Step |
Hyp |
Ref |
Expression |
1 |
|
frlmval.f |
⊢ 𝐹 = ( 𝑅 freeLMod 𝐼 ) |
2 |
|
frlmbasfsupp.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
3 |
|
frlmbasfsupp.b |
⊢ 𝐵 = ( Base ‘ 𝐹 ) |
4 |
|
simpr |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
5 |
1 3
|
frlmrcl |
⊢ ( 𝑋 ∈ 𝐵 → 𝑅 ∈ V ) |
6 |
|
simpl |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → 𝐼 ∈ 𝑊 ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
8 |
1 7 2 3
|
frlmelbas |
⊢ ( ( 𝑅 ∈ V ∧ 𝐼 ∈ 𝑊 ) → ( 𝑋 ∈ 𝐵 ↔ ( 𝑋 ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) ∧ 𝑋 finSupp 0 ) ) ) |
9 |
5 6 8
|
syl2an2 |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ 𝐵 ↔ ( 𝑋 ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) ∧ 𝑋 finSupp 0 ) ) ) |
10 |
4 9
|
mpbid |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) ∧ 𝑋 finSupp 0 ) ) |
11 |
10
|
simprd |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 finSupp 0 ) |