This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )