Theorem mnringbaserd

Description: The base set of a monoid ring. Converse of mnringbased . (Contributed by Rohan Ridenour, 14-May-2024)

	Ref	Expression
Hypotheses	mnringbaserd.1	⊢ 𝐹 = ( 𝑅 MndRing 𝑀 )
	mnringbaserd.2	⊢ 𝐵 = ( Base ‘ 𝐹 )
	mnringbaserd.3	⊢ 𝐴 = ( Base ‘ 𝑀 )
	mnringbaserd.4	⊢ 𝑉 = ( 𝑅 freeLMod 𝐴 )
	mnringbaserd.5	⊢ ( 𝜑 → 𝑅 ∈ 𝑈 )
	mnringbaserd.6	⊢ ( 𝜑 → 𝑀 ∈ 𝑊 )
Assertion	mnringbaserd	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑉 ) )

Proof

Step	Hyp	Ref	Expression
1		mnringbaserd.1	⊢ 𝐹 = ( 𝑅 MndRing 𝑀 )
2		mnringbaserd.2	⊢ 𝐵 = ( Base ‘ 𝐹 )
3		mnringbaserd.3	⊢ 𝐴 = ( Base ‘ 𝑀 )
4		mnringbaserd.4	⊢ 𝑉 = ( 𝑅 freeLMod 𝐴 )
5		mnringbaserd.5	⊢ ( 𝜑 → 𝑅 ∈ 𝑈 )
6		mnringbaserd.6	⊢ ( 𝜑 → 𝑀 ∈ 𝑊 )
7		eqid	⊢ ( Base ‘ 𝑉 ) = ( Base ‘ 𝑉 )
8	1 3 4 7 5 6	mnringbased	⊢ ( 𝜑 → ( Base ‘ 𝑉 ) = ( Base ‘ 𝐹 ) )
9	2 8	eqtr4id	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑉 ) )