Theorem ringlidmd

Description: The unit element of a ring is a left multiplicative identity. (Contributed by SN, 14-Aug-2024)

	Ref	Expression
Hypotheses	ringlidmd.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	ringlidmd.t	⊢ · = ( .r ‘ 𝑅 )
	ringlidmd.u	⊢ 1 = ( 1r ‘ 𝑅 )
	ringlidmd.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
	ringlidmd.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion	ringlidmd	⊢ ( 𝜑 → ( 1 · 𝑋 ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		ringlidmd.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		ringlidmd.t	⊢ · = ( .r ‘ 𝑅 )
3		ringlidmd.u	⊢ 1 = ( 1r ‘ 𝑅 )
4		ringlidmd.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
5		ringlidmd.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6	1 2 3	ringlidm	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 1 · 𝑋 ) = 𝑋 )
7	4 5 6	syl2anc	⊢ ( 𝜑 → ( 1 · 𝑋 ) = 𝑋 )