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	⊢ · = ( .r ‘ 𝑅 )
	ringcld.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
	ringcld.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	ringcld.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	ringcld	⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ringcld.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		ringcld.t	⊢ · = ( .r ‘ 𝑅 )
3		ringcld.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
4		ringcld.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		ringcld.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6	1 2	ringcl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 · 𝑌 ) ∈ 𝐵 )
7	3 4 5 6	syl3anc	⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) ∈ 𝐵 )