Theorem retancl

Description: The closure of the tangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014)

	Ref	Expression
Assertion	retancl	⊢ ( ( 𝐴 ∈ ℝ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( tan ‘ 𝐴 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
2		tanval	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( tan ‘ 𝐴 ) = ( ( sin ‘ 𝐴 ) / ( cos ‘ 𝐴 ) ) )
3	1 2	sylan	⊢ ( ( 𝐴 ∈ ℝ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( tan ‘ 𝐴 ) = ( ( sin ‘ 𝐴 ) / ( cos ‘ 𝐴 ) ) )
4		recoscl	⊢ ( 𝐴 ∈ ℝ → ( cos ‘ 𝐴 ) ∈ ℝ )
5		resincl	⊢ ( 𝐴 ∈ ℝ → ( sin ‘ 𝐴 ) ∈ ℝ )
6		redivcl	⊢ ( ( ( sin ‘ 𝐴 ) ∈ ℝ ∧ ( cos ‘ 𝐴 ) ∈ ℝ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( sin ‘ 𝐴 ) / ( cos ‘ 𝐴 ) ) ∈ ℝ )
7	5 6	syl3an1	⊢ ( ( 𝐴 ∈ ℝ ∧ ( cos ‘ 𝐴 ) ∈ ℝ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( sin ‘ 𝐴 ) / ( cos ‘ 𝐴 ) ) ∈ ℝ )
8	4 7	syl3an2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( sin ‘ 𝐴 ) / ( cos ‘ 𝐴 ) ) ∈ ℝ )
9	8	3anidm12	⊢ ( ( 𝐴 ∈ ℝ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( sin ‘ 𝐴 ) / ( cos ‘ 𝐴 ) ) ∈ ℝ )
10	3 9	eqeltrd	⊢ ( ( 𝐴 ∈ ℝ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( tan ‘ 𝐴 ) ∈ ℝ )