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 6556 |
. . . . . . . . 9
⊢ (𝑥 = 𝑀 → (𝑥 + 1) = (𝑀 + 1)) |
4 | 3 | oveq2d 6565 |
. . . . . . . 8
⊢ (𝑥 = 𝑀 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑀 + 1))) |
5 | 4 | raleqdv 3121 |
. . . . . . 7
⊢ (𝑥 = 𝑀 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵)) |
6 | 5 | anbi2d 736 |
. . . . . 6
⊢ (𝑥 = 𝑀 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵))) |
7 | | oveq2 6557 |
. . . . . . . . 9
⊢ (𝑥 = 𝑀 → (𝑀...𝑥) = (𝑀...𝑀)) |
8 | 7 | mpteq1d 4666 |
. . . . . . . 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 6565 |
. . . . . . 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 3506 |
. . . . . . . 8
⊢ (𝑥 = 𝑀 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑀 + 1) / 𝑘⦌𝐶) |
11 | 10 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑥 = 𝑀 → (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑀 + 1) /
𝑘⦌𝐶)) |
12 | 9, 11 | eqeq12d 2625 |
. . . . . 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 333 |
. . . . 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 6556 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1)) |
15 | 14 | oveq2d 6565 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑦 + 1))) |
16 | 15 | raleqdv 3121 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵)) |
17 | 16 | anbi2d 736 |
. . . . . 6
⊢ (𝑥 = 𝑦 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵))) |
18 | | oveq2 6557 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝑀...𝑥) = (𝑀...𝑦)) |
19 | 18 | mpteq1d 4666 |
. . . . . . . 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 6565 |
. . . . . . 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 3506 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑦 + 1) / 𝑘⦌𝐶) |
22 | 21 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑦 + 1) /
𝑘⦌𝐶)) |
23 | 20, 22 | eqeq12d 2625 |
. . . . . 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 333 |
. . . . 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 6556 |
. . . . . . . . 9
⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1)) |
26 | 25 | oveq2d 6565 |
. . . . . . . 8
⊢ (𝑥 = (𝑦 + 1) → (𝑀...(𝑥 + 1)) = (𝑀...((𝑦 + 1) + 1))) |
27 | 26 | raleqdv 3121 |
. . . . . . 7
⊢ (𝑥 = (𝑦 + 1) → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) |
28 | 27 | anbi2d 736 |
. . . . . 6
⊢ (𝑥 = (𝑦 + 1) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵))) |
29 | | oveq2 6557 |
. . . . . . . . 9
⊢ (𝑥 = (𝑦 + 1) → (𝑀...𝑥) = (𝑀...(𝑦 + 1))) |
30 | 29 | mpteq1d 4666 |
. . . . . . . 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 6565 |
. . . . . . 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 3506 |
. . . . . . . 8
⊢ (𝑥 = (𝑦 + 1) → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶) |
33 | 32 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑥 = (𝑦 + 1) → (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶)) |
34 | 31, 33 | eqeq12d 2625 |
. . . . . 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 333 |
. . . . 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 6556 |
. . . . . . . . 9
⊢ (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1)) |
37 | 36 | oveq2d 6565 |
. . . . . . . 8
⊢ (𝑥 = 𝑁 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑁 + 1))) |
38 | 37 | raleqdv 3121 |
. . . . . . 7
⊢ (𝑥 = 𝑁 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵)) |
39 | 38 | anbi2d 736 |
. . . . . 6
⊢ (𝑥 = 𝑁 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵))) |
40 | | oveq2 6557 |
. . . . . . . . 9
⊢ (𝑥 = 𝑁 → (𝑀...𝑥) = (𝑀...𝑁)) |
41 | 40 | mpteq1d 4666 |
. . . . . . . 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 6565 |
. . . . . . 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 3506 |
. . . . . . . 8
⊢ (𝑥 = 𝑁 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑁 + 1) / 𝑘⦌𝐶) |
44 | 43 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑥 = 𝑁 → (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑁 + 1) /
𝑘⦌𝐶)) |
45 | 42, 44 | eqeq12d 2625 |
. . . . . 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 333 |
. . . . 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 11568 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
48 | 2, 47 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℤ) |
49 | 48 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝑀 ∈ ℤ) |
50 | | fzsn 12254 |
. . . . . . . . . 10
⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) |
51 | 49, 50 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝑀...𝑀) = {𝑀}) |
52 | 51 | mpteq1d 4666 |
. . . . . . . 8
⊢ ((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 6565 |
. . . . . . 7
⊢ ((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 |
. . . . . . . 8
⊢ 𝐵 = (Base‘𝐺) |
55 | | telgsumfzs.g |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺 ∈ Abel) |
56 | | ablgrp 18021 |
. . . . . . . . . . 11
⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) |
57 | 55, 56 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺 ∈ Grp) |
58 | | grpmnd 17252 |
. . . . . . . . . 10
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) |
59 | 57, 58 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 ∈ Mnd) |
60 | 59 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ Mnd) |
61 | 57 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ Grp) |
62 | | uzid 11578 |
. . . . . . . . . . . . 13
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
(ℤ≥‘𝑀)) |
63 | 49, 62 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝑀 ∈ (ℤ≥‘𝑀)) |
64 | | peano2uz 11617 |
. . . . . . . . . . . 12
⊢ (𝑀 ∈
(ℤ≥‘𝑀) → (𝑀 + 1) ∈
(ℤ≥‘𝑀)) |
65 | 63, 64 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝑀 + 1) ∈
(ℤ≥‘𝑀)) |
66 | | eluzfz1 12219 |
. . . . . . . . . . 11
⊢ ((𝑀 + 1) ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...(𝑀 + 1))) |
67 | 65, 66 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝑀 ∈ (𝑀...(𝑀 + 1))) |
68 | | rspcsbela 3958 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ (𝑀...(𝑀 + 1)) ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵) |
69 | 67, 68 | sylancom 698 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵) |
70 | | eluzfz2 12220 |
. . . . . . . . . . 11
⊢ ((𝑀 + 1) ∈
(ℤ≥‘𝑀) → (𝑀 + 1) ∈ (𝑀...(𝑀 + 1))) |
71 | 65, 70 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝑀 + 1) ∈ (𝑀...(𝑀 + 1))) |
72 | | rspcsbela 3958 |
. . . . . . . . . 10
⊢ (((𝑀 + 1) ∈ (𝑀...(𝑀 + 1)) ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋(𝑀 + 1) / 𝑘⦌𝐶 ∈ 𝐵) |
73 | 71, 72 | sylancom 698 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋(𝑀 + 1) / 𝑘⦌𝐶 ∈ 𝐵) |
74 | | telgsumfzs.m |
. . . . . . . . . 10
⊢ − =
(-g‘𝐺) |
75 | 54, 74 | grpsubcl 17318 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Grp ∧
⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵 ∧ ⦋(𝑀 + 1) / 𝑘⦌𝐶 ∈ 𝐵) → (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑀 + 1) /
𝑘⦌𝐶) ∈ 𝐵) |
76 | 61, 69, 73, 75 | syl3anc 1318 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑀 + 1) /
𝑘⦌𝐶) ∈ 𝐵) |
77 | | csbeq1 3502 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑀 → ⦋𝑖 / 𝑘⦌𝐶 = ⦋𝑀 / 𝑘⦌𝐶) |
78 | | oveq1 6556 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑀 → (𝑖 + 1) = (𝑀 + 1)) |
79 | 78 | csbeq1d 3506 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑀 → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = ⦋(𝑀 + 1) / 𝑘⦌𝐶) |
80 | 77, 79 | oveq12d 6567 |
. . . . . . . . 9
⊢ (𝑖 = 𝑀 → (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑀 + 1) /
𝑘⦌𝐶)) |
81 | 80 | adantl 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) ∧ 𝑖 = 𝑀) → (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑀 + 1) /
𝑘⦌𝐶)) |
82 | 54, 60, 49, 76, 81 | gsumsnd 18175 |
. . . . . . 7
⊢ ((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)) |
83 | 53, 82 | eqtrd 2644 |
. . . . . 6
⊢ ((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)) |
84 | 83 | a1i 11 |
. . . . 5
⊢ (𝑀 ∈ ℤ → ((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))) |
85 | 54, 55, 74 | telgsumfzslem 18208 |
. . . . . . 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))) |
86 | 85 | ex 449 |
. . . . . 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)))) |
87 | | eluzelz 11573 |
. . . . . . . . . . 11
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → 𝑦 ∈ ℤ) |
88 | 87 | peano2zd 11361 |
. . . . . . . . . 10
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑦 + 1) ∈ ℤ) |
89 | 88 | peano2zd 11361 |
. . . . . . . . . 10
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → ((𝑦 + 1) + 1) ∈ ℤ) |
90 | | peano2z 11295 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℤ → (𝑦 + 1) ∈
ℤ) |
91 | 90 | zred 11358 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℤ → (𝑦 + 1) ∈
ℝ) |
92 | 87, 91 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑦 + 1) ∈ ℝ) |
93 | 92 | lep1d 10834 |
. . . . . . . . . 10
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑦 + 1) ≤ ((𝑦 + 1) + 1)) |
94 | | eluz2 11569 |
. . . . . . . . . 10
⊢ (((𝑦 + 1) + 1) ∈
(ℤ≥‘(𝑦 + 1)) ↔ ((𝑦 + 1) ∈ ℤ ∧ ((𝑦 + 1) + 1) ∈ ℤ ∧
(𝑦 + 1) ≤ ((𝑦 + 1) + 1))) |
95 | 88, 89, 93, 94 | syl3anbrc 1239 |
. . . . . . . . 9
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘(𝑦 + 1))) |
96 | | fzss2 12252 |
. . . . . . . . 9
⊢ (((𝑦 + 1) + 1) ∈
(ℤ≥‘(𝑦 + 1)) → (𝑀...(𝑦 + 1)) ⊆ (𝑀...((𝑦 + 1) + 1))) |
97 | 95, 96 | syl 17 |
. . . . . . . 8
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑀...(𝑦 + 1)) ⊆ (𝑀...((𝑦 + 1) + 1))) |
98 | | ssralv 3629 |
. . . . . . . 8
⊢ ((𝑀...(𝑦 + 1)) ⊆ (𝑀...((𝑦 + 1) + 1)) → (∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵 → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵)) |
99 | 97, 98 | syl 17 |
. . . . . . 7
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵 → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵)) |
100 | 99 | adantld 482 |
. . . . . 6
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵)) |
101 | 86, 100 | a2and 849 |
. . . . 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)))) |
102 | 13, 24, 35, 46, 84, 101 | uzind4 11622 |
. . . 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))) |
103 | 102 | expd 451 |
. . 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)))) |
104 | 2, 103 | mpcom 37 |
. 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))) |
105 | 1, 104 | mpd 15 |
1
⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑁 + 1) /
𝑘⦌𝐶)) |