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Metamath Proof Explorer

Theorem asinrecl

Description: The arcsine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015)

		Ref	Expression
	Assertion	asinrecl	$⊢ A \in [- 1, 1] \to arcsin (A) \in ℝ$

Step	Hyp	Ref	Expression
1		halfpire	$⊢ \frac{π}{2} \in ℝ$
2	1	renegcli	$⊢ - \frac{π}{2} \in ℝ$
3		iccssre	$⊢ (- \frac{π}{2} \in ℝ \land \frac{π}{2} \in ℝ) \to [- \frac{π}{2}, \frac{π}{2}] \subseteq ℝ$
4	2 1 3	mp2an	$⊢ [- \frac{π}{2}, \frac{π}{2}] \subseteq ℝ$
5		asinrebnd	$⊢ A \in [- 1, 1] \to arcsin (A) \in [- \frac{π}{2}, \frac{π}{2}]$
6	4 5	sselid	$⊢ A \in [- 1, 1] \to arcsin (A) \in ℝ$