Axiom ax-addcl 10789

Description: Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by Theorem axaddcl 10765. Proofs should normally use addcl 10811 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 10727	. . . 4 class ℂ
3	1, 2	wcel 2110	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2110	. . 3 wff 𝐵 ∈ ℂ
6	3, 5	wa 399	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7		caddc 10732	. . . 4 class +
8	1, 4, 7	co 7213	. . 3 class (𝐴 + 𝐵)
9	8, 2	wcel 2110	. 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  10811