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Axiom ax-addcl 7739
Description: Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by theorem axaddcl 7695. Proofs should normally use addcl 7768 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 7641 . . . 4 class
31, 2wcel 1481 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1481 . . 3 wff 𝐵 ∈ ℂ
63, 5wa 103 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7 caddc 7646 . . . 4 class +
81, 4, 7co 5781 . . 3 class (𝐴 + 𝐵)
98, 2wcel 1481 . 2 wff (𝐴 + 𝐵) ∈ ℂ
106, 9wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
Colors of variables: wff set class
This axiom is referenced by:  addcl  7768
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