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Metamath Proof Explorer

Theorem submgmcl

Description: Submagmas are closed under the magma operation. (Contributed by AV, 26-Feb-2020)

Ref Expression
Hypothesis submgmcl.p ⊢ + = ( +g ‘ 𝑀 )
Assertion submgmcl ( ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) → ( 𝑋 + 𝑌 ) ∈ 𝑆 )

Proof

Step Hyp Ref Expression
1 submgmcl.p ⊢ + = ( +g ‘ 𝑀 )
2 submgmrcl ⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → 𝑀 ∈ Mgm )
3 eqid ⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 )
4 3 1 issubmgm ⊢ ( 𝑀 ∈ Mgm → ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ↔ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) ) )
5 2 4 syl ⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ↔ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) ) )
6 5 ibi ⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) )
7 6 simprd ⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 )
8 ovrspc2v ⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) → ( 𝑋 + 𝑌 ) ∈ 𝑆 )
9 7 8 sylan2 ⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) ∧ 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ) → ( 𝑋 + 𝑌 ) ∈ 𝑆 )
10 9 ancoms ⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) ) → ( 𝑋 + 𝑌 ) ∈ 𝑆 )
11 10 3impb ⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) → ( 𝑋 + 𝑌 ) ∈ 𝑆 )