Step |
Hyp |
Ref |
Expression |
1 |
|
frlmval.f |
⊢ 𝐹 = ( 𝑅 freeLMod 𝐼 ) |
2 |
|
frlmelbas.n |
⊢ 𝑁 = ( Base ‘ 𝑅 ) |
3 |
|
frlmelbas.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
4 |
|
frlmelbas.b |
⊢ 𝐵 = ( Base ‘ 𝐹 ) |
5 |
|
eqid |
⊢ { 𝑘 ∈ ( 𝑁 ↑m 𝐼 ) ∣ 𝑘 finSupp 0 } = { 𝑘 ∈ ( 𝑁 ↑m 𝐼 ) ∣ 𝑘 finSupp 0 } |
6 |
1 2 3 5
|
frlmbas |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → { 𝑘 ∈ ( 𝑁 ↑m 𝐼 ) ∣ 𝑘 finSupp 0 } = ( Base ‘ 𝐹 ) ) |
7 |
4 6
|
eqtr4id |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → 𝐵 = { 𝑘 ∈ ( 𝑁 ↑m 𝐼 ) ∣ 𝑘 finSupp 0 } ) |
8 |
7
|
eleq2d |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → ( 𝑋 ∈ 𝐵 ↔ 𝑋 ∈ { 𝑘 ∈ ( 𝑁 ↑m 𝐼 ) ∣ 𝑘 finSupp 0 } ) ) |
9 |
|
breq1 |
⊢ ( 𝑘 = 𝑋 → ( 𝑘 finSupp 0 ↔ 𝑋 finSupp 0 ) ) |
10 |
9
|
elrab |
⊢ ( 𝑋 ∈ { 𝑘 ∈ ( 𝑁 ↑m 𝐼 ) ∣ 𝑘 finSupp 0 } ↔ ( 𝑋 ∈ ( 𝑁 ↑m 𝐼 ) ∧ 𝑋 finSupp 0 ) ) |
11 |
8 10
|
bitrdi |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → ( 𝑋 ∈ 𝐵 ↔ ( 𝑋 ∈ ( 𝑁 ↑m 𝐼 ) ∧ 𝑋 finSupp 0 ) ) ) |