Theorem mnringvscad

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

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

Proof

Step	Hyp	Ref	Expression
1		mnringvscad.1	⊢ 𝐹 = ( 𝑅 MndRing 𝑀 )
2		mnringvscad.2	⊢ 𝐵 = ( Base ‘ 𝑀 )
3		mnringvscad.3	⊢ 𝑉 = ( 𝑅 freeLMod 𝐵 )
4		mnringvscad.4	⊢ ( 𝜑 → 𝑅 ∈ 𝑈 )
5		mnringvscad.5	⊢ ( 𝜑 → 𝑀 ∈ 𝑊 )
6		vscaid	⊢ ·𝑠 = Slot ( ·𝑠 ‘ ndx )
7		vscandxnmulrndx	⊢ ( ·𝑠 ‘ ndx ) ≠ ( .r ‘ ndx )
8	1 6 7 2 3 4 5	mnringnmulrd	⊢ ( 𝜑 → ( ·𝑠 ‘ 𝑉 ) = ( ·𝑠 ‘ 𝐹 ) )