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Axiom ax-cnre 8806
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 8782. For naming consistency, use cnre 8830 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 8731 . . 3  class  CC
31, 2wcel 1685 . 2  wff  A  e.  CC
4 vx . . . . . . 7  set  x
54cv 1623 . . . . . 6  class  x
6 ci 8735 . . . . . . 7  class  _i
7 vy . . . . . . . 8  set  y
87cv 1623 . . . . . . 7  class  y
9 cmul 8738 . . . . . . 7  class  x.
106, 8, 9co 5820 . . . . . 6  class  ( _i  x.  y )
11 caddc 8736 . . . . . 6  class  +
125, 10, 11co 5820 . . . . 5  class  ( x  +  ( _i  x.  y ) )
131, 12wceq 1624 . . . 4  wff  A  =  ( x  +  ( _i  x.  y ) )
14 cr 8732 . . . 4  class  RR
1513, 7, 14wrex 2546 . . 3  wff  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) )
1615, 4, 14wrex 2546 . 2  wff  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) )
173, 16wi 6 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  8830
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