This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Axiom ax-addcl

Description: Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by Theorem axaddcl . Proofs should normally use addcl instead, which asserts the same thing but follows our naming conventions for closures. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994)

		Ref	Expression
	Assertion	ax-addcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) ∈ ℂ )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cA	⊢ 𝐴
1		cc	⊢ ℂ
2	0 1	wcel	⊢ 𝐴 ∈ ℂ
3		cB	⊢ 𝐵
4	3 1	wcel	⊢ 𝐵 ∈ ℂ
5	2 4	wa	⊢ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ )
6		caddc	⊢ +
7	0 3 6	co	⊢ ( 𝐴 + 𝐵 )
8	7 1	wcel	⊢ ( 𝐴 + 𝐵 ) ∈ ℂ
9	5 8	wi	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) ∈ ℂ )