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Metamath Proof Explorer

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 )