Theorem nmulcld

Description: Closure law for natural multiplication. Deduction form. (Contributed by Scott Fenton, 12-Jun-2026)

	Ref	Expression
Hypotheses	nmulcld.1	⊢ ( 𝜑 → 𝐴 ∈ On )
	nmulcld.2	⊢ ( 𝜑 → 𝐵 ∈ On )
Assertion	nmulcld	⊢ ( 𝜑 → ( 𝐴 ·no 𝐵 ) ∈ On )

Proof

Step	Hyp	Ref	Expression
1		nmulcld.1	⊢ ( 𝜑 → 𝐴 ∈ On )
2		nmulcld.2	⊢ ( 𝜑 → 𝐵 ∈ On )
3		nmulcl	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ·no 𝐵 ) ∈ On )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ·no 𝐵 ) ∈ On )