Theorem ringlidmd

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

	Ref	Expression
Hypotheses	ringlidmd.b	$$⊢B={\mathrm{Base}}_R$$
	ringlidmd.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
	ringlidmd.u	$$⊢\underset{\dot{}}{1}=1_R$$
	ringlidmd.r	$$⊢\phi \to R\in \mathrm{Ring}$$
	ringlidmd.x	$$⊢\phi \to X\in B$$
Assertion	ringlidmd	$$⊢\phi \to \underset{\dot{}}{1}\underset{\dot{}}{·}X=X$$

Proof

Step	Hyp	Ref	Expression
1		ringlidmd.b	$$⊢B={\mathrm{Base}}_R$$
2		ringlidmd.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
3		ringlidmd.u	$$⊢\underset{\dot{}}{1}=1_R$$
4		ringlidmd.r	$$⊢\phi \to R\in \mathrm{Ring}$$
5		ringlidmd.x	$$⊢\phi \to X\in B$$
6	1 2 3	ringlidm	$$⊢\left(R\in \mathrm{Ring}\wedge X\in B\right)\to \underset{\dot{}}{1}\underset{\dot{}}{·}X=X$$
7	4 5 6	syl2anc	$$⊢\phi \to \underset{\dot{}}{1}\underset{\dot{}}{·}X=X$$