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Axiom ax-cnre 9994
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 9970. For naming consistency, use cnre 10021 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
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cc 9919 . . 3 class
31, 2wcel 1988 . 2 wff 𝐴 ∈ ℂ
4 vx . . . . . . 7 setvar 𝑥
54cv 1480 . . . . . 6 class 𝑥
6 ci 9923 . . . . . . 7 class i
7 vy . . . . . . . 8 setvar 𝑦
87cv 1480 . . . . . . 7 class 𝑦
9 cmul 9926 . . . . . . 7 class ·
106, 8, 9co 6635 . . . . . 6 class (i · 𝑦)
11 caddc 9924 . . . . . 6 class +
125, 10, 11co 6635 . . . . 5 class (𝑥 + (i · 𝑦))
131, 12wceq 1481 . . . 4 wff 𝐴 = (𝑥 + (i · 𝑦))
14 cr 9920 . . . 4 class
1513, 7, 14wrex 2910 . . 3 wff 𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))
1615, 4, 14wrex 2910 . 2 wff 𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))
173, 16wi 4 1 wff (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
Colors of variables: wff setvar class
This axiom is referenced by:  cnre  10021
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