Step | Hyp | Ref
| Expression |
1 | | telgsumfzs.f |
. 2
⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵) |
2 | | telgsumfzs.n |
. . 3
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
3 | | oveq1 7282 |
. . . . . . . . 9
⊢ (𝑥 = 𝑀 → (𝑥 + 1) = (𝑀 + 1)) |
4 | 3 | oveq2d 7291 |
. . . . . . . 8
⊢ (𝑥 = 𝑀 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑀 + 1))) |
5 | 4 | raleqdv 3348 |
. . . . . . 7
⊢ (𝑥 = 𝑀 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵)) |
6 | 5 | anbi2d 629 |
. . . . . 6
⊢ (𝑥 = 𝑀 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵))) |
7 | | oveq2 7283 |
. . . . . . . . 9
⊢ (𝑥 = 𝑀 → (𝑀...𝑥) = (𝑀...𝑀)) |
8 | 7 | mpteq1d 5169 |
. . . . . . . 8
⊢ (𝑥 = 𝑀 → (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶)) = (𝑖 ∈ (𝑀...𝑀) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) |
9 | 8 | oveq2d 7291 |
. . . . . . 7
⊢ (𝑥 = 𝑀 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...𝑀) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶)))) |
10 | 3 | csbeq1d 3836 |
. . . . . . . 8
⊢ (𝑥 = 𝑀 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑀 + 1) / 𝑘⦌𝐶) |
11 | 10 | oveq2d 7291 |
. . . . . . 7
⊢ (𝑥 = 𝑀 → (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑀 + 1) /
𝑘⦌𝐶)) |
12 | 9, 11 | eqeq12d 2754 |
. . . . . 6
⊢ (𝑥 = 𝑀 → ((𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶) ↔ (𝐺 Σg (𝑖 ∈ (𝑀...𝑀) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑀 + 1) /
𝑘⦌𝐶))) |
13 | 6, 12 | imbi12d 345 |
. . . . 5
⊢ (𝑥 = 𝑀 → (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶)) ↔ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑀) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑀 + 1) /
𝑘⦌𝐶)))) |
14 | | oveq1 7282 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1)) |
15 | 14 | oveq2d 7291 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑦 + 1))) |
16 | 15 | raleqdv 3348 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵)) |
17 | 16 | anbi2d 629 |
. . . . . 6
⊢ (𝑥 = 𝑦 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵))) |
18 | | oveq2 7283 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝑀...𝑥) = (𝑀...𝑦)) |
19 | 18 | mpteq1d 5169 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶)) = (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) |
20 | 19 | oveq2d 7291 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶)))) |
21 | 14 | csbeq1d 3836 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑦 + 1) / 𝑘⦌𝐶) |
22 | 21 | oveq2d 7291 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑦 + 1) /
𝑘⦌𝐶)) |
23 | 20, 22 | eqeq12d 2754 |
. . . . . 6
⊢ (𝑥 = 𝑦 → ((𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶) ↔ (𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑦 + 1) /
𝑘⦌𝐶))) |
24 | 17, 23 | imbi12d 345 |
. . . . 5
⊢ (𝑥 = 𝑦 → (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶)) ↔ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑦 + 1) /
𝑘⦌𝐶)))) |
25 | | oveq1 7282 |
. . . . . . . . 9
⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1)) |
26 | 25 | oveq2d 7291 |
. . . . . . . 8
⊢ (𝑥 = (𝑦 + 1) → (𝑀...(𝑥 + 1)) = (𝑀...((𝑦 + 1) + 1))) |
27 | 26 | raleqdv 3348 |
. . . . . . 7
⊢ (𝑥 = (𝑦 + 1) → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) |
28 | 27 | anbi2d 629 |
. . . . . 6
⊢ (𝑥 = (𝑦 + 1) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵))) |
29 | | oveq2 7283 |
. . . . . . . . 9
⊢ (𝑥 = (𝑦 + 1) → (𝑀...𝑥) = (𝑀...(𝑦 + 1))) |
30 | 29 | mpteq1d 5169 |
. . . . . . . 8
⊢ (𝑥 = (𝑦 + 1) → (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶)) = (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) |
31 | 30 | oveq2d 7291 |
. . . . . . 7
⊢ (𝑥 = (𝑦 + 1) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶)))) |
32 | 25 | csbeq1d 3836 |
. . . . . . . 8
⊢ (𝑥 = (𝑦 + 1) → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶) |
33 | 32 | oveq2d 7291 |
. . . . . . 7
⊢ (𝑥 = (𝑦 + 1) → (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶)) |
34 | 31, 33 | eqeq12d 2754 |
. . . . . 6
⊢ (𝑥 = (𝑦 + 1) → ((𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶) ↔ (𝐺 Σg (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶))) |
35 | 28, 34 | imbi12d 345 |
. . . . 5
⊢ (𝑥 = (𝑦 + 1) → (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶)) ↔ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶)))) |
36 | | oveq1 7282 |
. . . . . . . . 9
⊢ (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1)) |
37 | 36 | oveq2d 7291 |
. . . . . . . 8
⊢ (𝑥 = 𝑁 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑁 + 1))) |
38 | 37 | raleqdv 3348 |
. . . . . . 7
⊢ (𝑥 = 𝑁 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵)) |
39 | 38 | anbi2d 629 |
. . . . . 6
⊢ (𝑥 = 𝑁 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵))) |
40 | | oveq2 7283 |
. . . . . . . . 9
⊢ (𝑥 = 𝑁 → (𝑀...𝑥) = (𝑀...𝑁)) |
41 | 40 | mpteq1d 5169 |
. . . . . . . 8
⊢ (𝑥 = 𝑁 → (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶)) = (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) |
42 | 41 | oveq2d 7291 |
. . . . . . 7
⊢ (𝑥 = 𝑁 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶)))) |
43 | 36 | csbeq1d 3836 |
. . . . . . . 8
⊢ (𝑥 = 𝑁 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑁 + 1) / 𝑘⦌𝐶) |
44 | 43 | oveq2d 7291 |
. . . . . . 7
⊢ (𝑥 = 𝑁 → (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑁 + 1) /
𝑘⦌𝐶)) |
45 | 42, 44 | eqeq12d 2754 |
. . . . . 6
⊢ (𝑥 = 𝑁 → ((𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶) ↔ (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑁 + 1) /
𝑘⦌𝐶))) |
46 | 39, 45 | imbi12d 345 |
. . . . 5
⊢ (𝑥 = 𝑁 → (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶)) ↔ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑁 + 1) /
𝑘⦌𝐶)))) |
47 | | eluzel2 12587 |
. . . . . . . . . . 11
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
48 | 2, 47 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ ℤ) |
49 | 48 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝑀 ∈ ℤ) |
50 | | fzsn 13298 |
. . . . . . . . 9
⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) |
51 | 49, 50 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝑀...𝑀) = {𝑀}) |
52 | 51 | mpteq1d 5169 |
. . . . . . 7
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝑖 ∈ (𝑀...𝑀) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶)) = (𝑖 ∈ {𝑀} ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) |
53 | 52 | oveq2d 7291 |
. . . . . 6
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑀) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (𝐺 Σg (𝑖 ∈ {𝑀} ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶)))) |
54 | | telgsumfzs.b |
. . . . . . 7
⊢ 𝐵 = (Base‘𝐺) |
55 | | telgsumfzs.g |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺 ∈ Abel) |
56 | | ablgrp 19391 |
. . . . . . . . . 10
⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) |
57 | 55, 56 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 ∈ Grp) |
58 | 57 | grpmndd 18589 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ Mnd) |
59 | 58 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ Mnd) |
60 | 57 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ Grp) |
61 | | uzid 12597 |
. . . . . . . . . . . 12
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
(ℤ≥‘𝑀)) |
62 | 49, 61 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝑀 ∈ (ℤ≥‘𝑀)) |
63 | | peano2uz 12641 |
. . . . . . . . . . 11
⊢ (𝑀 ∈
(ℤ≥‘𝑀) → (𝑀 + 1) ∈
(ℤ≥‘𝑀)) |
64 | 62, 63 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝑀 + 1) ∈
(ℤ≥‘𝑀)) |
65 | | eluzfz1 13263 |
. . . . . . . . . 10
⊢ ((𝑀 + 1) ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...(𝑀 + 1))) |
66 | 64, 65 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝑀 ∈ (𝑀...(𝑀 + 1))) |
67 | | rspcsbela 4369 |
. . . . . . . . 9
⊢ ((𝑀 ∈ (𝑀...(𝑀 + 1)) ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵) |
68 | 66, 67 | sylancom 588 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵) |
69 | | eluzfz2 13264 |
. . . . . . . . . 10
⊢ ((𝑀 + 1) ∈
(ℤ≥‘𝑀) → (𝑀 + 1) ∈ (𝑀...(𝑀 + 1))) |
70 | 64, 69 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝑀 + 1) ∈ (𝑀...(𝑀 + 1))) |
71 | | rspcsbela 4369 |
. . . . . . . . 9
⊢ (((𝑀 + 1) ∈ (𝑀...(𝑀 + 1)) ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋(𝑀 + 1) / 𝑘⦌𝐶 ∈ 𝐵) |
72 | 70, 71 | sylancom 588 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋(𝑀 + 1) / 𝑘⦌𝐶 ∈ 𝐵) |
73 | | telgsumfzs.m |
. . . . . . . . 9
⊢ − =
(-g‘𝐺) |
74 | 54, 73 | grpsubcl 18655 |
. . . . . . . 8
⊢ ((𝐺 ∈ Grp ∧
⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵 ∧ ⦋(𝑀 + 1) / 𝑘⦌𝐶 ∈ 𝐵) → (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑀 + 1) /
𝑘⦌𝐶) ∈ 𝐵) |
75 | 60, 68, 72, 74 | syl3anc 1370 |
. . . . . . 7
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑀 + 1) /
𝑘⦌𝐶) ∈ 𝐵) |
76 | | csbeq1 3835 |
. . . . . . . . 9
⊢ (𝑖 = 𝑀 → ⦋𝑖 / 𝑘⦌𝐶 = ⦋𝑀 / 𝑘⦌𝐶) |
77 | | oveq1 7282 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑀 → (𝑖 + 1) = (𝑀 + 1)) |
78 | 77 | csbeq1d 3836 |
. . . . . . . . 9
⊢ (𝑖 = 𝑀 → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = ⦋(𝑀 + 1) / 𝑘⦌𝐶) |
79 | 76, 78 | oveq12d 7293 |
. . . . . . . 8
⊢ (𝑖 = 𝑀 → (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑀 + 1) /
𝑘⦌𝐶)) |
80 | 79 | adantl 482 |
. . . . . . 7
⊢ (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) ∧ 𝑖 = 𝑀) → (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑀 + 1) /
𝑘⦌𝐶)) |
81 | 54, 59, 49, 75, 80 | gsumsnd 19553 |
. . . . . 6
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝐺 Σg (𝑖 ∈ {𝑀} ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑀 + 1) /
𝑘⦌𝐶)) |
82 | 53, 81 | eqtrd 2778 |
. . . . 5
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑀) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑀 + 1) /
𝑘⦌𝐶)) |
83 | 54, 55, 73 | telgsumfzslem 19589 |
. . . . . . 7
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ((𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑦 + 1) /
𝑘⦌𝐶) → (𝐺 Σg (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶))) |
84 | 83 | ex 413 |
. . . . . 6
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ((𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑦 + 1) /
𝑘⦌𝐶) → (𝐺 Σg (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶)))) |
85 | | eluzelz 12592 |
. . . . . . . . . . 11
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → 𝑦 ∈ ℤ) |
86 | 85 | peano2zd 12429 |
. . . . . . . . . 10
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑦 + 1) ∈ ℤ) |
87 | 86 | peano2zd 12429 |
. . . . . . . . . 10
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → ((𝑦 + 1) + 1) ∈ ℤ) |
88 | | peano2z 12361 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℤ → (𝑦 + 1) ∈
ℤ) |
89 | 88 | zred 12426 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℤ → (𝑦 + 1) ∈
ℝ) |
90 | 85, 89 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑦 + 1) ∈ ℝ) |
91 | 90 | lep1d 11906 |
. . . . . . . . . 10
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑦 + 1) ≤ ((𝑦 + 1) + 1)) |
92 | | eluz2 12588 |
. . . . . . . . . 10
⊢ (((𝑦 + 1) + 1) ∈
(ℤ≥‘(𝑦 + 1)) ↔ ((𝑦 + 1) ∈ ℤ ∧ ((𝑦 + 1) + 1) ∈ ℤ ∧
(𝑦 + 1) ≤ ((𝑦 + 1) + 1))) |
93 | 86, 87, 91, 92 | syl3anbrc 1342 |
. . . . . . . . 9
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘(𝑦 + 1))) |
94 | | fzss2 13296 |
. . . . . . . . 9
⊢ (((𝑦 + 1) + 1) ∈
(ℤ≥‘(𝑦 + 1)) → (𝑀...(𝑦 + 1)) ⊆ (𝑀...((𝑦 + 1) + 1))) |
95 | 93, 94 | syl 17 |
. . . . . . . 8
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑀...(𝑦 + 1)) ⊆ (𝑀...((𝑦 + 1) + 1))) |
96 | | ssralv 3987 |
. . . . . . . 8
⊢ ((𝑀...(𝑦 + 1)) ⊆ (𝑀...((𝑦 + 1) + 1)) → (∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵 → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵)) |
97 | 95, 96 | syl 17 |
. . . . . . 7
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵 → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵)) |
98 | 97 | adantld 491 |
. . . . . 6
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵)) |
99 | 84, 98 | a2and 842 |
. . . . 5
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑦 + 1) /
𝑘⦌𝐶)) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶)))) |
100 | 13, 24, 35, 46, 82, 99 | uzind4i 12650 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑁 + 1) /
𝑘⦌𝐶))) |
101 | 100 | expd 416 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝜑 → (∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑁 + 1) /
𝑘⦌𝐶)))) |
102 | 2, 101 | mpcom 38 |
. 2
⊢ (𝜑 → (∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑁 + 1) /
𝑘⦌𝐶))) |
103 | 1, 102 | mpd 15 |
1
⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑁 + 1) /
𝑘⦌𝐶)) |