Theorem idlss

Description: An ideal of R is a subset of R . (Contributed by Jeff Madsen, 10-Jun-2010)

	Ref	Expression
Hypotheses	idlss.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
	idlss.2	⊢ 𝑋 = ran 𝐺
Assertion	idlss	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → 𝐼 ⊆ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		idlss.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
2		idlss.2	⊢ 𝑋 = ran 𝐺
3		eqid	⊢ ( 2nd ‘ 𝑅 ) = ( 2nd ‘ 𝑅 )
4		eqid	⊢ ( GId ‘ 𝐺 ) = ( GId ‘ 𝐺 )
5	1 3 2 4	isidl	⊢ ( 𝑅 ∈ RingOps → ( 𝐼 ∈ ( Idl ‘ 𝑅 ) ↔ ( 𝐼 ⊆ 𝑋 ∧ ( GId ‘ 𝐺 ) ∈ 𝐼 ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ 𝐼 ( 𝑥 𝐺 𝑦 ) ∈ 𝐼 ∧ ∀ 𝑧 ∈ 𝑋 ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ 𝐼 ∧ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ 𝐼 ) ) ) ) )
6	5	biimpa	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → ( 𝐼 ⊆ 𝑋 ∧ ( GId ‘ 𝐺 ) ∈ 𝐼 ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ 𝐼 ( 𝑥 𝐺 𝑦 ) ∈ 𝐼 ∧ ∀ 𝑧 ∈ 𝑋 ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ 𝐼 ∧ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ 𝐼 ) ) ) )
7	6	simp1d	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → 𝐼 ⊆ 𝑋 )