Step |
Hyp |
Ref |
Expression |
1 |
|
rpvmasum.z |
⊢ 𝑍 = ( ℤ/nℤ ‘ 𝑁 ) |
2 |
|
rpvmasum.l |
⊢ 𝐿 = ( ℤRHom ‘ 𝑍 ) |
3 |
|
rpvmasum.a |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
4 |
|
rpvmasum2.g |
⊢ 𝐺 = ( DChr ‘ 𝑁 ) |
5 |
|
rpvmasum2.d |
⊢ 𝐷 = ( Base ‘ 𝐺 ) |
6 |
|
rpvmasum2.1 |
⊢ 1 = ( 0g ‘ 𝐺 ) |
7 |
|
dchrisum0f.f |
⊢ 𝐹 = ( 𝑏 ∈ ℕ ↦ Σ 𝑣 ∈ { 𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏 } ( 𝑋 ‘ ( 𝐿 ‘ 𝑣 ) ) ) |
8 |
|
breq2 |
⊢ ( 𝑏 = 𝐴 → ( 𝑞 ∥ 𝑏 ↔ 𝑞 ∥ 𝐴 ) ) |
9 |
8
|
rabbidv |
⊢ ( 𝑏 = 𝐴 → { 𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏 } = { 𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴 } ) |
10 |
9
|
sumeq1d |
⊢ ( 𝑏 = 𝐴 → Σ 𝑣 ∈ { 𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏 } ( 𝑋 ‘ ( 𝐿 ‘ 𝑣 ) ) = Σ 𝑣 ∈ { 𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴 } ( 𝑋 ‘ ( 𝐿 ‘ 𝑣 ) ) ) |
11 |
|
2fveq3 |
⊢ ( 𝑣 = 𝑡 → ( 𝑋 ‘ ( 𝐿 ‘ 𝑣 ) ) = ( 𝑋 ‘ ( 𝐿 ‘ 𝑡 ) ) ) |
12 |
11
|
cbvsumv |
⊢ Σ 𝑣 ∈ { 𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴 } ( 𝑋 ‘ ( 𝐿 ‘ 𝑣 ) ) = Σ 𝑡 ∈ { 𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴 } ( 𝑋 ‘ ( 𝐿 ‘ 𝑡 ) ) |
13 |
10 12
|
eqtrdi |
⊢ ( 𝑏 = 𝐴 → Σ 𝑣 ∈ { 𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏 } ( 𝑋 ‘ ( 𝐿 ‘ 𝑣 ) ) = Σ 𝑡 ∈ { 𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴 } ( 𝑋 ‘ ( 𝐿 ‘ 𝑡 ) ) ) |
14 |
|
sumex |
⊢ Σ 𝑡 ∈ { 𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴 } ( 𝑋 ‘ ( 𝐿 ‘ 𝑡 ) ) ∈ V |
15 |
13 7 14
|
fvmpt |
⊢ ( 𝐴 ∈ ℕ → ( 𝐹 ‘ 𝐴 ) = Σ 𝑡 ∈ { 𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴 } ( 𝑋 ‘ ( 𝐿 ‘ 𝑡 ) ) ) |