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Mirrors > Home > ILE Home > Th. List > ax-addrcl | GIF version |
Description: Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axaddrcl 7864. Proofs should normally use readdcl 7937 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Ref | Expression |
---|---|
ax-addrcl | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class 𝐴 | |
2 | cr 7810 | . . . 4 class ℝ | |
3 | 1, 2 | wcel 2148 | . . 3 wff 𝐴 ∈ ℝ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 2148 | . . 3 wff 𝐵 ∈ ℝ |
6 | 3, 5 | wa 104 | . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) |
7 | caddc 7814 | . . . 4 class + | |
8 | 1, 4, 7 | co 5875 | . . 3 class (𝐴 + 𝐵) |
9 | 8, 2 | wcel 2148 | . 2 wff (𝐴 + 𝐵) ∈ ℝ |
10 | 6, 9 | wi 4 | 1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) |
Colors of variables: wff set class |
This axiom is referenced by: readdcl 7937 |
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