Theorem ringcl

Description: Closure of the multiplication operation of a ring. (Contributed by NM, 26-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015)

	Ref	Expression
Hypotheses	ringcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	ringcl.t	⊢ · = ( .r ‘ 𝑅 )
Assertion	ringcl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 · 𝑌 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ringcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		ringcl.t	⊢ · = ( .r ‘ 𝑅 )
3		eqid	⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
4	3	ringmgp	⊢ ( 𝑅 ∈ Ring → ( mulGrp ‘ 𝑅 ) ∈ Mnd )
5	3 1	mgpbas	⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) )
6	3 2	mgpplusg	⊢ · = ( +g ‘ ( mulGrp ‘ 𝑅 ) )
7	5 6	mndcl	⊢ ( ( ( mulGrp ‘ 𝑅 ) ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 · 𝑌 ) ∈ 𝐵 )
8	4 7	syl3an1	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 · 𝑌 ) ∈ 𝐵 )