Theorem grpmnd

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

	Ref	Expression
Assertion	grpmnd	$$⊢G\in \mathrm{Grp}\to G\in \mathrm{Mnd}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢{\mathrm{Base}}_G={\mathrm{Base}}_G$$
2		eqid	$$⊢{+}_G={+}_G$$
3		eqid	$$⊢0_G=0_G$$
4	1 2 3	isgrp	$$⊢G\in \mathrm{Grp}\leftrightarrow \left(G\in \mathrm{Mnd}\wedge \forall a\in {\mathrm{Base}}_G\exists m\in {\mathrm{Base}}_{G}m{+}_{G}a=0_G\right)$$
5	4	simplbi	$$⊢G\in \mathrm{Grp}\to G\in \mathrm{Mnd}$$