Theorem mnringvscad

Description: The scalar product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-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		df-vsca	⊢ ·𝑠 = Slot 6
7		6nn	⊢ 6 ∈ ℕ
8		3re	⊢ 3 ∈ ℝ
9		3lt6	⊢ 3 < 6
10	8 9	gtneii	⊢ 6 ≠ 3
11		mulrndx	⊢ ( .r ‘ ndx ) = 3
12	10 11	neeqtrri	⊢ 6 ≠ ( .r ‘ ndx )
13	1 6 7 12 2 3 4 5	mnringnmulrd	⊢ ( 𝜑 → ( ·𝑠 ‘ 𝑉 ) = ( ·𝑠 ‘ 𝐹 ) )