Theorem gagrpid

Description: The identity of the group does not alter the base set. (Contributed by Jeff Hankins, 11-Aug-2009) (Revised by Mario Carneiro, 13-Jan-2015)

	Ref	Expression
Hypothesis	gagrpid.1	$$⊢\underset{\dot{}}{0}=0_G$$
Assertion	gagrpid	$$⊢\left(\underset{\dot{}}{\oplus }\in \left(G\mathrm{GrpAct}Y\right)\wedge A\in Y\right)\to \underset{\dot{}}{0}\underset{\dot{}}{\oplus }A=A$$

Proof

Step	Hyp	Ref	Expression
1		gagrpid.1	$$⊢\underset{\dot{}}{0}=0_G$$
2		eqid	$$⊢{\mathrm{Base}}_G={\mathrm{Base}}_G$$
3		eqid	$$⊢{+}_G={+}_G$$
4	2 3 1	isga	$$⊢\underset{\dot{}}{\oplus }\in \left(G\mathrm{GrpAct}Y\right)\leftrightarrow \left(\left(G\in \mathrm{Grp}\wedge Y\in \mathrm{V}\right)\wedge \left(\underset{\dot{}}{\oplus }:{\mathrm{Base}}_G\times Y⟶Y\wedge \forall x\in Y\left(\underset{\dot{}}{0}\underset{\dot{}}{\oplus }x=x\wedge \forall y\in {\mathrm{Base}}_G\forall z\in {\mathrm{Base}}_G\left(y{+}_{G}z\right)\underset{\dot{}}{\oplus }x=y\underset{\dot{}}{\oplus }\left(z\underset{\dot{}}{\oplus }x\right)\right)\right)\right)$$
5	4	simprbi	$$⊢\underset{\dot{}}{\oplus }\in \left(G\mathrm{GrpAct}Y\right)\to \left(\underset{\dot{}}{\oplus }:{\mathrm{Base}}_G\times Y⟶Y\wedge \forall x\in Y\left(\underset{\dot{}}{0}\underset{\dot{}}{\oplus }x=x\wedge \forall y\in {\mathrm{Base}}_G\forall z\in {\mathrm{Base}}_G\left(y{+}_{G}z\right)\underset{\dot{}}{\oplus }x=y\underset{\dot{}}{\oplus }\left(z\underset{\dot{}}{\oplus }x\right)\right)\right)$$
6		simpl	$$⊢\left(\underset{\dot{}}{0}\underset{\dot{}}{\oplus }x=x\wedge \forall y\in {\mathrm{Base}}_G\forall z\in {\mathrm{Base}}_G\left(y{+}_{G}z\right)\underset{\dot{}}{\oplus }x=y\underset{\dot{}}{\oplus }\left(z\underset{\dot{}}{\oplus }x\right)\right)\to \underset{\dot{}}{0}\underset{\dot{}}{\oplus }x=x$$
7	6	ralimi	$$⊢\forall x\in Y\left(\underset{\dot{}}{0}\underset{\dot{}}{\oplus }x=x\wedge \forall y\in {\mathrm{Base}}_G\forall z\in {\mathrm{Base}}_G\left(y{+}_{G}z\right)\underset{\dot{}}{\oplus }x=y\underset{\dot{}}{\oplus }\left(z\underset{\dot{}}{\oplus }x\right)\right)\to \forall x\in Y\underset{\dot{}}{0}\underset{\dot{}}{\oplus }x=x$$
8	5 7	simpl2im	$$⊢\underset{\dot{}}{\oplus }\in \left(G\mathrm{GrpAct}Y\right)\to \forall x\in Y\underset{\dot{}}{0}\underset{\dot{}}{\oplus }x=x$$
9		oveq2	$$⊢x=A\to \underset{\dot{}}{0}\underset{\dot{}}{\oplus }x=\underset{\dot{}}{0}\underset{\dot{}}{\oplus }A$$
10		id	$$⊢x=A\to x=A$$
11	9 10	eqeq12d	$$⊢x=A\to \left(\underset{\dot{}}{0}\underset{\dot{}}{\oplus }x=x\leftrightarrow \underset{\dot{}}{0}\underset{\dot{}}{\oplus }A=A\right)$$
12	11	rspccva	$$⊢\left(\forall x\in Y\underset{\dot{}}{0}\underset{\dot{}}{\oplus }x=x\wedge A\in Y\right)\to \underset{\dot{}}{0}\underset{\dot{}}{\oplus }A=A$$
13	8 12	sylan	$$⊢\left(\underset{\dot{}}{\oplus }\in \left(G\mathrm{GrpAct}Y\right)\wedge A\in Y\right)\to \underset{\dot{}}{0}\underset{\dot{}}{\oplus }A=A$$