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Metamath Proof Explorer

Theorem subrgmcl

Description: A subring is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014) (Proof shortened by AV, 30-Mar-2025)

Ref Expression
Hypothesis subrgmcl.p ⊢ · = ( .r ‘ 𝑅 )
Assertion subrgmcl ( ( 𝐴 ∈ ( SubRing ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 · 𝑌 ) ∈ 𝐴 )

Proof

Step Hyp Ref Expression
1 subrgmcl.p ⊢ · = ( .r ‘ 𝑅 )
2 subrgsubrng ⊢ ( 𝐴 ∈ ( SubRing ‘ 𝑅 ) → 𝐴 ∈ ( SubRng ‘ 𝑅 ) )
3 1 subrngmcl ⊢ ( ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 · 𝑌 ) ∈ 𝐴 )
4 2 3 syl3an1 ⊢ ( ( 𝐴 ∈ ( SubRing ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 · 𝑌 ) ∈ 𝐴 )