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Metamath Proof Explorer

Theorem slmdsrg

Description: The scalar component of a semimodule is a semiring. (Contributed by NM, 8-Dec-2013) (Revised by Mario Carneiro, 19-Jun-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

Ref Expression
Hypothesis slmdsrg.1 ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
Assertion slmdsrg ( 𝑊 ∈ SLMod → 𝐹 ∈ SRing )

Proof

Step Hyp Ref Expression
1 slmdsrg.1 ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2 eqid ⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
3 eqid ⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 )
4 eqid ⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 )
5 eqid ⊢ ( 0g ‘ 𝑊 ) = ( 0g ‘ 𝑊 )
6 eqid ⊢ ( Base ‘ 𝐹 ) = ( Base ‘ 𝐹 )
7 eqid ⊢ ( +g ‘ 𝐹 ) = ( +g ‘ 𝐹 )
8 eqid ⊢ ( .r ‘ 𝐹 ) = ( .r ‘ 𝐹 )
9 eqid ⊢ ( 1r ‘ 𝐹 ) = ( 1r ‘ 𝐹 )
10 eqid ⊢ ( 0g ‘ 𝐹 ) = ( 0g ‘ 𝐹 )
11 2 3 4 5 1 6 7 8 9 10 isslmd ⊢ ( 𝑊 ∈ SLMod ↔ ( 𝑊 ∈ CMnd ∧ 𝐹 ∈ SRing ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐹 ) ∀ 𝑧 ∈ ( Base ‘ 𝐹 ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑦 ∈ ( Base ‘ 𝑊 ) ( ( ( 𝑧 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑧 ( ·𝑠 ‘ 𝑊 ) ( 𝑦 ( +g ‘ 𝑊 ) 𝑥 ) ) = ( ( 𝑧 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) ( 𝑧 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ) ∧ ( ( 𝑤 ( +g ‘ 𝐹 ) 𝑧 ) ( ·𝑠 ‘ 𝑊 ) 𝑦 ) = ( ( 𝑤 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) ( 𝑧 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) ) ∧ ( ( ( 𝑤 ( .r ‘ 𝐹 ) 𝑧 ) ( ·𝑠 ‘ 𝑊 ) 𝑦 ) = ( 𝑤 ( ·𝑠 ‘ 𝑊 ) ( 𝑧 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ) ∧ ( ( 1r ‘ 𝐹 ) ( ·𝑠 ‘ 𝑊 ) 𝑦 ) = 𝑦 ∧ ( ( 0g ‘ 𝐹 ) ( ·𝑠 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ 𝑊 ) ) ) ) )
12 11 simp2bi ⊢ ( 𝑊 ∈ SLMod → 𝐹 ∈ SRing )