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Axiom ax-addcl 11204
Description: Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by Theorem axaddcl 11180. Proofs should normally use addcl 11226 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
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cc 11142 . . . 4 class
31, 2wcel 2098 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 2098 . . 3 wff 𝐵 ∈ ℂ
63, 5wa 394 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7 caddc 11147 . . . 4 class +
81, 4, 7co 7424 . . 3 class (𝐴 + 𝐵)
98, 2wcel 2098 . 2 wff (𝐴 + 𝐵) ∈ ℂ
106, 9wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
Colors of variables: wff setvar class
This axiom is referenced by:  addcl  11226
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