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Metamath Proof Explorer

Theorem 2idllidld

Description: A two-sided ideal is a left ideal. (Contributed by Thierry Arnoux, 9-Mar-2025)

		Ref	Expression
	Hypothesis	2idllidld.1	$⊢ φ \to I \in 2Ideal (R)$
	Assertion	2idllidld	$⊢ φ \to I \in LIdeal (R)$

Step	Hyp	Ref	Expression
1		2idllidld.1	$⊢ φ \to I \in 2Ideal (R)$
2		eqid	$⊢ LIdeal (R) = LIdeal (R)$
3		eqid	$⊢ {opp}_{r} (R) = {opp}_{r} (R)$
4		eqid	$⊢ LIdeal ({opp}_{r} (R)) = LIdeal ({opp}_{r} (R))$
5		eqid	$⊢ 2Ideal (R) = 2Ideal (R)$
6	2 3 4 5	2idlval	$⊢ 2Ideal (R) = LIdeal (R) \cap LIdeal ({opp}_{r} (R))$
7	1 6	eleqtrdi	$⊢ φ \to I \in (LIdeal (R) \cap LIdeal ({opp}_{r} (R)))$
8	7	elin1d	$⊢ φ \to I \in LIdeal (R)$