Axiom ax-addcl 10443

Description: Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 10419. Proofs should normally use addcl 10465 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 10381	. . . 4 class ℂ
3	1, 2	wcel 2081	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2081	. . 3 wff 𝐵 ∈ ℂ
6	3, 5	wa 396	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7		caddc 10386	. . . 4 class +
8	1, 4, 7	co 7016	. . 3 class (𝐴 + 𝐵)
9	8, 2	wcel 2081	. 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  10465