Theorem sincos1sgn

Description: The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008)

	Ref	Expression
Assertion	sincos1sgn	⊢ ( 0 < ( sin ‘ 1 ) ∧ 0 < ( cos ‘ 1 ) )

Proof

Step	Hyp	Ref	Expression
1		1re	⊢ 1 ∈ ℝ
2		0lt1	⊢ 0 < 1
3		1le1	⊢ 1 ≤ 1
4		0xr	⊢ 0 ∈ ℝ*
5		elioc2	⊢ ( ( 0 ∈ ℝ* ∧ 1 ∈ ℝ ) → ( 1 ∈ ( 0 (,] 1 ) ↔ ( 1 ∈ ℝ ∧ 0 < 1 ∧ 1 ≤ 1 ) ) )
6	4 1 5	mp2an	⊢ ( 1 ∈ ( 0 (,] 1 ) ↔ ( 1 ∈ ℝ ∧ 0 < 1 ∧ 1 ≤ 1 ) )
7	1 2 3 6	mpbir3an	⊢ 1 ∈ ( 0 (,] 1 )
8		sin01gt0	⊢ ( 1 ∈ ( 0 (,] 1 ) → 0 < ( sin ‘ 1 ) )
9		cos01gt0	⊢ ( 1 ∈ ( 0 (,] 1 ) → 0 < ( cos ‘ 1 ) )
10	8 9	jca	⊢ ( 1 ∈ ( 0 (,] 1 ) → ( 0 < ( sin ‘ 1 ) ∧ 0 < ( cos ‘ 1 ) ) )
11	7 10	ax-mp	⊢ ( 0 < ( sin ‘ 1 ) ∧ 0 < ( cos ‘ 1 ) )