Theorem ringcld

Description: Closure of the multiplication operation of a ring. (Contributed by SN, 29-Jul-2024)

	Ref	Expression
Hypotheses	ringcld.b	|- B = ( Base ` R )
	ringcld.t	|- .x. = ( .r ` R )
	ringcld.r	|- ( ph -> R e. Ring )
	ringcld.x	|- ( ph -> X e. B )
	ringcld.y	|- ( ph -> Y e. B )
Assertion	ringcld	|- ( ph -> ( X .x. Y ) e. B )

Proof

Step	Hyp	Ref	Expression
1		ringcld.b	|- B = ( Base ` R )
2		ringcld.t	|- .x. = ( .r ` R )
3		ringcld.r	|- ( ph -> R e. Ring )
4		ringcld.x	|- ( ph -> X e. B )
5		ringcld.y	|- ( ph -> Y e. B )
6	1 2	ringcl	|- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
7	3 4 5 6	syl3anc	|- ( ph -> ( X .x. Y ) e. B )