Metamath Proof Explorer

Theorem slmdcmn

Description: A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018)

		Ref	Expression
	Assertion	slmdcmn	⊢ ( 𝑊 ∈ SLMod → 𝑊 ∈ CMnd )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
2		eqid	⊢ ( +_g ‘ 𝑊 ) = ( +_g ‘ 𝑊 )
3		eqid	⊢ ( ·_𝑠 ‘ 𝑊 ) = ( ·_𝑠 ‘ 𝑊 )
4		eqid	⊢ ( 0_g ‘ 𝑊 ) = ( 0_g ‘ 𝑊 )
5		eqid	⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
6		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
7		eqid	⊢ ( +_g ‘ ( Scalar ‘ 𝑊 ) ) = ( +_g ‘ ( Scalar ‘ 𝑊 ) )
8		eqid	⊢ ( ._r ‘ ( Scalar ‘ 𝑊 ) ) = ( ._r ‘ ( Scalar ‘ 𝑊 ) )
9		eqid	⊢ ( 1_r ‘ ( Scalar ‘ 𝑊 ) ) = ( 1_r ‘ ( Scalar ‘ 𝑊 ) )
10		eqid	⊢ ( 0_g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0_g ‘ ( Scalar ‘ 𝑊 ) )
11	1 2 3 4 5 6 7 8 9 10	isslmd	⊢ ( 𝑊 ∈ SLMod ↔ ( 𝑊 ∈ CMnd ∧ ( Scalar ‘ 𝑊 ) ∈ SRing ∧ ∀ 𝑤 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑧 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 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 ‘ ( Scalar ‘ 𝑊 ) ) 𝑧 ) ( ·_𝑠 ‘ 𝑊 ) 𝑦 ) = ( ( 𝑤 ( ·_𝑠 ‘ 𝑊 ) 𝑦 ) ( +_g ‘ 𝑊 ) ( 𝑧 ( ·_𝑠 ‘ 𝑊 ) 𝑦 ) ) ) ∧ ( ( ( 𝑤 ( ._r ‘ ( Scalar ‘ 𝑊 ) ) 𝑧 ) ( ·_𝑠 ‘ 𝑊 ) 𝑦 ) = ( 𝑤 ( ·_𝑠 ‘ 𝑊 ) ( 𝑧 ( ·_𝑠 ‘ 𝑊 ) 𝑦 ) ) ∧ ( ( 1_r ‘ ( Scalar ‘ 𝑊 ) ) ( ·_𝑠 ‘ 𝑊 ) 𝑦 ) = 𝑦 ∧ ( ( 0_g ‘ ( Scalar ‘ 𝑊 ) ) ( ·_𝑠 ‘ 𝑊 ) 𝑦 ) = ( 0_g ‘ 𝑊 ) ) ) ) )
12	11	simp1bi	⊢ ( 𝑊 ∈ SLMod → 𝑊 ∈ CMnd )