Theorem subm0cl

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

	Ref	Expression
Hypothesis	subm0cl.z	$$⊢\underset{\dot{}}{0}=0_M$$
Assertion	subm0cl	$$⊢S\in \mathrm{SubMnd}\left(M\right)\to \underset{\dot{}}{0}\in S$$

Proof

Step	Hyp	Ref	Expression
1		subm0cl.z	$$⊢\underset{\dot{}}{0}=0_M$$
2		submrcl	$$⊢S\in \mathrm{SubMnd}\left(M\right)\to M\in \mathrm{Mnd}$$
3		eqid	$$⊢{\mathrm{Base}}_M={\mathrm{Base}}_M$$
4		eqid	$$⊢M{\upharpoonright }_{𝑠}S=M{\upharpoonright }_{𝑠}S$$
5	3 1 4	issubm2	$$⊢M\in \mathrm{Mnd}\to \left(S\in \mathrm{SubMnd}\left(M\right)\leftrightarrow \left(S\subseteq {\mathrm{Base}}_M\wedge \underset{\dot{}}{0}\in S\wedge M{\upharpoonright }_{𝑠}S\in \mathrm{Mnd}\right)\right)$$
6	2 5	syl	$$⊢S\in \mathrm{SubMnd}\left(M\right)\to \left(S\in \mathrm{SubMnd}\left(M\right)\leftrightarrow \left(S\subseteq {\mathrm{Base}}_M\wedge \underset{\dot{}}{0}\in S\wedge M{\upharpoonright }_{𝑠}S\in \mathrm{Mnd}\right)\right)$$
7	6	ibi	$$⊢S\in \mathrm{SubMnd}\left(M\right)\to \left(S\subseteq {\mathrm{Base}}_M\wedge \underset{\dot{}}{0}\in S\wedge M{\upharpoonright }_{𝑠}S\in \mathrm{Mnd}\right)$$
8	7	simp2d	$$⊢S\in \mathrm{SubMnd}\left(M\right)\to \underset{\dot{}}{0}\in S$$