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 e. CC
2		acosval	|- ( 1 e. CC -> ( arccos ` 1 ) = ( ( _pi / 2 ) - ( arcsin ` 1 ) ) )
3	1 2	ax-mp	|- ( arccos ` 1 ) = ( ( _pi / 2 ) - ( arcsin ` 1 ) )
4		asin1	|- ( arcsin ` 1 ) = ( _pi / 2 )
5	4	oveq2i	|- ( ( _pi / 2 ) - ( arcsin ` 1 ) ) = ( ( _pi / 2 ) - ( _pi / 2 ) )
6		picn	|- _pi e. CC
7		halfcl	|- ( _pi e. CC -> ( _pi / 2 ) e. CC )
8	6 7	ax-mp	|- ( _pi / 2 ) e. CC
9	8	subidi	|- ( ( _pi / 2 ) - ( _pi / 2 ) ) = 0
10	3 5 9	3eqtri	|- ( arccos ` 1 ) = 0