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 = ran 𝐺
	grpoidval.2	⊢ 𝑈 = ( GId ‘ 𝐺 )
Assertion	grporid	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐺 𝑈 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		grpoidval.1	⊢ 𝑋 = ran 𝐺
2		grpoidval.2	⊢ 𝑈 = ( GId ‘ 𝐺 )
3	1 2	grpoidinv2	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( ( ( 𝑈 𝐺 𝐴 ) = 𝐴 ∧ ( 𝐴 𝐺 𝑈 ) = 𝐴 ) ∧ ∃ 𝑥 ∈ 𝑋 ( ( 𝑥 𝐺 𝐴 ) = 𝑈 ∧ ( 𝐴 𝐺 𝑥 ) = 𝑈 ) ) )
4		simplr	⊢ ( ( ( ( 𝑈 𝐺 𝐴 ) = 𝐴 ∧ ( 𝐴 𝐺 𝑈 ) = 𝐴 ) ∧ ∃ 𝑥 ∈ 𝑋 ( ( 𝑥 𝐺 𝐴 ) = 𝑈 ∧ ( 𝐴 𝐺 𝑥 ) = 𝑈 ) ) → ( 𝐴 𝐺 𝑈 ) = 𝐴 )
5	3 4	syl	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐺 𝑈 ) = 𝐴 )