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Metamath Proof Explorer

Theorem mndcl

Description: Closure of the operation of a monoid. (Contributed by NM, 14-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015) (Proof shortened by AV, 8-Feb-2020)

Ref Expression
Hypotheses mndcl.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
mndcl.p ⊢ + = ( +g ‘ 𝐺 )
Assertion mndcl ( ( 𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 mndcl.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 mndcl.p ⊢ + = ( +g ‘ 𝐺 )
3 mndmgm ⊢ ( 𝐺 ∈ Mnd → 𝐺 ∈ Mgm )
4 1 2 mgmcl ⊢ ( ( 𝐺 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )
5 3 4 syl3an1 ⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )