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 )