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Metamath Proof Explorer

Theorem mndidcl

Description: The identity element of a monoid belongs to the monoid. (Contributed by NM, 27-Aug-2011) (Revised by Mario Carneiro, 27-Dec-2014)

Ref Expression
Hypotheses mndidcl.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
mndidcl.o ⊢ 0 = ( 0g ‘ 𝐺 )
Assertion mndidcl ( 𝐺 ∈ Mnd → 0 ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 mndidcl.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 mndidcl.o ⊢ 0 = ( 0g ‘ 𝐺 )
3 eqid ⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
4 1 3 mndid ⊢ ( 𝐺 ∈ Mnd → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = 𝑦 ) )
5 1 2 3 4 mgmidcl ⊢ ( 𝐺 ∈ Mnd → 0 ∈ 𝐵 )