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Metamath Proof Explorer

Theorem idl0cl

Description: An ideal contains 0 . (Contributed by Jeff Madsen, 10-Jun-2010)

Ref Expression
Hypotheses idl0cl.1 ⊢ 𝐺 = ( 1st ‘ 𝑅 )
idl0cl.2 ⊢ 𝑍 = ( GId ‘ 𝐺 )
Assertion idl0cl ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → 𝑍 ∈ 𝐼 )

Proof

Step Hyp Ref Expression
1 idl0cl.1 ⊢ 𝐺 = ( 1st ‘ 𝑅 )
2 idl0cl.2 ⊢ 𝑍 = ( GId ‘ 𝐺 )
3 eqid ⊢ ( 2nd ‘ 𝑅 ) = ( 2nd ‘ 𝑅 )
4 eqid ⊢ ran 𝐺 = ran 𝐺
5 1 3 4 2 isidl ⊢ ( 𝑅 ∈ RingOps → ( 𝐼 ∈ ( Idl ‘ 𝑅 ) ↔ ( 𝐼 ⊆ ran 𝐺 ∧ 𝑍 ∈ 𝐼 ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ 𝐼 ( 𝑥 𝐺 𝑦 ) ∈ 𝐼 ∧ ∀ 𝑧 ∈ ran 𝐺 ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ 𝐼 ∧ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ 𝐼 ) ) ) ) )
6 5 biimpa ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → ( 𝐼 ⊆ ran 𝐺 ∧ 𝑍 ∈ 𝐼 ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ 𝐼 ( 𝑥 𝐺 𝑦 ) ∈ 𝐼 ∧ ∀ 𝑧 ∈ ran 𝐺 ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ 𝐼 ∧ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ 𝐼 ) ) ) )
7 6 simp2d ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → 𝑍 ∈ 𝐼 )