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Axiom ax-cnre 8826
Description: A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom 17 of 22 for real and complex numbers, justified by theorem axcnre 8802. For naming consistency, use cnre 8850 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.)
Assertion
Ref Expression
ax-cnre  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
Distinct variable group:    x, y, A

Detailed syntax breakdown of Axiom ax-cnre
StepHypRef Expression
1 cA . . 3  class  A
2 cc 8751 . . 3  class  CC
31, 2wcel 1696 . 2  wff  A  e.  CC
4 vx . . . . . . 7  set  x
54cv 1631 . . . . . 6  class  x
6 ci 8755 . . . . . . 7  class  _i
7 vy . . . . . . . 8  set  y
87cv 1631 . . . . . . 7  class  y
9 cmul 8758 . . . . . . 7  class  x.
106, 8, 9co 5874 . . . . . 6  class  ( _i  x.  y )
11 caddc 8756 . . . . . 6  class  +
125, 10, 11co 5874 . . . . 5  class  ( x  +  ( _i  x.  y ) )
131, 12wceq 1632 . . . 4  wff  A  =  ( x  +  ( _i  x.  y ) )
14 cr 8752 . . . 4  class  RR
1513, 7, 14wrex 2557 . . 3  wff  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) )
1615, 4, 14wrex 2557 . 2  wff  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) )
173, 16wi 4 1  wff  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
Colors of variables: wff set class
This axiom is referenced by:  cnre  8850
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