Metamath Proof Explorer


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 ( 𝜑𝑅𝑈 )
mnringbaserd.6 ( 𝜑𝑀𝑊 )
Assertion mnringbaserd ( 𝜑𝐵 = ( 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 ( 𝜑𝑅𝑈 )
6 mnringbaserd.6 ( 𝜑𝑀𝑊 )
7 eqid ( Base ‘ 𝑉 ) = ( Base ‘ 𝑉 )
8 1 3 4 7 5 6 mnringbased ( 𝜑 → ( Base ‘ 𝑉 ) = ( Base ‘ 𝐹 ) )
9 2 8 eqtr4id ( 𝜑𝐵 = ( Base ‘ 𝑉 ) )