| 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 7438 |
. . . . . . . . 9
⊢ (𝑥 = 𝑀 → (𝑥 + 1) = (𝑀 + 1)) |
| 4 | 3 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑥 = 𝑀 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑀 + 1))) |
| 5 | 4 | raleqdv 3326 |
. . . . . . 7
⊢ (𝑥 = 𝑀 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵)) |
| 6 | 5 | anbi2d 630 |
. . . . . 6
⊢ (𝑥 = 𝑀 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵))) |
| 7 | | oveq2 7439 |
. . . . . . . . 9
⊢ (𝑥 = 𝑀 → (𝑀...𝑥) = (𝑀...𝑀)) |
| 8 | 7 | mpteq1d 5237 |
. . . . . . . 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 7447 |
. . . . . . 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 3903 |
. . . . . . . 8
⊢ (𝑥 = 𝑀 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑀 + 1) / 𝑘⦌𝐶) |
| 11 | 10 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑥 = 𝑀 → (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑀 + 1) /
𝑘⦌𝐶)) |
| 12 | 9, 11 | eqeq12d 2753 |
. . . . . 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 344 |
. . . . 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 7438 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1)) |
| 15 | 14 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑦 + 1))) |
| 16 | 15 | raleqdv 3326 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵)) |
| 17 | 16 | anbi2d 630 |
. . . . . 6
⊢ (𝑥 = 𝑦 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵))) |
| 18 | | oveq2 7439 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝑀...𝑥) = (𝑀...𝑦)) |
| 19 | 18 | mpteq1d 5237 |
. . . . . . . 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 7447 |
. . . . . . 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 3903 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑦 + 1) / 𝑘⦌𝐶) |
| 22 | 21 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑦 + 1) /
𝑘⦌𝐶)) |
| 23 | 20, 22 | eqeq12d 2753 |
. . . . . 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 344 |
. . . . 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 7438 |
. . . . . . . . 9
⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1)) |
| 26 | 25 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑥 = (𝑦 + 1) → (𝑀...(𝑥 + 1)) = (𝑀...((𝑦 + 1) + 1))) |
| 27 | 26 | raleqdv 3326 |
. . . . . . 7
⊢ (𝑥 = (𝑦 + 1) → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) |
| 28 | 27 | anbi2d 630 |
. . . . . 6
⊢ (𝑥 = (𝑦 + 1) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵))) |
| 29 | | oveq2 7439 |
. . . . . . . . 9
⊢ (𝑥 = (𝑦 + 1) → (𝑀...𝑥) = (𝑀...(𝑦 + 1))) |
| 30 | 29 | mpteq1d 5237 |
. . . . . . . 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 7447 |
. . . . . . 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 3903 |
. . . . . . . 8
⊢ (𝑥 = (𝑦 + 1) → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶) |
| 33 | 32 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑥 = (𝑦 + 1) → (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶)) |
| 34 | 31, 33 | eqeq12d 2753 |
. . . . . 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 344 |
. . . . 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 7438 |
. . . . . . . . 9
⊢ (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1)) |
| 37 | 36 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑥 = 𝑁 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑁 + 1))) |
| 38 | 37 | raleqdv 3326 |
. . . . . . 7
⊢ (𝑥 = 𝑁 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵)) |
| 39 | 38 | anbi2d 630 |
. . . . . 6
⊢ (𝑥 = 𝑁 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵))) |
| 40 | | oveq2 7439 |
. . . . . . . . 9
⊢ (𝑥 = 𝑁 → (𝑀...𝑥) = (𝑀...𝑁)) |
| 41 | 40 | mpteq1d 5237 |
. . . . . . . 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 7447 |
. . . . . . 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 3903 |
. . . . . . . 8
⊢ (𝑥 = 𝑁 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑁 + 1) / 𝑘⦌𝐶) |
| 44 | 43 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑥 = 𝑁 → (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑥 + 1) /
𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑁 + 1) /
𝑘⦌𝐶)) |
| 45 | 42, 44 | eqeq12d 2753 |
. . . . . 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 344 |
. . . . 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 12883 |
. . . . . . . . . . 11
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
| 48 | 2, 47 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 49 | 48 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝑀 ∈ ℤ) |
| 50 | | fzsn 13606 |
. . . . . . . . 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 5237 |
. . . . . . 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 7447 |
. . . . . 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 19803 |
. . . . . . . . . 10
⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) |
| 57 | 55, 56 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 ∈ Grp) |
| 58 | 57 | grpmndd 18964 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ Mnd) |
| 59 | 58 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ Mnd) |
| 60 | 57 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ Grp) |
| 61 | | uzid 12893 |
. . . . . . . . . . . 12
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
(ℤ≥‘𝑀)) |
| 62 | 49, 61 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝑀 ∈ (ℤ≥‘𝑀)) |
| 63 | | peano2uz 12943 |
. . . . . . . . . . 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 13571 |
. . . . . . . . . 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 4438 |
. . . . . . . . 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 13572 |
. . . . . . . . . 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 4438 |
. . . . . . . . 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 19038 |
. . . . . . . 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 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑀 + 1) /
𝑘⦌𝐶) ∈ 𝐵) |
| 76 | | csbeq1 3902 |
. . . . . . . . 9
⊢ (𝑖 = 𝑀 → ⦋𝑖 / 𝑘⦌𝐶 = ⦋𝑀 / 𝑘⦌𝐶) |
| 77 | | oveq1 7438 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑀 → (𝑖 + 1) = (𝑀 + 1)) |
| 78 | 77 | csbeq1d 3903 |
. . . . . . . . 9
⊢ (𝑖 = 𝑀 → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = ⦋(𝑀 + 1) / 𝑘⦌𝐶) |
| 79 | 76, 78 | oveq12d 7449 |
. . . . . . . 8
⊢ (𝑖 = 𝑀 → (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑀 + 1) /
𝑘⦌𝐶)) |
| 80 | 79 | adantl 481 |
. . . . . . 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 19970 |
. . . . . 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 2777 |
. . . . 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 20006 |
. . . . . . 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 412 |
. . . . . 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 12888 |
. . . . . . . . . . 11
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → 𝑦 ∈ ℤ) |
| 86 | 85 | peano2zd 12725 |
. . . . . . . . . 10
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑦 + 1) ∈ ℤ) |
| 87 | 86 | peano2zd 12725 |
. . . . . . . . . 10
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → ((𝑦 + 1) + 1) ∈ ℤ) |
| 88 | | peano2z 12658 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℤ → (𝑦 + 1) ∈
ℤ) |
| 89 | 88 | zred 12722 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℤ → (𝑦 + 1) ∈
ℝ) |
| 90 | 85, 89 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑦 + 1) ∈ ℝ) |
| 91 | 90 | lep1d 12199 |
. . . . . . . . . 10
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑦 + 1) ≤ ((𝑦 + 1) + 1)) |
| 92 | | eluz2 12884 |
. . . . . . . . . 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 1344 |
. . . . . . . . 9
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘(𝑦 + 1))) |
| 94 | | fzss2 13604 |
. . . . . . . . 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 4052 |
. . . . . . . 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 490 |
. . . . . 6
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵)) |
| 99 | 84, 98 | a2and 846 |
. . . . 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 12952 |
. . . 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 415 |
. . 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)) |