Theorem grplid

Description: The identity element of a group is a left identity. (Contributed by NM, 18-Aug-2011)

	Ref	Expression
Hypotheses	grpbn0.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grplid.p	⊢ + = ( +g ‘ 𝐺 )
	grplid.o	⊢ 0 = ( 0g ‘ 𝐺 )
Assertion	grplid	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 0 + 𝑋 ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		grpbn0.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grplid.p	⊢ + = ( +g ‘ 𝐺 )
3		grplid.o	⊢ 0 = ( 0g ‘ 𝐺 )
4		grpmnd	⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd )
5	1 2 3	mndlid	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ) → ( 0 + 𝑋 ) = 𝑋 )
6	4 5	sylan	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 0 + 𝑋 ) = 𝑋 )