Theorem gagrp

Description: The left argument of a group action is a group. (Contributed by Jeff Hankins, 11-Aug-2009) (Revised by Mario Carneiro, 30-Apr-2015)

	Ref	Expression
Assertion	gagrp	$$⊢\underset{\dot{}}{\oplus }\in \left(G\mathrm{GrpAct}Y\right)\to G\in \mathrm{Grp}$$

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	isga	$$⊢\underset{\dot{}}{\oplus }\in \left(G\mathrm{GrpAct}Y\right)\leftrightarrow \left(\left(G\in \mathrm{Grp}\wedge Y\in \mathrm{V}\right)\wedge \left(\underset{\dot{}}{\oplus }:{\mathrm{Base}}_G\times Y⟶Y\wedge \forall x\in Y\left(0_G\underset{\dot{}}{\oplus }x=x\wedge \forall y\in {\mathrm{Base}}_G\forall z\in {\mathrm{Base}}_G\left(y{+}_{G}z\right)\underset{\dot{}}{\oplus }x=y\underset{\dot{}}{\oplus }\left(z\underset{\dot{}}{\oplus }x\right)\right)\right)\right)$$
5	4	simplbi	$$⊢\underset{\dot{}}{\oplus }\in \left(G\mathrm{GrpAct}Y\right)\to \left(G\in \mathrm{Grp}\wedge Y\in \mathrm{V}\right)$$
6	5	simpld	$$⊢\underset{\dot{}}{\oplus }\in \left(G\mathrm{GrpAct}Y\right)\to G\in \mathrm{Grp}$$