Metamath Proof Explorer

Theorem abssinbd

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

Ref Expression
Assertion abssinbd ⊢ A ∈ ℝ → sin ⁡ A ≤ 1

Proof

Step Hyp Ref Expression
1 sinbnd ⊢ A ∈ ℝ → − 1 ≤ sin ⁡ A ∧ sin ⁡ A ≤ 1
2 resincl ⊢ A ∈ ℝ → sin ⁡ A ∈ ℝ
3 1red ⊢ A ∈ ℝ → 1 ∈ ℝ
4 2 3 absled ⊢ A ∈ ℝ → sin ⁡ A ≤ 1 ↔ − 1 ≤ sin ⁡ A ∧ sin ⁡ A ≤ 1
5 1 4 mpbird ⊢ A ∈ ℝ → sin ⁡ A ≤ 1