Theorem grporid

Description: The identity element of a group is a right identity. (Contributed by NM, 24-Oct-2006) (Revised by Mario Carneiro, 15-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	grpoidval.1	$$⊢X=\mathrm{ran}G$$
	grpoidval.2	$$⊢U=\mathrm{GId}\left(G\right)$$
Assertion	grporid	$$⊢\left(G\in \mathrm{GrpOp}\wedge A\in X\right)\to AGU=A$$

Proof

Step	Hyp	Ref	Expression
1		grpoidval.1	$$⊢X=\mathrm{ran}G$$
2		grpoidval.2	$$⊢U=\mathrm{GId}\left(G\right)$$
3	1 2	grpoidinv2	$$⊢\left(G\in \mathrm{GrpOp}\wedge A\in X\right)\to \left(\left(UGA=A\wedge AGU=A\right)\wedge \exists x\in X\left(xGA=U\wedge AGx=U\right)\right)$$
4		simplr	$$⊢\left(\left(UGA=A\wedge AGU=A\right)\wedge \exists x\in X\left(xGA=U\wedge AGx=U\right)\right)\to AGU=A$$
5	3 4	syl	$$⊢\left(G\in \mathrm{GrpOp}\wedge A\in X\right)\to AGU=A$$