Theorem mul0ord

Description: If a product is zero, one of its factors must be zero. Theorem I.11 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	msq0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	mul0ord.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	mul0ord	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) = 0 ↔ ( 𝐴 = 0 ∨ 𝐵 = 0 ) ) )

Proof

Step	Hyp	Ref	Expression
1		msq0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		mul0ord.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		mul0or	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) = 0 ↔ ( 𝐴 = 0 ∨ 𝐵 = 0 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) = 0 ↔ ( 𝐴 = 0 ∨ 𝐵 = 0 ) ) )