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Metamath Proof Explorer

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 )