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Metamath Proof Explorer

Theorem axaddcl

Description: Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcl be used later. Instead, in most cases use addcl . (Contributed by NM, 14-Jun-1995) (New usage is discouraged.)

		Ref	Expression
	Assertion	axaddcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) ∈ ℂ )

Proof

Step	Hyp	Ref	Expression
1		axaddf	⊢ + : ( ℂ × ℂ ) ⟶ ℂ
2	1	fovcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) ∈ ℂ )