Axiom ax-cnre 10009

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 9985. For naming consistency, use cnre 10036 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

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		cc 9934	. . 3 class CC
3	1, 2	wcel 1990	. 2 wff A e. CC
4		vx	. . . . . . 7 setvar x
5	4	cv 1482	. . . . . 6 class x
6		ci 9938	. . . . . . 7 class _i
7		vy	. . . . . . . 8 setvar y
8	7	cv 1482	. . . . . . 7 class y
9		cmul 9941	. . . . . . 7 class x.
10	6, 8, 9	co 6650	. . . . . 6 class ( _i x. y )
11		caddc 9939	. . . . . 6 class +
12	5, 10, 11	co 6650	. . . . 5 class ( x + ( _i x. y ) )
13	1, 12	wceq 1483	. . . 4 wff A = ( x + ( _i x. y ) )
14		cr 9935	. . . 4 class RR
15	13, 7, 14	wrex 2913	. . 3 wff E. y e. RR A = ( x + ( _i x. y ) )
16	15, 4, 14	wrex 2913	. 2 wff E. x e. RR E. y e. RR A = ( x + ( _i x. y ) )
17	3, 16	wi 4	1 wff ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )

This axiom is referenced by:  cnre  10036