Theorem mnringbased

Description: The base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024) (Proof shortened by AV, 1-Nov-2024)

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

Proof

Step	Hyp	Ref	Expression
1		mnringbased.1	⊢ 𝐹 = ( 𝑅 MndRing 𝑀 )
2		mnringbased.2	⊢ 𝐴 = ( Base ‘ 𝑀 )
3		mnringbased.3	⊢ 𝑉 = ( 𝑅 freeLMod 𝐴 )
4		mnringbased.4	⊢ 𝐵 = ( Base ‘ 𝑉 )
5		mnringbased.5	⊢ ( 𝜑 → 𝑅 ∈ 𝑈 )
6		mnringbased.6	⊢ ( 𝜑 → 𝑀 ∈ 𝑊 )
7		baseid	⊢ Base = Slot ( Base ‘ ndx )
8		basendxnmulrndx	⊢ ( Base ‘ ndx ) ≠ ( .r ‘ ndx )
9	1 7 8 2 3 5 6	mnringnmulrd	⊢ ( 𝜑 → ( Base ‘ 𝑉 ) = ( Base ‘ 𝐹 ) )
10	4 9	eqtrid	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐹 ) )