Metamath Proof Explorer

Theorem idladdcl

Description: An ideal is closed under addition. (Contributed by Jeff Madsen, 10-Jun-2010)

		Ref	Expression
	Hypothesis	idladdcl.1	⊢ 𝐺 = ( 1^st ‘ 𝑅 )
	Assertion	idladdcl	⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) ∧ ( 𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼 ) ) → ( 𝐴 𝐺 𝐵 ) ∈ 𝐼 )

Proof

Step	Hyp	Ref	Expression
1		idladdcl.1	⊢ 𝐺 = ( 1^st ‘ 𝑅 )
2		eqid	⊢ ( 2^nd ‘ 𝑅 ) = ( 2^nd ‘ 𝑅 )
3		eqid	⊢ ran 𝐺 = ran 𝐺
4		eqid	⊢ ( GId ‘ 𝐺 ) = ( GId ‘ 𝐺 )
5	1 2 3 4	isidl	⊢ ( 𝑅 ∈ RingOps → ( 𝐼 ∈ ( Idl ‘ 𝑅 ) ↔ ( 𝐼 ⊆ ran 𝐺 ∧ ( GId ‘ 𝐺 ) ∈ 𝐼 ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ 𝐼 ( 𝑥 𝐺 𝑦 ) ∈ 𝐼 ∧ ∀ 𝑧 ∈ ran 𝐺 ( ( 𝑧 ( 2^nd ‘ 𝑅 ) 𝑥 ) ∈ 𝐼 ∧ ( 𝑥 ( 2^nd ‘ 𝑅 ) 𝑧 ) ∈ 𝐼 ) ) ) ) )
6	5	biimpa	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → ( 𝐼 ⊆ ran 𝐺 ∧ ( GId ‘ 𝐺 ) ∈ 𝐼 ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ 𝐼 ( 𝑥 𝐺 𝑦 ) ∈ 𝐼 ∧ ∀ 𝑧 ∈ ran 𝐺 ( ( 𝑧 ( 2^nd ‘ 𝑅 ) 𝑥 ) ∈ 𝐼 ∧ ( 𝑥 ( 2^nd ‘ 𝑅 ) 𝑧 ) ∈ 𝐼 ) ) ) )
7	6	simp3d	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ 𝐼 ( 𝑥 𝐺 𝑦 ) ∈ 𝐼 ∧ ∀ 𝑧 ∈ ran 𝐺 ( ( 𝑧 ( 2^nd ‘ 𝑅 ) 𝑥 ) ∈ 𝐼 ∧ ( 𝑥 ( 2^nd ‘ 𝑅 ) 𝑧 ) ∈ 𝐼 ) ) )
8		simpl	⊢ ( ( ∀ 𝑦 ∈ 𝐼 ( 𝑥 𝐺 𝑦 ) ∈ 𝐼 ∧ ∀ 𝑧 ∈ ran 𝐺 ( ( 𝑧 ( 2^nd ‘ 𝑅 ) 𝑥 ) ∈ 𝐼 ∧ ( 𝑥 ( 2^nd ‘ 𝑅 ) 𝑧 ) ∈ 𝐼 ) ) → ∀ 𝑦 ∈ 𝐼 ( 𝑥 𝐺 𝑦 ) ∈ 𝐼 )
9	8	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ 𝐼 ( 𝑥 𝐺 𝑦 ) ∈ 𝐼 ∧ ∀ 𝑧 ∈ ran 𝐺 ( ( 𝑧 ( 2^nd ‘ 𝑅 ) 𝑥 ) ∈ 𝐼 ∧ ( 𝑥 ( 2^nd ‘ 𝑅 ) 𝑧 ) ∈ 𝐼 ) ) → ∀ 𝑥 ∈ 𝐼 ∀ 𝑦 ∈ 𝐼 ( 𝑥 𝐺 𝑦 ) ∈ 𝐼 )
10	7 9	syl	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) → ∀ 𝑥 ∈ 𝐼 ∀ 𝑦 ∈ 𝐼 ( 𝑥 𝐺 𝑦 ) ∈ 𝐼 )
11		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐺 𝑦 ) = ( 𝐴 𝐺 𝑦 ) )
12	11	eleq1d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝐺 𝑦 ) ∈ 𝐼 ↔ ( 𝐴 𝐺 𝑦 ) ∈ 𝐼 ) )
13		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐺 𝑦 ) = ( 𝐴 𝐺 𝐵 ) )
14	13	eleq1d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝐺 𝑦 ) ∈ 𝐼 ↔ ( 𝐴 𝐺 𝐵 ) ∈ 𝐼 ) )
15	12 14	rspc2v	⊢ ( ( 𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼 ) → ( ∀ 𝑥 ∈ 𝐼 ∀ 𝑦 ∈ 𝐼 ( 𝑥 𝐺 𝑦 ) ∈ 𝐼 → ( 𝐴 𝐺 𝐵 ) ∈ 𝐼 ) )
16	10 15	mpan9	⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝐼 ∈ ( Idl ‘ 𝑅 ) ) ∧ ( 𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼 ) ) → ( 𝐴 𝐺 𝐵 ) ∈ 𝐼 )