Theorem grpideu

Description: The two-sided identity element of a group is unique. Lemma 2.2.1(a) of Herstein p. 55. (Contributed by NM, 16-Aug-2011) (Revised by Mario Carneiro, 8-Dec-2014)

	Ref	Expression
Hypotheses	grpcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grpcl.p	⊢ + = ( +g ‘ 𝐺 )
	grpinvex.p	⊢ 0 = ( 0g ‘ 𝐺 )
Assertion	grpideu	⊢ ( 𝐺 ∈ Grp → ∃! 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		grpcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpcl.p	⊢ + = ( +g ‘ 𝐺 )
3		grpinvex.p	⊢ 0 = ( 0g ‘ 𝐺 )
4		grpmnd	⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd )
5	1 2	mndideu	⊢ ( 𝐺 ∈ Mnd → ∃! 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) )
6	4 5	syl	⊢ ( 𝐺 ∈ Grp → ∃! 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) )