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Metamath Proof Explorer

Theorem opsrlmod

Description: Ordered power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015)

Ref Expression
Hypotheses opsrring.o ⊢ 𝑂 = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 )
opsrring.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
opsrring.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
opsrring.t ⊢ ( 𝜑 → 𝑇 ⊆ ( 𝐼 × 𝐼 ) )
Assertion opsrlmod ( 𝜑 → 𝑂 ∈ LMod )

Proof

Step Hyp Ref Expression
1 opsrring.o ⊢ 𝑂 = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 )
2 opsrring.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
3 opsrring.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
4 opsrring.t ⊢ ( 𝜑 → 𝑇 ⊆ ( 𝐼 × 𝐼 ) )
5 eqid ⊢ ( 𝐼 mPwSer 𝑅 ) = ( 𝐼 mPwSer 𝑅 )
6 5 2 3 psrlmod ⊢ ( 𝜑 → ( 𝐼 mPwSer 𝑅 ) ∈ LMod )
7 eqidd ⊢ ( 𝜑 → ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) )
8 5 1 4 opsrbas ⊢ ( 𝜑 → ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Base ‘ 𝑂 ) )
9 5 1 4 opsrplusg ⊢ ( 𝜑 → ( +g ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( +g ‘ 𝑂 ) )
10 9 oveqdr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑦 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ) → ( 𝑥 ( +g ‘ ( 𝐼 mPwSer 𝑅 ) ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑂 ) 𝑦 ) )
11 5 2 3 psrsca ⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ ( 𝐼 mPwSer 𝑅 ) ) )
12 5 1 4 2 3 opsrsca ⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ 𝑂 ) )
13 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
14 5 1 4 opsrvsca ⊢ ( 𝜑 → ( ·𝑠 ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( ·𝑠 ‘ 𝑂 ) )
15 14 oveqdr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) ) → ( 𝑥 ( ·𝑠 ‘ ( 𝐼 mPwSer 𝑅 ) ) 𝑦 ) = ( 𝑥 ( ·𝑠 ‘ 𝑂 ) 𝑦 ) )
16 7 8 10 11 12 13 15 lmodpropd ⊢ ( 𝜑 → ( ( 𝐼 mPwSer 𝑅 ) ∈ LMod ↔ 𝑂 ∈ LMod ) )
17 6 16 mpbid ⊢ ( 𝜑 → 𝑂 ∈ LMod )