Theorem subm0cl

Description: Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015)

	Ref	Expression
Hypothesis	subm0cl.z	⊢ 0 = ( 0g ‘ 𝑀 )
Assertion	subm0cl	⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) → 0 ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		subm0cl.z	⊢ 0 = ( 0g ‘ 𝑀 )
2		submrcl	⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) → 𝑀 ∈ Mnd )
3		eqid	⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 )
4		eqid	⊢ ( 𝑀 ↾s 𝑆 ) = ( 𝑀 ↾s 𝑆 )
5	3 1 4	issubm2	⊢ ( 𝑀 ∈ Mnd → ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) ↔ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ 0 ∈ 𝑆 ∧ ( 𝑀 ↾s 𝑆 ) ∈ Mnd ) ) )
6	2 5	syl	⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) → ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) ↔ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ 0 ∈ 𝑆 ∧ ( 𝑀 ↾s 𝑆 ) ∈ Mnd ) ) )
7	6	ibi	⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) → ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ 0 ∈ 𝑆 ∧ ( 𝑀 ↾s 𝑆 ) ∈ Mnd ) )
8	7	simp2d	⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) → 0 ∈ 𝑆 )