Theorem sqabsaddi

Description: Square of absolute value of sum. Proposition 10-3.7(g) of Gleason p. 133. (Contributed by NM, 2-Oct-1999)

	Ref	Expression
Hypotheses	absvalsqi.1	|- A e. CC
	abssub.2	|- B e. CC
Assertion	sqabsaddi	|- ( ( abs ` ( A + B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) + ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		absvalsqi.1	|- A e. CC
2		abssub.2	|- B e. CC
3		sqabsadd	|- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A + B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) + ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) )
4	1 2 3	mp2an	|- ( ( abs ` ( A + B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) + ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) )