Axiom ax-addcl 11076

Description: Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by Theorem axaddcl 11052. Proofs should normally use addcl 11098 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 11014	. . . 4 class ℂ
3	1, 2	wcel 2113	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2113	. . 3 wff 𝐵 ∈ ℂ
6	3, 5	wa 395	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7		caddc 11019	. . . 4 class +
8	1, 4, 7	co 7355	. . 3 class (𝐴 + 𝐵)
9	8, 2	wcel 2113	. 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  11098