Theorem mnringaddgd

Description: The additive operation of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024)

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

Proof

Step	Hyp	Ref	Expression
1		mnringaddgd.1	⊢ 𝐹 = ( 𝑅 MndRing 𝑀 )
2		mnringaddgd.2	⊢ 𝐴 = ( Base ‘ 𝑀 )
3		mnringaddgd.3	⊢ 𝑉 = ( 𝑅 freeLMod 𝐴 )
4		mnringaddgd.4	⊢ ( 𝜑 → 𝑅 ∈ 𝑈 )
5		mnringaddgd.5	⊢ ( 𝜑 → 𝑀 ∈ 𝑊 )
6		df-plusg	⊢ +g = Slot 2
7		2nn	⊢ 2 ∈ ℕ
8		2re	⊢ 2 ∈ ℝ
9		2lt3	⊢ 2 < 3
10	8 9	ltneii	⊢ 2 ≠ 3
11		mulrndx	⊢ ( .r ‘ ndx ) = 3
12	10 11	neeqtrri	⊢ 2 ≠ ( .r ‘ ndx )
13	1 6 7 12 2 3 4 5	mnringnmulrd	⊢ ( 𝜑 → ( +g ‘ 𝑉 ) = ( +g ‘ 𝐹 ) )