Theorem mnringnmulrd

Description: Components of a monoid ring other than its ring product match its underlying free module. (Contributed by Rohan Ridenour, 14-May-2024) (Revised by AV, 1-Nov-2024)

	Ref	Expression
Hypotheses	mnringnmulrd.1	⊢ 𝐹 = ( 𝑅 MndRing 𝑀 )
	mnringnmulrd.2	⊢ 𝐸 = Slot ( 𝐸 ‘ ndx )
	mnringnmulrd.4	⊢ ( 𝐸 ‘ ndx ) ≠ ( .r ‘ ndx )
	mnringnmulrd.5	⊢ 𝐴 = ( Base ‘ 𝑀 )
	mnringnmulrd.6	⊢ 𝑉 = ( 𝑅 freeLMod 𝐴 )
	mnringnmulrd.7	⊢ ( 𝜑 → 𝑅 ∈ 𝑈 )
	mnringnmulrd.8	⊢ ( 𝜑 → 𝑀 ∈ 𝑊 )
Assertion	mnringnmulrd	⊢ ( 𝜑 → ( 𝐸 ‘ 𝑉 ) = ( 𝐸 ‘ 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		mnringnmulrd.1	⊢ 𝐹 = ( 𝑅 MndRing 𝑀 )
2		mnringnmulrd.2	⊢ 𝐸 = Slot ( 𝐸 ‘ ndx )
3		mnringnmulrd.4	⊢ ( 𝐸 ‘ ndx ) ≠ ( .r ‘ ndx )
4		mnringnmulrd.5	⊢ 𝐴 = ( Base ‘ 𝑀 )
5		mnringnmulrd.6	⊢ 𝑉 = ( 𝑅 freeLMod 𝐴 )
6		mnringnmulrd.7	⊢ ( 𝜑 → 𝑅 ∈ 𝑈 )
7		mnringnmulrd.8	⊢ ( 𝜑 → 𝑀 ∈ 𝑊 )
8	2 3	setsnid	⊢ ( 𝐸 ‘ 𝑉 ) = ( 𝐸 ‘ ( 𝑉 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑉 ) , 𝑦 ∈ ( Base ‘ 𝑉 ) ↦ ( 𝑉 Σg ( 𝑎 ∈ 𝐴 , 𝑏 ∈ 𝐴 ↦ ( 𝑖 ∈ 𝐴 ↦ if ( 𝑖 = ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) , ( ( 𝑥 ‘ 𝑎 ) ( .r ‘ 𝑅 ) ( 𝑦 ‘ 𝑏 ) ) , ( 0g ‘ 𝑅 ) ) ) ) ) ) ⟩ ) )
9		eqid	⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
10		eqid	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
11		eqid	⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 )
12		eqid	⊢ ( Base ‘ 𝑉 ) = ( Base ‘ 𝑉 )
13	1 9 10 4 11 5 12 6 7	mnringvald	⊢ ( 𝜑 → 𝐹 = ( 𝑉 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑉 ) , 𝑦 ∈ ( Base ‘ 𝑉 ) ↦ ( 𝑉 Σg ( 𝑎 ∈ 𝐴 , 𝑏 ∈ 𝐴 ↦ ( 𝑖 ∈ 𝐴 ↦ if ( 𝑖 = ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) , ( ( 𝑥 ‘ 𝑎 ) ( .r ‘ 𝑅 ) ( 𝑦 ‘ 𝑏 ) ) , ( 0g ‘ 𝑅 ) ) ) ) ) ) ⟩ ) )
14	13	fveq2d	⊢ ( 𝜑 → ( 𝐸 ‘ 𝐹 ) = ( 𝐸 ‘ ( 𝑉 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑉 ) , 𝑦 ∈ ( Base ‘ 𝑉 ) ↦ ( 𝑉 Σg ( 𝑎 ∈ 𝐴 , 𝑏 ∈ 𝐴 ↦ ( 𝑖 ∈ 𝐴 ↦ if ( 𝑖 = ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) , ( ( 𝑥 ‘ 𝑎 ) ( .r ‘ 𝑅 ) ( 𝑦 ‘ 𝑏 ) ) , ( 0g ‘ 𝑅 ) ) ) ) ) ) ⟩ ) ) )
15	8 14	eqtr4id	⊢ ( 𝜑 → ( 𝐸 ‘ 𝑉 ) = ( 𝐸 ‘ 𝐹 ) )