Metamath Proof Explorer
Description: The additive operation of a monoid ring. (Contributed by Rohan
Ridenour, 14-May-2024) (Proof shortened by AV, 1-Nov-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 |
|
plusgid |
⊢ +g = Slot ( +g ‘ ndx ) |
7 |
|
plusgndxnmulrndx |
⊢ ( +g ‘ ndx ) ≠ ( .r ‘ ndx ) |
8 |
1 6 7 2 3 4 5
|
mnringnmulrd |
⊢ ( 𝜑 → ( +g ‘ 𝑉 ) = ( +g ‘ 𝐹 ) ) |