Theorem nmulcld

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

	Ref	Expression
Hypotheses	nmulcld.1	|- ( ph -> A e. On )
	nmulcld.2	|- ( ph -> B e. On )
Assertion	nmulcld	|- ( ph -> ( A .no B ) e. On )

Proof

Step	Hyp	Ref	Expression
1		nmulcld.1	|- ( ph -> A e. On )
2		nmulcld.2	|- ( ph -> B e. On )
3		nmulcl	|- ( ( A e. On /\ B e. On ) -> ( A .no B ) e. On )
4	1 2 3	syl2anc	|- ( ph -> ( A .no B ) e. On )