Theorem subm0

Description: Submonoids have the same identity. (Contributed by Mario Carneiro, 7-Mar-2015)

	Ref	Expression
Hypotheses	submmnd.h	$$⊢H=M{\upharpoonright }_{𝑠}S$$
	subm0.z	$$⊢\underset{\dot{}}{0}=0_M$$
Assertion	subm0	$$⊢S\in \mathrm{SubMnd}\left(M\right)\to \underset{\dot{}}{0}=0_H$$

Proof

Step	Hyp	Ref	Expression
1		submmnd.h	$$⊢H=M{\upharpoonright }_{𝑠}S$$
2		subm0.z	$$⊢\underset{\dot{}}{0}=0_M$$
3		submrcl	$$⊢S\in \mathrm{SubMnd}\left(M\right)\to M\in \mathrm{Mnd}$$
4	1	submmnd	$$⊢S\in \mathrm{SubMnd}\left(M\right)\to H\in \mathrm{Mnd}$$
5		eqid	$$⊢{\mathrm{Base}}_M={\mathrm{Base}}_M$$
6	5	submss	$$⊢S\in \mathrm{SubMnd}\left(M\right)\to S\subseteq {\mathrm{Base}}_M$$
7	2	subm0cl	$$⊢S\in \mathrm{SubMnd}\left(M\right)\to \underset{\dot{}}{0}\in S$$
8	5 2 1	submnd0	$$⊢\left(\left(M\in \mathrm{Mnd}\wedge H\in \mathrm{Mnd}\right)\wedge \left(S\subseteq {\mathrm{Base}}_M\wedge \underset{\dot{}}{0}\in S\right)\right)\to \underset{\dot{}}{0}=0_H$$
9	3 4 6 7 8	syl22anc	$$⊢S\in \mathrm{SubMnd}\left(M\right)\to \underset{\dot{}}{0}=0_H$$