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Metamath Proof Explorer

Theorem 2idlelb

Description: Membership in a two-sided ideal. Formerly part of proof for 2idlcpbl . (Contributed by Mario Carneiro, 14-Jun-2015) (Revised by AV, 20-Feb-2025)

Ref Expression
Hypotheses 2idlel.i ⊢ 𝐼 = ( LIdeal ‘ 𝑅 )
2idlel.o ⊢ 𝑂 = ( oppr ‘ 𝑅 )
2idlel.j ⊢ 𝐽 = ( LIdeal ‘ 𝑂 )
2idlel.t ⊢ 𝑇 = ( 2Ideal ‘ 𝑅 )
Assertion 2idlelb ( 𝑈 ∈ 𝑇 ↔ ( 𝑈 ∈ 𝐼 ∧ 𝑈 ∈ 𝐽 ) )

Proof

Step Hyp Ref Expression
1 2idlel.i ⊢ 𝐼 = ( LIdeal ‘ 𝑅 )
2 2idlel.o ⊢ 𝑂 = ( oppr ‘ 𝑅 )
3 2idlel.j ⊢ 𝐽 = ( LIdeal ‘ 𝑂 )
4 2idlel.t ⊢ 𝑇 = ( 2Ideal ‘ 𝑅 )
5 1 2 3 4 2idlval ⊢ 𝑇 = ( 𝐼 ∩ 𝐽 )
6 5 elin2 ⊢ ( 𝑈 ∈ 𝑇 ↔ ( 𝑈 ∈ 𝐼 ∧ 𝑈 ∈ 𝐽 ) )