Theorem submid

Description: Every monoid is trivially a submonoid of itself. (Contributed by Stefan O'Rear, 15-Aug-2015)

	Ref	Expression
Hypothesis	submss.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
Assertion	submid	⊢ ( 𝑀 ∈ Mnd → 𝐵 ∈ ( SubMnd ‘ 𝑀 ) )

Proof

Step	Hyp	Ref	Expression
1		submss.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
2		ssidd	⊢ ( 𝑀 ∈ Mnd → 𝐵 ⊆ 𝐵 )
3		eqid	⊢ ( 0g ‘ 𝑀 ) = ( 0g ‘ 𝑀 )
4	1 3	mndidcl	⊢ ( 𝑀 ∈ Mnd → ( 0g ‘ 𝑀 ) ∈ 𝐵 )
5	1	ressid	⊢ ( 𝑀 ∈ Mnd → ( 𝑀 ↾s 𝐵 ) = 𝑀 )
6		id	⊢ ( 𝑀 ∈ Mnd → 𝑀 ∈ Mnd )
7	5 6	eqeltrd	⊢ ( 𝑀 ∈ Mnd → ( 𝑀 ↾s 𝐵 ) ∈ Mnd )
8		eqid	⊢ ( 𝑀 ↾s 𝐵 ) = ( 𝑀 ↾s 𝐵 )
9	1 3 8	issubm2	⊢ ( 𝑀 ∈ Mnd → ( 𝐵 ∈ ( SubMnd ‘ 𝑀 ) ↔ ( 𝐵 ⊆ 𝐵 ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ∧ ( 𝑀 ↾s 𝐵 ) ∈ Mnd ) ) )
10	2 4 7 9	mpbir3and	⊢ ( 𝑀 ∈ Mnd → 𝐵 ∈ ( SubMnd ‘ 𝑀 ) )