Theorem mul0ori

Description: If a product is zero, one of its factors must be zero. Theorem I.11 of Apostol p. 18. (Contributed by NM, 7-Oct-1999)

	Ref	Expression
Hypotheses	mul0or.1	⊢ 𝐴 ∈ ℂ
	mul0or.2	⊢ 𝐵 ∈ ℂ
Assertion	mul0ori	⊢ ( ( 𝐴 · 𝐵 ) = 0 ↔ ( 𝐴 = 0 ∨ 𝐵 = 0 ) )

Proof

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