Theorem sigarid

Description: Signed area of a flat parallelogram is zero. (Contributed by Saveliy Skresanov, 20-Sep-2017)

	Ref	Expression
Hypothesis	sigar	|- G = ( x e. CC , y e. CC |-> ( Im ` ( ( * ` x ) x. y ) ) )
Assertion	sigarid	|- ( A e. CC -> ( A G A ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		sigar	|- G = ( x e. CC , y e. CC |-> ( Im ` ( ( * ` x ) x. y ) ) )
2	1	sigarval	|- ( ( A e. CC /\ A e. CC ) -> ( A G A ) = ( Im ` ( ( * ` A ) x. A ) ) )
3	2	anidms	|- ( A e. CC -> ( A G A ) = ( Im ` ( ( * ` A ) x. A ) ) )
4		cjcl	|- ( A e. CC -> ( * ` A ) e. CC )
5		id	|- ( A e. CC -> A e. CC )
6	4 5	mulcomd	|- ( A e. CC -> ( ( * ` A ) x. A ) = ( A x. ( * ` A ) ) )
7		cjmulrcl	|- ( A e. CC -> ( A x. ( * ` A ) ) e. RR )
8	6 7	eqeltrd	|- ( A e. CC -> ( ( * ` A ) x. A ) e. RR )
9	8	reim0d	|- ( A e. CC -> ( Im ` ( ( * ` A ) x. A ) ) = 0 )
10	3 9	eqtrd	|- ( A e. CC -> ( A G A ) = 0 )