Theorem msq0d

Description: A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypothesis	msq0d.1	|- ( ph -> A e. CC )
Assertion	msq0d	|- ( ph -> ( ( A x. A ) = 0 <-> A = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		msq0d.1	|- ( ph -> A e. CC )
2		mul0or	|- ( ( A e. CC /\ A e. CC ) -> ( ( A x. A ) = 0 <-> ( A = 0 \/ A = 0 ) ) )
3	1 1 2	syl2anc	|- ( ph -> ( ( A x. A ) = 0 <-> ( A = 0 \/ A = 0 ) ) )
4		oridm	|- ( ( A = 0 \/ A = 0 ) <-> A = 0 )
5	3 4	bitrdi	|- ( ph -> ( ( A x. A ) = 0 <-> A = 0 ) )