . . . 6  class  y
1513, 9, 14wrex 2557 . . . . 5  wff  E. u  e.  y  w  =  ( v  +Q  u
)
162cv 1631 . . . . 5  class  x
1715, 7, 16wrex 2557 . . . 4  wff  E. v  e.  x  E. u  e.  y  w  =  ( v  +Q  u
)
1817, 5cab 2282 . . 3  class  { w  |  E. v  e.  x  E. u  e.  y  w  =  ( v  +Q  u ) }
192, 3, 4, 4, 18cmpt2 5876 . 2  class  ( x  e.  P. ,  y  e.  P.  |->  { w  |  E. v  e.  x  E. u  e.  y  w  =  ( v  +Q  u ) } )
201, 19wceq 1632 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:  plpv  8650  dmplp  8652  addclprlem2  8657  addclpr  8658  addasspr  8662  distrlem1pr  8665  distrlem4pr  8666  distrlem5pr  8667  ltaddpr  8674  ltexprlem6  8681  ltexprlem7  8682
  Copyright terms: Public domain W3C validator