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Metamath Proof Explorer

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 )