Metamath Proof Explorer
Description: The base set of a monoid ring. Converse of mnringbased .
(Contributed by Rohan Ridenour, 14-May-2024)
|
|
Ref |
Expression |
|
Hypotheses |
mnringbaserd.1 |
⊢ 𝐹 = ( 𝑅 MndRing 𝑀 ) |
|
|
mnringbaserd.2 |
⊢ 𝐵 = ( Base ‘ 𝐹 ) |
|
|
mnringbaserd.3 |
⊢ 𝐴 = ( Base ‘ 𝑀 ) |
|
|
mnringbaserd.4 |
⊢ 𝑉 = ( 𝑅 freeLMod 𝐴 ) |
|
|
mnringbaserd.5 |
⊢ ( 𝜑 → 𝑅 ∈ 𝑈 ) |
|
|
mnringbaserd.6 |
⊢ ( 𝜑 → 𝑀 ∈ 𝑊 ) |
|
Assertion |
mnringbaserd |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑉 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
mnringbaserd.1 |
⊢ 𝐹 = ( 𝑅 MndRing 𝑀 ) |
2 |
|
mnringbaserd.2 |
⊢ 𝐵 = ( Base ‘ 𝐹 ) |
3 |
|
mnringbaserd.3 |
⊢ 𝐴 = ( Base ‘ 𝑀 ) |
4 |
|
mnringbaserd.4 |
⊢ 𝑉 = ( 𝑅 freeLMod 𝐴 ) |
5 |
|
mnringbaserd.5 |
⊢ ( 𝜑 → 𝑅 ∈ 𝑈 ) |
6 |
|
mnringbaserd.6 |
⊢ ( 𝜑 → 𝑀 ∈ 𝑊 ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝑉 ) = ( Base ‘ 𝑉 ) |
8 |
1 3 4 7 5 6
|
mnringbased |
⊢ ( 𝜑 → ( Base ‘ 𝑉 ) = ( Base ‘ 𝐹 ) ) |
9 |
2 8
|
eqtr4id |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑉 ) ) |