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Axiom ax-addcl 10034
Description: Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 10010. Proofs should normally use addcl 10056 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 9972 . . . 4 class
31, 2wcel 2030 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 2030 . . 3 wff 𝐵 ∈ ℂ
63, 5wa 383 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7 caddc 9977 . . . 4 class +
81, 4, 7co 6690 . . 3 class (𝐴 + 𝐵)
98, 2wcel 2030 . 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  10056
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