Theorem abscosbd

Description: Bound for the absolute value of the cosine of a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	abscosbd	⊢ ( 𝐴 ∈ ℝ → ( abs ‘ ( cos ‘ 𝐴 ) ) ≤ 1 )

Proof

Step	Hyp	Ref	Expression
1		cosbnd	⊢ ( 𝐴 ∈ ℝ → ( - 1 ≤ ( cos ‘ 𝐴 ) ∧ ( cos ‘ 𝐴 ) ≤ 1 ) )
2		recoscl	⊢ ( 𝐴 ∈ ℝ → ( cos ‘ 𝐴 ) ∈ ℝ )
3		1red	⊢ ( 𝐴 ∈ ℝ → 1 ∈ ℝ )
4	2 3	absled	⊢ ( 𝐴 ∈ ℝ → ( ( abs ‘ ( cos ‘ 𝐴 ) ) ≤ 1 ↔ ( - 1 ≤ ( cos ‘ 𝐴 ) ∧ ( cos ‘ 𝐴 ) ≤ 1 ) ) )
5	1 4	mpbird	⊢ ( 𝐴 ∈ ℝ → ( abs ‘ ( cos ‘ 𝐴 ) ) ≤ 1 )