Theorem gaf

Description: The mapping of the group action operation. (Contributed by Jeff Hankins, 11-Aug-2009) (Revised by Mario Carneiro, 13-Jan-2015)

	Ref	Expression
Hypothesis	gaf.1	$$⊢X={\mathrm{Base}}_G$$
Assertion	gaf	$$⊢\underset{\dot{}}{\oplus }\in \left(G\mathrm{GrpAct}Y\right)\to \underset{\dot{}}{\oplus }:X\times Y⟶Y$$

Proof

Step	Hyp	Ref	Expression
1		gaf.1	$$⊢X={\mathrm{Base}}_G$$
2		eqid	$$⊢{+}_G={+}_G$$
3		eqid	$$⊢0_G=0_G$$
4	1 2 3	isga	$$⊢\underset{\dot{}}{\oplus }\in \left(G\mathrm{GrpAct}Y\right)\leftrightarrow \left(\left(G\in \mathrm{Grp}\wedge Y\in \mathrm{V}\right)\wedge \left(\underset{\dot{}}{\oplus }:X\times Y⟶Y\wedge \forall x\in Y\left(0_G\underset{\dot{}}{\oplus }x=x\wedge \forall y\in X\forall z\in X\left(y{+}_{G}z\right)\underset{\dot{}}{\oplus }x=y\underset{\dot{}}{\oplus }\left(z\underset{\dot{}}{\oplus }x\right)\right)\right)\right)$$
5	4	simprbi	$$⊢\underset{\dot{}}{\oplus }\in \left(G\mathrm{GrpAct}Y\right)\to \left(\underset{\dot{}}{\oplus }:X\times Y⟶Y\wedge \forall x\in Y\left(0_G\underset{\dot{}}{\oplus }x=x\wedge \forall y\in X\forall z\in X\left(y{+}_{G}z\right)\underset{\dot{}}{\oplus }x=y\underset{\dot{}}{\oplus }\left(z\underset{\dot{}}{\oplus }x\right)\right)\right)$$
6	5	simpld	$$⊢\underset{\dot{}}{\oplus }\in \left(G\mathrm{GrpAct}Y\right)\to \underset{\dot{}}{\oplus }:X\times Y⟶Y$$