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Metamath Proof Explorer

Theorem reccl

Description: Closure law for reciprocal. (Contributed by NM, 30-Apr-2005)

		Ref	Expression
	Assertion	reccl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / 𝐴 ) ∈ ℂ )

Step	Hyp	Ref	Expression
1		ax-1cn	⊢ 1 ∈ ℂ
2		divcl	⊢ ( ( 1 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / 𝐴 ) ∈ ℂ )
3	1 2	mp3an1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / 𝐴 ) ∈ ℂ )