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 )