Theorem ringlidm

Description: The unit element of a ring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011)

	Ref	Expression
Hypotheses	rngidm.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	rngidm.t	⊢ · = ( .r ‘ 𝑅 )
	rngidm.u	⊢ 1 = ( 1r ‘ 𝑅 )
Assertion	ringlidm	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 1 · 𝑋 ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		rngidm.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		rngidm.t	⊢ · = ( .r ‘ 𝑅 )
3		rngidm.u	⊢ 1 = ( 1r ‘ 𝑅 )
4	1 2 3	ringidmlem	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( ( 1 · 𝑋 ) = 𝑋 ∧ ( 𝑋 · 1 ) = 𝑋 ) )
5	4	simpld	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 1 · 𝑋 ) = 𝑋 )