Axiom ax-addcl 7388

Description: Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by theorem axaddcl 7348. Proofs should normally use addcl 7414 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 7295	. . . 4 class ℂ
3	1, 2	wcel 1436	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 1436	. . 3 wff 𝐵 ∈ ℂ
6	3, 5	wa 102	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7		caddc 7300	. . . 4 class +
8	1, 4, 7	co 5615	. . 3 class (𝐴 + 𝐵)
9	8, 2	wcel 1436	. 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  7414