Theorem cntzcmnf

Description: Discharge the centralizer assumption in a commutative monoid. (Contributed by Mario Carneiro, 24-Apr-2016)

	Ref	Expression
Hypotheses	cntzcmnf.b	$$⊢B={\mathrm{Base}}_G$$
	cntzcmnf.z	$$⊢Z=\mathrm{Cntz}\left(G\right)$$
	cntzcmnf.g	$$⊢\phi \to G\in \mathrm{CMnd}$$
	cntzcmnf.f	$$⊢\phi \to F:A⟶B$$
Assertion	cntzcmnf	$$⊢\phi \to \mathrm{ran}F\subseteq Z\left(\mathrm{ran}F\right)$$

Proof

Step	Hyp	Ref	Expression
1		cntzcmnf.b	$$⊢B={\mathrm{Base}}_G$$
2		cntzcmnf.z	$$⊢Z=\mathrm{Cntz}\left(G\right)$$
3		cntzcmnf.g	$$⊢\phi \to G\in \mathrm{CMnd}$$
4		cntzcmnf.f	$$⊢\phi \to F:A⟶B$$
5	4	frnd	$$⊢\phi \to \mathrm{ran}F\subseteq B$$
6	1 2	cntzcmn	$$⊢\left(G\in \mathrm{CMnd}\wedge \mathrm{ran}F\subseteq B\right)\to Z\left(\mathrm{ran}F\right)=B$$
7	3 5 6	syl2anc	$$⊢\phi \to Z\left(\mathrm{ran}F\right)=B$$
8	5 7	sseqtrrd	$$⊢\phi \to \mathrm{ran}F\subseteq Z\left(\mathrm{ran}F\right)$$