Theorem bru2

Description: The basic 2nd root of unity is -u 1 . (Contributed by AV, 24-Oct-2026)

	Ref	Expression
Assertion	bru2	|- ( bRoU ` 2 ) = -u 1

Proof

Step	Hyp	Ref	Expression
1		2nn	|- 2 e. NN
2		bruval2	|- ( 2 e. NN -> ( bRoU ` 2 ) = ( -u 1 ^c ( 2 / 2 ) ) )
3	1 2	ax-mp	|- ( bRoU ` 2 ) = ( -u 1 ^c ( 2 / 2 ) )
4		2div2e1	|- ( 2 / 2 ) = 1
5	4	oveq2i	|- ( -u 1 ^c ( 2 / 2 ) ) = ( -u 1 ^c 1 )
6		neg1cn	|- -u 1 e. CC
7		cxp1	|- ( -u 1 e. CC -> ( -u 1 ^c 1 ) = -u 1 )
8	6 7	ax-mp	|- ( -u 1 ^c 1 ) = -u 1
9	3 5 8	3eqtri	|- ( bRoU ` 2 ) = -u 1