Step |
Hyp |
Ref |
Expression |
1 |
|
tdeglem.a |
⊢ 𝐴 = { 𝑚 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑚 “ ℕ ) ∈ Fin } |
2 |
|
tdeglem.h |
⊢ 𝐻 = ( ℎ ∈ 𝐴 ↦ ( ℂfld Σg ℎ ) ) |
3 |
|
cnfld0 |
⊢ 0 = ( 0g ‘ ℂfld ) |
4 |
|
cnring |
⊢ ℂfld ∈ Ring |
5 |
|
ringcmn |
⊢ ( ℂfld ∈ Ring → ℂfld ∈ CMnd ) |
6 |
4 5
|
mp1i |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ℎ ∈ 𝐴 ) → ℂfld ∈ CMnd ) |
7 |
|
simpl |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ℎ ∈ 𝐴 ) → 𝐼 ∈ 𝑉 ) |
8 |
|
nn0subm |
⊢ ℕ0 ∈ ( SubMnd ‘ ℂfld ) |
9 |
8
|
a1i |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ℎ ∈ 𝐴 ) → ℕ0 ∈ ( SubMnd ‘ ℂfld ) ) |
10 |
1
|
psrbagfOLD |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ℎ ∈ 𝐴 ) → ℎ : 𝐼 ⟶ ℕ0 ) |
11 |
1
|
psrbagfsuppOLD |
⊢ ( ( ℎ ∈ 𝐴 ∧ 𝐼 ∈ 𝑉 ) → ℎ finSupp 0 ) |
12 |
11
|
ancoms |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ℎ ∈ 𝐴 ) → ℎ finSupp 0 ) |
13 |
3 6 7 9 10 12
|
gsumsubmcl |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ℎ ∈ 𝐴 ) → ( ℂfld Σg ℎ ) ∈ ℕ0 ) |
14 |
13 2
|
fmptd |
⊢ ( 𝐼 ∈ 𝑉 → 𝐻 : 𝐴 ⟶ ℕ0 ) |