Theorem mnring0g2d

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

	Ref	Expression
Hypotheses	mnring0g2d.1	⊢ 𝐹 = ( 𝑅 MndRing 𝑀 )
	mnring0g2d.2	⊢ 0 = ( 0g ‘ 𝑅 )
	mnring0g2d.3	⊢ 𝐴 = ( Base ‘ 𝑀 )
	mnring0g2d.4	⊢ ( 𝜑 → 𝑅 ∈ Ring )
	mnring0g2d.5	⊢ ( 𝜑 → 𝑀 ∈ 𝑊 )
Assertion	mnring0g2d	⊢ ( 𝜑 → ( 𝐴 × { 0 } ) = ( 0g ‘ 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		mnring0g2d.1	⊢ 𝐹 = ( 𝑅 MndRing 𝑀 )
2		mnring0g2d.2	⊢ 0 = ( 0g ‘ 𝑅 )
3		mnring0g2d.3	⊢ 𝐴 = ( Base ‘ 𝑀 )
4		mnring0g2d.4	⊢ ( 𝜑 → 𝑅 ∈ Ring )
5		mnring0g2d.5	⊢ ( 𝜑 → 𝑀 ∈ 𝑊 )
6	3	fvexi	⊢ 𝐴 ∈ V
7		eqid	⊢ ( 𝑅 freeLMod 𝐴 ) = ( 𝑅 freeLMod 𝐴 )
8	7 2	frlm0	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐴 ∈ V ) → ( 𝐴 × { 0 } ) = ( 0g ‘ ( 𝑅 freeLMod 𝐴 ) ) )
9	4 6 8	sylancl	⊢ ( 𝜑 → ( 𝐴 × { 0 } ) = ( 0g ‘ ( 𝑅 freeLMod 𝐴 ) ) )
10	1 3 7 4 5	mnring0gd	⊢ ( 𝜑 → ( 0g ‘ ( 𝑅 freeLMod 𝐴 ) ) = ( 0g ‘ 𝐹 ) )
11	9 10	eqtrd	⊢ ( 𝜑 → ( 𝐴 × { 0 } ) = ( 0g ‘ 𝐹 ) )