Theorem islidl

Description: Predicate of being a (left) ideal. (Contributed by Stefan O'Rear, 1-Apr-2015)

	Ref	Expression
Hypotheses	islidl.s	⊢ 𝑈 = ( LIdeal ‘ 𝑅 )
	islidl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	islidl.p	⊢ + = ( +g ‘ 𝑅 )
	islidl.t	⊢ · = ( .r ‘ 𝑅 )
Assertion	islidl	⊢ ( 𝐼 ∈ 𝑈 ↔ ( 𝐼 ⊆ 𝐵 ∧ 𝐼 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑎 ∈ 𝐼 ∀ 𝑏 ∈ 𝐼 ( ( 𝑥 · 𝑎 ) + 𝑏 ) ∈ 𝐼 ) )

Proof

Step	Hyp	Ref	Expression
1		islidl.s	⊢ 𝑈 = ( LIdeal ‘ 𝑅 )
2		islidl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
3		islidl.p	⊢ + = ( +g ‘ 𝑅 )
4		islidl.t	⊢ · = ( .r ‘ 𝑅 )
5		rlmsca2	⊢ ( I ‘ 𝑅 ) = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) )
6		baseid	⊢ Base = Slot ( Base ‘ ndx )
7	6 2	strfvi	⊢ 𝐵 = ( Base ‘ ( I ‘ 𝑅 ) )
8		rlmbas	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ ( ringLMod ‘ 𝑅 ) )
9	2 8	eqtri	⊢ 𝐵 = ( Base ‘ ( ringLMod ‘ 𝑅 ) )
10		rlmplusg	⊢ ( +g ‘ 𝑅 ) = ( +g ‘ ( ringLMod ‘ 𝑅 ) )
11	3 10	eqtri	⊢ + = ( +g ‘ ( ringLMod ‘ 𝑅 ) )
12		rlmvsca	⊢ ( .r ‘ 𝑅 ) = ( ·𝑠 ‘ ( ringLMod ‘ 𝑅 ) )
13	4 12	eqtri	⊢ · = ( ·𝑠 ‘ ( ringLMod ‘ 𝑅 ) )
14		lidlval	⊢ ( LIdeal ‘ 𝑅 ) = ( LSubSp ‘ ( ringLMod ‘ 𝑅 ) )
15	1 14	eqtri	⊢ 𝑈 = ( LSubSp ‘ ( ringLMod ‘ 𝑅 ) )
16	5 7 9 11 13 15	islss	⊢ ( 𝐼 ∈ 𝑈 ↔ ( 𝐼 ⊆ 𝐵 ∧ 𝐼 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑎 ∈ 𝐼 ∀ 𝑏 ∈ 𝐼 ( ( 𝑥 · 𝑎 ) + 𝑏 ) ∈ 𝐼 ) )