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Metamath Proof Explorer

Theorem rngacl

Description: Closure of the addition operation of a non-unital ring. (Contributed by AV, 16-Feb-2025)

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

Proof

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