Theorem msq0i

Description: A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by NM, 28-Jul-1999)

	Ref	Expression
Hypothesis	mul0or.1	⊢ 𝐴 ∈ ℂ
Assertion	msq0i	⊢ ( ( 𝐴 · 𝐴 ) = 0 ↔ 𝐴 = 0 )

Proof

Step	Hyp	Ref	Expression
1		mul0or.1	⊢ 𝐴 ∈ ℂ
2		mul0or	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( ( 𝐴 · 𝐴 ) = 0 ↔ ( 𝐴 = 0 ∨ 𝐴 = 0 ) ) )
3	1 1 2	mp2an	⊢ ( ( 𝐴 · 𝐴 ) = 0 ↔ ( 𝐴 = 0 ∨ 𝐴 = 0 ) )
4		oridm	⊢ ( ( 𝐴 = 0 ∨ 𝐴 = 0 ) ↔ 𝐴 = 0 )
5	3 4	bitri	⊢ ( ( 𝐴 · 𝐴 ) = 0 ↔ 𝐴 = 0 )