| Step |
Hyp |
Ref |
Expression |
| 1 |
|
gsumzrsum.1 |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
| 2 |
|
gsumzrsum.2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℤ ) |
| 3 |
|
cnfldbas |
⊢ ℂ = ( Base ‘ ℂfld ) |
| 4 |
|
cnfldadd |
⊢ + = ( +g ‘ ℂfld ) |
| 5 |
|
df-zring |
⊢ ℤring = ( ℂfld ↾s ℤ ) |
| 6 |
|
cnfldex |
⊢ ℂfld ∈ V |
| 7 |
6
|
a1i |
⊢ ( 𝜑 → ℂfld ∈ V ) |
| 8 |
|
zsscn |
⊢ ℤ ⊆ ℂ |
| 9 |
8
|
a1i |
⊢ ( 𝜑 → ℤ ⊆ ℂ ) |
| 10 |
2
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℤ ) |
| 11 |
|
0zd |
⊢ ( 𝜑 → 0 ∈ ℤ ) |
| 12 |
|
addlid |
⊢ ( 𝑘 ∈ ℂ → ( 0 + 𝑘 ) = 𝑘 ) |
| 13 |
|
addrid |
⊢ ( 𝑘 ∈ ℂ → ( 𝑘 + 0 ) = 𝑘 ) |
| 14 |
12 13
|
jca |
⊢ ( 𝑘 ∈ ℂ → ( ( 0 + 𝑘 ) = 𝑘 ∧ ( 𝑘 + 0 ) = 𝑘 ) ) |
| 15 |
14
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℂ ) → ( ( 0 + 𝑘 ) = 𝑘 ∧ ( 𝑘 + 0 ) = 𝑘 ) ) |
| 16 |
3 4 5 7 1 9 10 11 15
|
gsumress |
⊢ ( 𝜑 → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = ( ℤring Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) |
| 17 |
2
|
zcnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
| 18 |
1 17
|
gsumfsum |
⊢ ( 𝜑 → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = Σ 𝑘 ∈ 𝐴 𝐵 ) |
| 19 |
16 18
|
eqtr3d |
⊢ ( 𝜑 → ( ℤring Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = Σ 𝑘 ∈ 𝐴 𝐵 ) |