Theorem bru3sqbru

Description: The square of the basic 3rd root of unity expressed by the basic 3rd root of unity. (Contributed by AV, 24-Oct-2026)

	Ref	Expression
Hypothesis	bru3eq.w	|- W = ( bRoU ` 3 )
Assertion	bru3sqbru	|- ( W ^ 2 ) = ( -u 1 - W )

Proof

Step	Hyp	Ref	Expression
1		bru3eq.w	|- W = ( bRoU ` 3 )
2		ax-1cn	|- 1 e. CC
3		3nn	|- 3 e. NN
4		brucn	|- ( 3 e. NN -> ( bRoU ` 3 ) e. CC )
5	3 4	ax-mp	|- ( bRoU ` 3 ) e. CC
6	1 5	eqeltri	|- W e. CC
7	2 6	addcomi	|- ( 1 + W ) = ( W + 1 )
8	7	oveq2i	|- ( ( W ^ 2 ) + ( 1 + W ) ) = ( ( W ^ 2 ) + ( W + 1 ) )
9	6	sqcli	|- ( W ^ 2 ) e. CC
10	9 6 2	addassi	|- ( ( ( W ^ 2 ) + W ) + 1 ) = ( ( W ^ 2 ) + ( W + 1 ) )
11	1	bru3eq	|- ( ( ( W ^ 2 ) + W ) + 1 ) = 0
12	8 10 11	3eqtr2i	|- ( ( W ^ 2 ) + ( 1 + W ) ) = 0
13	2 6	addcli	|- ( 1 + W ) e. CC
14		addeq0	|- ( ( ( W ^ 2 ) e. CC /\ ( 1 + W ) e. CC ) -> ( ( ( W ^ 2 ) + ( 1 + W ) ) = 0 <-> ( W ^ 2 ) = -u ( 1 + W ) ) )
15	9 13 14	mp2an	|- ( ( ( W ^ 2 ) + ( 1 + W ) ) = 0 <-> ( W ^ 2 ) = -u ( 1 + W ) )
16	12 15	mpbi	|- ( W ^ 2 ) = -u ( 1 + W )
17		negdi2	|- ( ( 1 e. CC /\ W e. CC ) -> -u ( 1 + W ) = ( -u 1 - W ) )
18	2 6 17	mp2an	|- -u ( 1 + W ) = ( -u 1 - W )
19	16 18	eqtri	|- ( W ^ 2 ) = ( -u 1 - W )