Theorem cmnmnd

Description: A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015)

	Ref	Expression
Assertion	cmnmnd	$$⊢G\in \mathrm{CMnd}\to G\in \mathrm{Mnd}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢{\mathrm{Base}}_G={\mathrm{Base}}_G$$
2		eqid	$$⊢{+}_G={+}_G$$
3	1 2	iscmn	$$⊢G\in \mathrm{CMnd}\leftrightarrow \left(G\in \mathrm{Mnd}\wedge \forall x\in {\mathrm{Base}}_G\forall y\in {\mathrm{Base}}_{G}x{+}_{G}y=y{+}_{G}x\right)$$
4	3	simplbi	$$⊢G\in \mathrm{CMnd}\to G\in \mathrm{Mnd}$$