Theorem lmodring

Description: The scalar component of a left module is a ring. (Contributed by NM, 8-Dec-2013) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypothesis	lmodring.1	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
Assertion	lmodring	⊢ ( 𝑊 ∈ LMod → 𝐹 ∈ Ring )

Proof

Step	Hyp	Ref	Expression
1		lmodring.1	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
3		eqid	⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 )
4		eqid	⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 )
5		eqid	⊢ ( Base ‘ 𝐹 ) = ( Base ‘ 𝐹 )
6		eqid	⊢ ( +g ‘ 𝐹 ) = ( +g ‘ 𝐹 )
7		eqid	⊢ ( .r ‘ 𝐹 ) = ( .r ‘ 𝐹 )
8		eqid	⊢ ( 1r ‘ 𝐹 ) = ( 1r ‘ 𝐹 )
9	2 3 4 1 5 6 7 8	islmod	⊢ ( 𝑊 ∈ LMod ↔ ( 𝑊 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀ 𝑞 ∈ ( Base ‘ 𝐹 ) ∀ 𝑟 ∈ ( Base ‘ 𝐹 ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑤 ∈ ( Base ‘ 𝑊 ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑤 ( +g ‘ 𝑊 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ 𝐹 ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ 𝐹 ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ∧ ( ( 1r ‘ 𝐹 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = 𝑤 ) ) ) )
10	9	simp2bi	⊢ ( 𝑊 ∈ LMod → 𝐹 ∈ Ring )