Theorem lidlmcl

Description: An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Stefan O'Rear, 3-Jan-2015) (Proof shortened by AV, 31-Mar-2025)

	Ref	Expression
Hypotheses	lidlcl.u	⊢ 𝑈 = ( LIdeal ‘ 𝑅 )
	lidlcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	lidlmcl.t	⊢ · = ( .r ‘ 𝑅 )
Assertion	lidlmcl	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼 ) ) → ( 𝑋 · 𝑌 ) ∈ 𝐼 )

Proof

Step	Hyp	Ref	Expression
1		lidlcl.u	⊢ 𝑈 = ( LIdeal ‘ 𝑅 )
2		lidlcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
3		lidlmcl.t	⊢ · = ( .r ‘ 𝑅 )
4		ringrng	⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Rng )
5	4	adantr	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ) → 𝑅 ∈ Rng )
6		simpr	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ) → 𝐼 ∈ 𝑈 )
7		eqid	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
8	1 7	lidl0cl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ) → ( 0g ‘ 𝑅 ) ∈ 𝐼 )
9	5 6 8	3jca	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ) → ( 𝑅 ∈ Rng ∧ 𝐼 ∈ 𝑈 ∧ ( 0g ‘ 𝑅 ) ∈ 𝐼 ) )
10	7 2 3 1	rnglidlmcl	⊢ ( ( ( 𝑅 ∈ Rng ∧ 𝐼 ∈ 𝑈 ∧ ( 0g ‘ 𝑅 ) ∈ 𝐼 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼 ) ) → ( 𝑋 · 𝑌 ) ∈ 𝐼 )
11	9 10	sylan	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼 ) ) → ( 𝑋 · 𝑌 ) ∈ 𝐼 )