Axiom ax-addcl 9981

Description: Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 9957. Proofs should normally use addcl 10003 instead, which asserts the same thing but follows our naming conventions for closures. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-addcl	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)

Detailed syntax breakdown of Axiom ax-addcl

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 9919	. . . 4 class ℂ
3	1, 2	wcel 1988	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 1988	. . 3 wff 𝐵 ∈ ℂ
6	3, 5	wa 384	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7		caddc 9924	. . . 4 class +
8	1, 4, 7	co 6635	. . 3 class (𝐴 + 𝐵)
9	8, 2	wcel 1988	. 2 wff (𝐴 + 𝐵) ∈ ℂ
10	6, 9	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)

This axiom is referenced by:  addcl  10003