Axiom ax-cnre 6774

Description: A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom for real and complex numbers, justified by theorem axcnre 6745. For naming consistency, use cnre 6801 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.)

Assertion

Ref	Expression
ax-cnre	⊢ (A ∈ ℂ → ∃x ∈ ℝ ∃y ∈ ℝ A = (x + (i · 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 6689	. . 3 class ℂ
3	1, 2	wcel 1390	. 2 wff A ∈ ℂ
4		vx	. . . . . . 7 setvar x
5	4	cv 1241	. . . . . 6 class x
6		ci 6693	. . . . . . 7 class i
7		vy	. . . . . . . 8 setvar y
8	7	cv 1241	. . . . . . 7 class y
9		cmul 6696	. . . . . . 7 class ·
10	6, 8, 9	co 5455	. . . . . 6 class (i · y)
11		caddc 6694	. . . . . 6 class +
12	5, 10, 11	co 5455	. . . . 5 class (x + (i · y))
13	1, 12	wceq 1242	. . . 4 wff A = (x + (i · y))
14		cr 6690	. . . 4 class ℝ
15	13, 7, 14	wrex 2301	. . 3 wff ∃y ∈ ℝ A = (x + (i · y))
16	15, 4, 14	wrex 2301	. 2 wff ∃x ∈ ℝ ∃y ∈ ℝ A = (x + (i · y))
17	3, 16	wi 4	1 wff (A ∈ ℂ → ∃x ∈ ℝ ∃y ∈ ℝ A = (x + (i · y)))

This axiom is referenced by:  cnre  6801