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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 6941. Proofs should normally use readdcl 7007 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 6888 | . . . 4 class ℝ | |
3 | 1, 2 | wcel 1393 | . . 3 wff 𝐴 ∈ ℝ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 1393 | . . 3 wff 𝐵 ∈ ℝ |
6 | 3, 5 | wa 97 | . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) |
7 | caddc 6892 | . . . 4 class + | |
8 | 1, 4, 7 | co 5512 | . . 3 class (𝐴 + 𝐵) |
9 | 8, 2 | wcel 1393 | . 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 7007 |
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