Metamath Proof Explorer


Axiom ax-addrcl

Description: Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by Theorem axaddrcl . Proofs should normally use readdcl instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994)

Ref Expression
Assertion ax-addrcl ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 + 𝐵 ) ∈ ℝ )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA 𝐴
1 cr
2 0 1 wcel 𝐴 ∈ ℝ
3 cB 𝐵
4 3 1 wcel 𝐵 ∈ ℝ
5 2 4 wa ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ )
6 caddc +
7 0 3 6 co ( 𝐴 + 𝐵 )
8 7 1 wcel ( 𝐴 + 𝐵 ) ∈ ℝ
9 5 8 wi ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 + 𝐵 ) ∈ ℝ )