Theorem sqval

Description: Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004)

	Ref	Expression
Assertion	sqval	|- ( A e. CC -> ( A ^ 2 ) = ( A x. A ) )

Proof

Step	Hyp	Ref	Expression
1		df-2	|- 2 = ( 1 + 1 )
2	1	oveq2i	|- ( A ^ 2 ) = ( A ^ ( 1 + 1 ) )
3		1nn0	|- 1 e. NN0
4		expp1	|- ( ( A e. CC /\ 1 e. NN0 ) -> ( A ^ ( 1 + 1 ) ) = ( ( A ^ 1 ) x. A ) )
5	3 4	mpan2	|- ( A e. CC -> ( A ^ ( 1 + 1 ) ) = ( ( A ^ 1 ) x. A ) )
6	2 5	syl5eq	|- ( A e. CC -> ( A ^ 2 ) = ( ( A ^ 1 ) x. A ) )
7		exp1	|- ( A e. CC -> ( A ^ 1 ) = A )
8	7	oveq1d	|- ( A e. CC -> ( ( A ^ 1 ) x. A ) = ( A x. A ) )
9	6 8	eqtrd	|- ( A e. CC -> ( A ^ 2 ) = ( A x. A ) )