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Mirrors > Home > MPE Home > Th. List > ax-cnre | Structured version Visualization version GIF version |
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 10803. For naming consistency, use cnre 10855 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.) |
Ref | Expression |
---|---|
ax-cnre | ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cc 10752 | . . 3 class ℂ | |
3 | 1, 2 | wcel 2111 | . 2 wff 𝐴 ∈ ℂ |
4 | vx | . . . . . . 7 setvar 𝑥 | |
5 | 4 | cv 1542 | . . . . . 6 class 𝑥 |
6 | ci 10756 | . . . . . . 7 class i | |
7 | vy | . . . . . . . 8 setvar 𝑦 | |
8 | 7 | cv 1542 | . . . . . . 7 class 𝑦 |
9 | cmul 10759 | . . . . . . 7 class · | |
10 | 6, 8, 9 | co 7232 | . . . . . 6 class (i · 𝑦) |
11 | caddc 10757 | . . . . . 6 class + | |
12 | 5, 10, 11 | co 7232 | . . . . 5 class (𝑥 + (i · 𝑦)) |
13 | 1, 12 | wceq 1543 | . . . 4 wff 𝐴 = (𝑥 + (i · 𝑦)) |
14 | cr 10753 | . . . 4 class ℝ | |
15 | 13, 7, 14 | wrex 3063 | . . 3 wff ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)) |
16 | 15, 4, 14 | wrex 3063 | . 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))) |
Colors of variables: wff setvar class |
This axiom is referenced by: cnre 10855 |
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