Theorem mnring0gd

Description: The additive identity of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024)

	Ref	Expression
Hypotheses	mnring0gd.1	⊢ 𝐹 = ( 𝑅 MndRing 𝑀 )
	mnring0gd.2	⊢ 𝐴 = ( Base ‘ 𝑀 )
	mnring0gd.3	⊢ 𝑉 = ( 𝑅 freeLMod 𝐴 )
	mnring0gd.4	⊢ ( 𝜑 → 𝑅 ∈ 𝑈 )
	mnring0gd.5	⊢ ( 𝜑 → 𝑀 ∈ 𝑊 )
Assertion	mnring0gd	⊢ ( 𝜑 → ( 0g ‘ 𝑉 ) = ( 0g ‘ 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		mnring0gd.1	⊢ 𝐹 = ( 𝑅 MndRing 𝑀 )
2		mnring0gd.2	⊢ 𝐴 = ( Base ‘ 𝑀 )
3		mnring0gd.3	⊢ 𝑉 = ( 𝑅 freeLMod 𝐴 )
4		mnring0gd.4	⊢ ( 𝜑 → 𝑅 ∈ 𝑈 )
5		mnring0gd.5	⊢ ( 𝜑 → 𝑀 ∈ 𝑊 )
6		eqidd	⊢ ( 𝜑 → ( Base ‘ 𝑉 ) = ( Base ‘ 𝑉 ) )
7		eqid	⊢ ( Base ‘ 𝑉 ) = ( Base ‘ 𝑉 )
8	1 2 3 7 4 5	mnringbased	⊢ ( 𝜑 → ( Base ‘ 𝑉 ) = ( Base ‘ 𝐹 ) )
9	1 2 3 4 5	mnringaddgd	⊢ ( 𝜑 → ( +g ‘ 𝑉 ) = ( +g ‘ 𝐹 ) )
10	9	oveqdr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑉 ) ∧ 𝑦 ∈ ( Base ‘ 𝑉 ) ) ) → ( 𝑥 ( +g ‘ 𝑉 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐹 ) 𝑦 ) )
11	6 8 10	grpidpropd	⊢ ( 𝜑 → ( 0g ‘ 𝑉 ) = ( 0g ‘ 𝐹 ) )