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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 7641. Proofs should normally use readdcl 7714 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 7587 | . . . 4 class ℝ | |
3 | 1, 2 | wcel 1465 | . . 3 wff 𝐴 ∈ ℝ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 1465 | . . 3 wff 𝐵 ∈ ℝ |
6 | 3, 5 | wa 103 | . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) |
7 | caddc 7591 | . . . 4 class + | |
8 | 1, 4, 7 | co 5742 | . . 3 class (𝐴 + 𝐵) |
9 | 8, 2 | wcel 1465 | . 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 7714 |
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