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Axiom ax-addcl 9981
Description: Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 9957. Proofs should normally use addcl 10003 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 9919 . . . 4 class
31, 2wcel 1988 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1988 . . 3 wff 𝐵 ∈ ℂ
63, 5wa 384 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7 caddc 9924 . . . 4 class +
81, 4, 7co 6635 . . 3 class (𝐴 + 𝐵)
98, 2wcel 1988 . 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  10003
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