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Mirrors > Home > ILE Home > Th. List > ax-cnre | GIF version |
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 6765. For naming consistency, use cnre 6821 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 6709 | . . 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 6713 | . . . . . . 7 class i | |
7 | vy | . . . . . . . 8 setvar y | |
8 | 7 | cv 1241 | . . . . . . 7 class y |
9 | cmul 6716 | . . . . . . 7 class · | |
10 | 6, 8, 9 | co 5455 | . . . . . 6 class (i · y) |
11 | caddc 6714 | . . . . . 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 6710 | . . . 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))) |
Colors of variables: wff set class |
This axiom is referenced by: cnre 6821 |
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