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 |
|
cnfldbas |
⊢ ℂ = ( Base ‘ ℂfld ) |
4 |
|
cnfld0 |
⊢ 0 = ( 0g ‘ ℂfld ) |
5 |
|
cnfldadd |
⊢ + = ( +g ‘ ℂfld ) |
6 |
|
cnring |
⊢ ℂfld ∈ Ring |
7 |
|
ringcmn |
⊢ ( ℂfld ∈ Ring → ℂfld ∈ CMnd ) |
8 |
6 7
|
mp1i |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ℂfld ∈ CMnd ) |
9 |
|
simpl |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑋 ∈ 𝐴 ) |
10 |
1
|
psrbagf |
⊢ ( 𝑋 ∈ 𝐴 → 𝑋 : 𝐼 ⟶ ℕ0 ) |
11 |
|
nn0sscn |
⊢ ℕ0 ⊆ ℂ |
12 |
|
fss |
⊢ ( ( 𝑋 : 𝐼 ⟶ ℕ0 ∧ ℕ0 ⊆ ℂ ) → 𝑋 : 𝐼 ⟶ ℂ ) |
13 |
10 11 12
|
sylancl |
⊢ ( 𝑋 ∈ 𝐴 → 𝑋 : 𝐼 ⟶ ℂ ) |
14 |
13
|
adantr |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑋 : 𝐼 ⟶ ℂ ) |
15 |
14
|
ffnd |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑋 Fn 𝐼 ) |
16 |
9 15
|
fndmexd |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝐼 ∈ V ) |
17 |
1
|
psrbagf |
⊢ ( 𝑌 ∈ 𝐴 → 𝑌 : 𝐼 ⟶ ℕ0 ) |
18 |
|
fss |
⊢ ( ( 𝑌 : 𝐼 ⟶ ℕ0 ∧ ℕ0 ⊆ ℂ ) → 𝑌 : 𝐼 ⟶ ℂ ) |
19 |
17 11 18
|
sylancl |
⊢ ( 𝑌 ∈ 𝐴 → 𝑌 : 𝐼 ⟶ ℂ ) |
20 |
19
|
adantl |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑌 : 𝐼 ⟶ ℂ ) |
21 |
1
|
psrbagfsupp |
⊢ ( 𝑋 ∈ 𝐴 → 𝑋 finSupp 0 ) |
22 |
21
|
adantr |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑋 finSupp 0 ) |
23 |
1
|
psrbagfsupp |
⊢ ( 𝑌 ∈ 𝐴 → 𝑌 finSupp 0 ) |
24 |
23
|
adantl |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑌 finSupp 0 ) |
25 |
3 4 5 8 16 14 20 22 24
|
gsumadd |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( ℂfld Σg ( 𝑋 ∘f + 𝑌 ) ) = ( ( ℂfld Σg 𝑋 ) + ( ℂfld Σg 𝑌 ) ) ) |
26 |
1
|
psrbagaddcl |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 ∘f + 𝑌 ) ∈ 𝐴 ) |
27 |
|
oveq2 |
⊢ ( ℎ = ( 𝑋 ∘f + 𝑌 ) → ( ℂfld Σg ℎ ) = ( ℂfld Σg ( 𝑋 ∘f + 𝑌 ) ) ) |
28 |
|
ovex |
⊢ ( ℂfld Σg ( 𝑋 ∘f + 𝑌 ) ) ∈ V |
29 |
27 2 28
|
fvmpt |
⊢ ( ( 𝑋 ∘f + 𝑌 ) ∈ 𝐴 → ( 𝐻 ‘ ( 𝑋 ∘f + 𝑌 ) ) = ( ℂfld Σg ( 𝑋 ∘f + 𝑌 ) ) ) |
30 |
26 29
|
syl |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝐻 ‘ ( 𝑋 ∘f + 𝑌 ) ) = ( ℂfld Σg ( 𝑋 ∘f + 𝑌 ) ) ) |
31 |
|
oveq2 |
⊢ ( ℎ = 𝑋 → ( ℂfld Σg ℎ ) = ( ℂfld Σg 𝑋 ) ) |
32 |
|
ovex |
⊢ ( ℂfld Σg 𝑋 ) ∈ V |
33 |
31 2 32
|
fvmpt |
⊢ ( 𝑋 ∈ 𝐴 → ( 𝐻 ‘ 𝑋 ) = ( ℂfld Σg 𝑋 ) ) |
34 |
|
oveq2 |
⊢ ( ℎ = 𝑌 → ( ℂfld Σg ℎ ) = ( ℂfld Σg 𝑌 ) ) |
35 |
|
ovex |
⊢ ( ℂfld Σg 𝑌 ) ∈ V |
36 |
34 2 35
|
fvmpt |
⊢ ( 𝑌 ∈ 𝐴 → ( 𝐻 ‘ 𝑌 ) = ( ℂfld Σg 𝑌 ) ) |
37 |
33 36
|
oveqan12d |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( ( 𝐻 ‘ 𝑋 ) + ( 𝐻 ‘ 𝑌 ) ) = ( ( ℂfld Σg 𝑋 ) + ( ℂfld Σg 𝑌 ) ) ) |
38 |
25 30 37
|
3eqtr4d |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝐻 ‘ ( 𝑋 ∘f + 𝑌 ) ) = ( ( 𝐻 ‘ 𝑋 ) + ( 𝐻 ‘ 𝑌 ) ) ) |