Axiom ax-cnre 11179

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 11155. For naming consistency, use cnre 11207 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 11104	. . 3 class ℂ
3	1, 2	wcel 2107	. 2 wff 𝐴 ∈ ℂ
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1541	. . . . . 6 class 𝑥
6		ci 11108	. . . . . . 7 class i
7		vy	. . . . . . . 8 setvar 𝑦
8	7	cv 1541	. . . . . . 7 class 𝑦
9		cmul 11111	. . . . . . 7 class ·
10	6, 8, 9	co 7404	. . . . . 6 class (i · 𝑦)
11		caddc 11109	. . . . . 6 class +
12	5, 10, 11	co 7404	. . . . 5 class (𝑥 + (i · 𝑦))
13	1, 12	wceq 1542	. . . 4 wff 𝐴 = (𝑥 + (i · 𝑦))
14		cr 11105	. . . 4 class ℝ
15	13, 7, 14	wrex 3071	. . 3 wff ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))
16	15, 4, 14	wrex 3071	. 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  11207