Metamath Proof Explorer
Description: The base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024)
|
|
Ref |
Expression |
|
Hypotheses |
mnringbased.1 |
⊢ 𝐹 = ( 𝑅 MndRing 𝑀 ) |
|
|
mnringbased.2 |
⊢ 𝐴 = ( Base ‘ 𝑀 ) |
|
|
mnringbased.3 |
⊢ 𝑉 = ( 𝑅 freeLMod 𝐴 ) |
|
|
mnringbased.4 |
⊢ 𝐵 = ( Base ‘ 𝑉 ) |
|
|
mnringbased.5 |
⊢ ( 𝜑 → 𝑅 ∈ 𝑈 ) |
|
|
mnringbased.6 |
⊢ ( 𝜑 → 𝑀 ∈ 𝑊 ) |
|
Assertion |
mnringbased |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐹 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
mnringbased.1 |
⊢ 𝐹 = ( 𝑅 MndRing 𝑀 ) |
2 |
|
mnringbased.2 |
⊢ 𝐴 = ( Base ‘ 𝑀 ) |
3 |
|
mnringbased.3 |
⊢ 𝑉 = ( 𝑅 freeLMod 𝐴 ) |
4 |
|
mnringbased.4 |
⊢ 𝐵 = ( Base ‘ 𝑉 ) |
5 |
|
mnringbased.5 |
⊢ ( 𝜑 → 𝑅 ∈ 𝑈 ) |
6 |
|
mnringbased.6 |
⊢ ( 𝜑 → 𝑀 ∈ 𝑊 ) |
7 |
|
df-base |
⊢ Base = Slot 1 |
8 |
|
1nn |
⊢ 1 ∈ ℕ |
9 |
|
1re |
⊢ 1 ∈ ℝ |
10 |
|
1lt3 |
⊢ 1 < 3 |
11 |
9 10
|
ltneii |
⊢ 1 ≠ 3 |
12 |
|
mulrndx |
⊢ ( .r ‘ ndx ) = 3 |
13 |
11 12
|
neeqtrri |
⊢ 1 ≠ ( .r ‘ ndx ) |
14 |
1 7 8 13 2 3 5 6
|
mnringnmulrd |
⊢ ( 𝜑 → ( Base ‘ 𝑉 ) = ( Base ‘ 𝐹 ) ) |
15 |
4 14
|
syl5eq |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐹 ) ) |