Theorem mndlid

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

	Ref	Expression
Hypotheses	mndlrid.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	mndlrid.p	⊢ + = ( +g ‘ 𝐺 )
	mndlrid.o	⊢ 0 = ( 0g ‘ 𝐺 )
Assertion	mndlid	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ) → ( 0 + 𝑋 ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		mndlrid.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		mndlrid.p	⊢ + = ( +g ‘ 𝐺 )
3		mndlrid.o	⊢ 0 = ( 0g ‘ 𝐺 )
4	1 2 3	mndlrid	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ) → ( ( 0 + 𝑋 ) = 𝑋 ∧ ( 𝑋 + 0 ) = 𝑋 ) )
5	4	simpld	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ) → ( 0 + 𝑋 ) = 𝑋 )