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Metamath Proof Explorer

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 ) )