Step |
Hyp |
Ref |
Expression |
1 |
|
rpvmasum.z |
⊢ 𝑍 = ( ℤ/nℤ ‘ 𝑁 ) |
2 |
|
rpvmasum.l |
⊢ 𝐿 = ( ℤRHom ‘ 𝑍 ) |
3 |
|
rpvmasum.a |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
4 |
|
dchrmusum.g |
⊢ 𝐺 = ( DChr ‘ 𝑁 ) |
5 |
|
dchrmusum.d |
⊢ 𝐷 = ( Base ‘ 𝐺 ) |
6 |
|
dchrmusum.1 |
⊢ 1 = ( 0g ‘ 𝐺 ) |
7 |
|
dchrmusum.b |
⊢ ( 𝜑 → 𝑋 ∈ 𝐷 ) |
8 |
|
dchrmusum.n1 |
⊢ ( 𝜑 → 𝑋 ≠ 1 ) |
9 |
|
eqid |
⊢ ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) = ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) |
10 |
1 2 3 4 5 6 7 8 9
|
dchrmusumlema |
⊢ ( 𝜑 → ∃ 𝑡 ∃ 𝑐 ∈ ( 0 [,) +∞ ) ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ⇝ 𝑡 ∧ ∀ 𝑦 ∈ ( 1 [,) +∞ ) ( abs ‘ ( ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ‘ ( ⌊ ‘ 𝑦 ) ) − 𝑡 ) ) ≤ ( 𝑐 / 𝑦 ) ) ) |
11 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑐 ∈ ( 0 [,) +∞ ) ∧ ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ⇝ 𝑡 ∧ ∀ 𝑦 ∈ ( 1 [,) +∞ ) ( abs ‘ ( ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ‘ ( ⌊ ‘ 𝑦 ) ) − 𝑡 ) ) ≤ ( 𝑐 / 𝑦 ) ) ) ) → 𝑁 ∈ ℕ ) |
12 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑐 ∈ ( 0 [,) +∞ ) ∧ ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ⇝ 𝑡 ∧ ∀ 𝑦 ∈ ( 1 [,) +∞ ) ( abs ‘ ( ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ‘ ( ⌊ ‘ 𝑦 ) ) − 𝑡 ) ) ≤ ( 𝑐 / 𝑦 ) ) ) ) → 𝑋 ∈ 𝐷 ) |
13 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑐 ∈ ( 0 [,) +∞ ) ∧ ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ⇝ 𝑡 ∧ ∀ 𝑦 ∈ ( 1 [,) +∞ ) ( abs ‘ ( ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ‘ ( ⌊ ‘ 𝑦 ) ) − 𝑡 ) ) ≤ ( 𝑐 / 𝑦 ) ) ) ) → 𝑋 ≠ 1 ) |
14 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑐 ∈ ( 0 [,) +∞ ) ∧ ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ⇝ 𝑡 ∧ ∀ 𝑦 ∈ ( 1 [,) +∞ ) ( abs ‘ ( ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ‘ ( ⌊ ‘ 𝑦 ) ) − 𝑡 ) ) ≤ ( 𝑐 / 𝑦 ) ) ) ) → 𝑐 ∈ ( 0 [,) +∞ ) ) |
15 |
|
simprrl |
⊢ ( ( 𝜑 ∧ ( 𝑐 ∈ ( 0 [,) +∞ ) ∧ ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ⇝ 𝑡 ∧ ∀ 𝑦 ∈ ( 1 [,) +∞ ) ( abs ‘ ( ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ‘ ( ⌊ ‘ 𝑦 ) ) − 𝑡 ) ) ≤ ( 𝑐 / 𝑦 ) ) ) ) → seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ⇝ 𝑡 ) |
16 |
|
simprrr |
⊢ ( ( 𝜑 ∧ ( 𝑐 ∈ ( 0 [,) +∞ ) ∧ ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ⇝ 𝑡 ∧ ∀ 𝑦 ∈ ( 1 [,) +∞ ) ( abs ‘ ( ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ‘ ( ⌊ ‘ 𝑦 ) ) − 𝑡 ) ) ≤ ( 𝑐 / 𝑦 ) ) ) ) → ∀ 𝑦 ∈ ( 1 [,) +∞ ) ( abs ‘ ( ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ‘ ( ⌊ ‘ 𝑦 ) ) − 𝑡 ) ) ≤ ( 𝑐 / 𝑦 ) ) |
17 |
1 2 11 4 5 6 12 13 9 14 15 16
|
dchrvmasumlem |
⊢ ( ( 𝜑 ∧ ( 𝑐 ∈ ( 0 [,) +∞ ) ∧ ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ⇝ 𝑡 ∧ ∀ 𝑦 ∈ ( 1 [,) +∞ ) ( abs ‘ ( ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ‘ ( ⌊ ‘ 𝑦 ) ) − 𝑡 ) ) ≤ ( 𝑐 / 𝑦 ) ) ) ) → ( 𝑥 ∈ ℝ+ ↦ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( Λ ‘ 𝑛 ) / 𝑛 ) ) ) ∈ 𝑂(1) ) |
18 |
17
|
rexlimdvaa |
⊢ ( 𝜑 → ( ∃ 𝑐 ∈ ( 0 [,) +∞ ) ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ⇝ 𝑡 ∧ ∀ 𝑦 ∈ ( 1 [,) +∞ ) ( abs ‘ ( ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ‘ ( ⌊ ‘ 𝑦 ) ) − 𝑡 ) ) ≤ ( 𝑐 / 𝑦 ) ) → ( 𝑥 ∈ ℝ+ ↦ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( Λ ‘ 𝑛 ) / 𝑛 ) ) ) ∈ 𝑂(1) ) ) |
19 |
18
|
exlimdv |
⊢ ( 𝜑 → ( ∃ 𝑡 ∃ 𝑐 ∈ ( 0 [,) +∞ ) ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ⇝ 𝑡 ∧ ∀ 𝑦 ∈ ( 1 [,) +∞ ) ( abs ‘ ( ( seq 1 ( + , ( 𝑎 ∈ ℕ ↦ ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑎 ) ) / 𝑎 ) ) ) ‘ ( ⌊ ‘ 𝑦 ) ) − 𝑡 ) ) ≤ ( 𝑐 / 𝑦 ) ) → ( 𝑥 ∈ ℝ+ ↦ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( Λ ‘ 𝑛 ) / 𝑛 ) ) ) ∈ 𝑂(1) ) ) |
20 |
10 19
|
mpd |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ+ ↦ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) · ( ( Λ ‘ 𝑛 ) / 𝑛 ) ) ) ∈ 𝑂(1) ) |