Theorem nmulcomd

Description: Natural multiplication commutes. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026)

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

Proof

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