Axiom ax-addcl 6980

Description: Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by theorem axaddcl 6940. Proofs should normally use addcl 7006 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 6887	. . . 4 class ℂ
3	1, 2	wcel 1393	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 1393	. . 3 wff 𝐵 ∈ ℂ
6	3, 5	wa 97	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7		caddc 6892	. . . 4 class +
8	1, 4, 7	co 5512	. . 3 class (𝐴 + 𝐵)
9	8, 2	wcel 1393	. 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  7006