Theorem acos1

Description: The arccosine of 1 is 0 . (Contributed by Mario Carneiro, 2-Apr-2015)

	Ref	Expression
Assertion	acos1	⊢ ( arccos ‘ 1 ) = 0

Proof

Step	Hyp	Ref	Expression
1		ax-1cn	⊢ 1 ∈ ℂ
2		acosval	⊢ ( 1 ∈ ℂ → ( arccos ‘ 1 ) = ( ( π / 2 ) − ( arcsin ‘ 1 ) ) )
3	1 2	ax-mp	⊢ ( arccos ‘ 1 ) = ( ( π / 2 ) − ( arcsin ‘ 1 ) )
4		asin1	⊢ ( arcsin ‘ 1 ) = ( π / 2 )
5	4	oveq2i	⊢ ( ( π / 2 ) − ( arcsin ‘ 1 ) ) = ( ( π / 2 ) − ( π / 2 ) )
6		picn	⊢ π ∈ ℂ
7		halfcl	⊢ ( π ∈ ℂ → ( π / 2 ) ∈ ℂ )
8	6 7	ax-mp	⊢ ( π / 2 ) ∈ ℂ
9	8	subidi	⊢ ( ( π / 2 ) − ( π / 2 ) ) = 0
10	3 5 9	3eqtri	⊢ ( arccos ‘ 1 ) = 0