Theorem mulsne0bd

Description: The product of two non-zero surreals is non-zero. (Contributed by Scott Fenton, 16-Apr-2025)

	Ref	Expression
Hypotheses	muls0ord.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	muls0ord.2	⊢ ( 𝜑 → 𝐵 ∈ No )
Assertion	mulsne0bd	⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐵 ) ≠ 0s ↔ ( 𝐴 ≠ 0s ∧ 𝐵 ≠ 0s ) ) )

Proof

Step	Hyp	Ref	Expression
1		muls0ord.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		muls0ord.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3	1 2	muls0ord	⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐵 ) = 0s ↔ ( 𝐴 = 0s ∨ 𝐵 = 0s ) ) )
4	3	necon3abid	⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐵 ) ≠ 0s ↔ ¬ ( 𝐴 = 0s ∨ 𝐵 = 0s ) ) )
5		neanior	⊢ ( ( 𝐴 ≠ 0s ∧ 𝐵 ≠ 0s ) ↔ ¬ ( 𝐴 = 0s ∨ 𝐵 = 0s ) )
6	4 5	bitr4di	⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐵 ) ≠ 0s ↔ ( 𝐴 ≠ 0s ∧ 𝐵 ≠ 0s ) ) )