Step |
Hyp |
Ref |
Expression |
1 |
|
nn0z |
⊢ ( 𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ ) |
2 |
|
sgmval2 |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 σ 𝐵 ) = Σ 𝑘 ∈ { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ( 𝑘 ↑ 𝐴 ) ) |
3 |
1 2
|
sylan |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 𝐴 σ 𝐵 ) = Σ 𝑘 ∈ { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ( 𝑘 ↑ 𝐴 ) ) |
4 |
|
fzfid |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 1 ... 𝐵 ) ∈ Fin ) |
5 |
|
dvdsssfz1 |
⊢ ( 𝐵 ∈ ℕ → { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ⊆ ( 1 ... 𝐵 ) ) |
6 |
5
|
adantl |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ⊆ ( 1 ... 𝐵 ) ) |
7 |
4 6
|
ssfid |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ∈ Fin ) |
8 |
|
elrabi |
⊢ ( 𝑘 ∈ { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } → 𝑘 ∈ ℕ ) |
9 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → 𝐴 ∈ ℕ0 ) |
10 |
|
nnexpcl |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝐴 ∈ ℕ0 ) → ( 𝑘 ↑ 𝐴 ) ∈ ℕ ) |
11 |
8 9 10
|
syl2anr |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) ∧ 𝑘 ∈ { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ) → ( 𝑘 ↑ 𝐴 ) ∈ ℕ ) |
12 |
11
|
nnzd |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) ∧ 𝑘 ∈ { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ) → ( 𝑘 ↑ 𝐴 ) ∈ ℤ ) |
13 |
7 12
|
fsumzcl |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → Σ 𝑘 ∈ { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ( 𝑘 ↑ 𝐴 ) ∈ ℤ ) |
14 |
|
nnz |
⊢ ( 𝐵 ∈ ℕ → 𝐵 ∈ ℤ ) |
15 |
|
iddvds |
⊢ ( 𝐵 ∈ ℤ → 𝐵 ∥ 𝐵 ) |
16 |
14 15
|
syl |
⊢ ( 𝐵 ∈ ℕ → 𝐵 ∥ 𝐵 ) |
17 |
|
breq1 |
⊢ ( 𝑝 = 𝐵 → ( 𝑝 ∥ 𝐵 ↔ 𝐵 ∥ 𝐵 ) ) |
18 |
17
|
rspcev |
⊢ ( ( 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐵 ) → ∃ 𝑝 ∈ ℕ 𝑝 ∥ 𝐵 ) |
19 |
16 18
|
mpdan |
⊢ ( 𝐵 ∈ ℕ → ∃ 𝑝 ∈ ℕ 𝑝 ∥ 𝐵 ) |
20 |
|
rabn0 |
⊢ ( { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ≠ ∅ ↔ ∃ 𝑝 ∈ ℕ 𝑝 ∥ 𝐵 ) |
21 |
19 20
|
sylibr |
⊢ ( 𝐵 ∈ ℕ → { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ≠ ∅ ) |
22 |
21
|
adantl |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ≠ ∅ ) |
23 |
11
|
nnrpd |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) ∧ 𝑘 ∈ { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ) → ( 𝑘 ↑ 𝐴 ) ∈ ℝ+ ) |
24 |
7 22 23
|
fsumrpcl |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → Σ 𝑘 ∈ { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ( 𝑘 ↑ 𝐴 ) ∈ ℝ+ ) |
25 |
24
|
rpgt0d |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → 0 < Σ 𝑘 ∈ { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ( 𝑘 ↑ 𝐴 ) ) |
26 |
|
elnnz |
⊢ ( Σ 𝑘 ∈ { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ( 𝑘 ↑ 𝐴 ) ∈ ℕ ↔ ( Σ 𝑘 ∈ { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ( 𝑘 ↑ 𝐴 ) ∈ ℤ ∧ 0 < Σ 𝑘 ∈ { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ( 𝑘 ↑ 𝐴 ) ) ) |
27 |
13 25 26
|
sylanbrc |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → Σ 𝑘 ∈ { 𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵 } ( 𝑘 ↑ 𝐴 ) ∈ ℕ ) |
28 |
3 27
|
eqeltrd |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 𝐴 σ 𝐵 ) ∈ ℕ ) |