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Metamath Proof Explorer

Theorem atancl

Description: Closure for the arctan function. (Contributed by Mario Carneiro, 31-Mar-2015)

		Ref	Expression
	Assertion	atancl	$⊢ A \in dom arctan \to arctan (A) \in ℂ$

Step	Hyp	Ref	Expression
1		atanf	$⊢ arctan : ℂ ∖ \{- i, i\} ⟶ ℂ$
2	1	ffvelcdmi	$⊢ A \in (ℂ ∖ \{- i, i\}) \to arctan (A) \in ℂ$
3	1	fdmi	$⊢ dom arctan = ℂ ∖ \{- i, i\}$
4	2 3	eleq2s	$⊢ A \in dom arctan \to arctan (A) \in ℂ$