Theorem cntzcmnss

Description: Any subset in a commutative monoid is a subset of its centralizer. (Contributed by AV, 12-Jan-2019)

	Ref	Expression
Hypotheses	cntzcmnss.b	$$⊢B={\mathrm{Base}}_G$$
	cntzcmnss.z	$$⊢Z=\mathrm{Cntz}\left(G\right)$$
Assertion	cntzcmnss	$$⊢\left(G\in \mathrm{CMnd}\wedge S\subseteq B\right)\to S\subseteq Z\left(S\right)$$

Proof

Step	Hyp	Ref	Expression
1		cntzcmnss.b	$$⊢B={\mathrm{Base}}_G$$
2		cntzcmnss.z	$$⊢Z=\mathrm{Cntz}\left(G\right)$$
3	1 2	cntzcmn	$$⊢\left(G\in \mathrm{CMnd}\wedge S\subseteq B\right)\to Z\left(S\right)=B$$
4		sseq2	$$⊢B=Z\left(S\right)\to \left(S\subseteq B\leftrightarrow S\subseteq Z\left(S\right)\right)$$
5	4	eqcoms	$$⊢Z\left(S\right)=B\to \left(S\subseteq B\leftrightarrow S\subseteq Z\left(S\right)\right)$$
6	5	biimpd	$$⊢Z\left(S\right)=B\to \left(S\subseteq B\to S\subseteq Z\left(S\right)\right)$$
7	6	adantld	$$⊢Z\left(S\right)=B\to \left(\left(G\in \mathrm{CMnd}\wedge S\subseteq B\right)\to S\subseteq Z\left(S\right)\right)$$
8	3 7	mpcom	$$⊢\left(G\in \mathrm{CMnd}\wedge S\subseteq B\right)\to S\subseteq Z\left(S\right)$$