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Metamath Proof Explorer

Theorem ringacl

Description: Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014)

Ref Expression
Hypotheses ringacl.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
ringacl.p ⊢ + = ( +g ‘ 𝑅 )
Assertion ringacl ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 ringacl.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 ringacl.p ⊢ + = ( +g ‘ 𝑅 )
3 ringgrp ⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp )
4 1 2 grpcl ⊢ ( ( 𝑅 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )
5 3 4 syl3an1 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )