Proof of Theorem telgsumfzslem
| Step | Hyp | Ref
| Expression |
| 1 | | telgsumfzs.b |
. . . . 5
⊢ 𝐵 = (Base‘𝐺) |
| 2 | | eqid 2737 |
. . . . 5
⊢
(+g‘𝐺) = (+g‘𝐺) |
| 3 | | telgsumfzs.g |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ Abel) |
| 4 | 3 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ Abel) |
| 5 | | ablcmn 19805 |
. . . . . . 7
⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) |
| 6 | 4, 5 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ CMnd) |
| 7 | 6 | adantl 481 |
. . . . 5
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → 𝐺 ∈ CMnd) |
| 8 | | fzfid 14014 |
. . . . 5
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (𝑀...(𝑦 + 1)) ∈ Fin) |
| 9 | | ablgrp 19803 |
. . . . . . . . 9
⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) |
| 10 | 3, 9 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ Grp) |
| 11 | 10 | ad2antrl 728 |
. . . . . . 7
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → 𝐺 ∈ Grp) |
| 12 | 11 | adantr 480 |
. . . . . 6
⊢ (((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) ∧ 𝑖 ∈ (𝑀...(𝑦 + 1))) → 𝐺 ∈ Grp) |
| 13 | | fzelp1 13616 |
. . . . . . 7
⊢ (𝑖 ∈ (𝑀...(𝑦 + 1)) → 𝑖 ∈ (𝑀...((𝑦 + 1) + 1))) |
| 14 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) |
| 15 | 14 | adantl 481 |
. . . . . . 7
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) |
| 16 | | rspcsbela 4438 |
. . . . . . 7
⊢ ((𝑖 ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋𝑖 / 𝑘⦌𝐶 ∈ 𝐵) |
| 17 | 13, 15, 16 | syl2anr 597 |
. . . . . 6
⊢ (((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) ∧ 𝑖 ∈ (𝑀...(𝑦 + 1))) → ⦋𝑖 / 𝑘⦌𝐶 ∈ 𝐵) |
| 18 | | fzp1elp1 13617 |
. . . . . . 7
⊢ (𝑖 ∈ (𝑀...(𝑦 + 1)) → (𝑖 + 1) ∈ (𝑀...((𝑦 + 1) + 1))) |
| 19 | | rspcsbela 4438 |
. . . . . . 7
⊢ (((𝑖 + 1) ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 ∈ 𝐵) |
| 20 | 18, 15, 19 | syl2anr 597 |
. . . . . 6
⊢ (((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) ∧ 𝑖 ∈ (𝑀...(𝑦 + 1))) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 ∈ 𝐵) |
| 21 | | telgsumfzs.m |
. . . . . . 7
⊢ − =
(-g‘𝐺) |
| 22 | 1, 21 | grpsubcl 19038 |
. . . . . 6
⊢ ((𝐺 ∈ Grp ∧
⦋𝑖 / 𝑘⦌𝐶 ∈ 𝐵 ∧ ⦋(𝑖 + 1) / 𝑘⦌𝐶 ∈ 𝐵) → (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶) ∈ 𝐵) |
| 23 | 12, 17, 20, 22 | syl3anc 1373 |
. . . . 5
⊢ (((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) ∧ 𝑖 ∈ (𝑀...(𝑦 + 1))) → (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶) ∈ 𝐵) |
| 24 | | fzp1disj 13623 |
. . . . . 6
⊢ ((𝑀...𝑦) ∩ {(𝑦 + 1)}) = ∅ |
| 25 | 24 | a1i 11 |
. . . . 5
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ((𝑀...𝑦) ∩ {(𝑦 + 1)}) = ∅) |
| 26 | | fzsuc 13611 |
. . . . . 6
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑀...(𝑦 + 1)) = ((𝑀...𝑦) ∪ {(𝑦 + 1)})) |
| 27 | 26 | adantr 480 |
. . . . 5
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (𝑀...(𝑦 + 1)) = ((𝑀...𝑦) ∪ {(𝑦 + 1)})) |
| 28 | 1, 2, 7, 8, 23, 25, 27 | gsummptfidmsplit 19948 |
. . . 4
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (𝐺 Σg (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = ((𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶)))(+g‘𝐺)(𝐺 Σg (𝑖 ∈ {(𝑦 + 1)} ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))))) |
| 29 | 28 | adantr 480 |
. . 3
⊢ (((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))) = ((𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶)))(+g‘𝐺)(𝐺 Σg (𝑖 ∈ {(𝑦 + 1)} ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))))) |
| 30 | | simpr 484 |
. . . 4
⊢ (((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) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑦 + 1) /
𝑘⦌𝐶)) |
| 31 | 10 | grpmndd 18964 |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ Mnd) |
| 32 | 31 | ad2antrl 728 |
. . . . . 6
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → 𝐺 ∈ Mnd) |
| 33 | | ovexd 7466 |
. . . . . 6
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (𝑦 + 1) ∈ V) |
| 34 | | peano2uz 12943 |
. . . . . . . . . 10
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑦 + 1) ∈
(ℤ≥‘𝑀)) |
| 35 | | eluzfz2 13572 |
. . . . . . . . . 10
⊢ ((𝑦 + 1) ∈
(ℤ≥‘𝑀) → (𝑦 + 1) ∈ (𝑀...(𝑦 + 1))) |
| 36 | 34, 35 | syl 17 |
. . . . . . . . 9
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑦 + 1) ∈ (𝑀...(𝑦 + 1))) |
| 37 | | fzelp1 13616 |
. . . . . . . . 9
⊢ ((𝑦 + 1) ∈ (𝑀...(𝑦 + 1)) → (𝑦 + 1) ∈ (𝑀...((𝑦 + 1) + 1))) |
| 38 | 36, 37 | syl 17 |
. . . . . . . 8
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑦 + 1) ∈ (𝑀...((𝑦 + 1) + 1))) |
| 39 | | rspcsbela 4438 |
. . . . . . . 8
⊢ (((𝑦 + 1) ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋(𝑦 + 1) / 𝑘⦌𝐶 ∈ 𝐵) |
| 40 | 38, 14, 39 | syl2an 596 |
. . . . . . 7
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ⦋(𝑦 + 1) / 𝑘⦌𝐶 ∈ 𝐵) |
| 41 | | peano2uz 12943 |
. . . . . . . . . 10
⊢ ((𝑦 + 1) ∈
(ℤ≥‘𝑀) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘𝑀)) |
| 42 | 34, 41 | syl 17 |
. . . . . . . . 9
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘𝑀)) |
| 43 | | eluzfz2 13572 |
. . . . . . . . 9
⊢ (((𝑦 + 1) + 1) ∈
(ℤ≥‘𝑀) → ((𝑦 + 1) + 1) ∈ (𝑀...((𝑦 + 1) + 1))) |
| 44 | 42, 43 | syl 17 |
. . . . . . . 8
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → ((𝑦 + 1) + 1) ∈ (𝑀...((𝑦 + 1) + 1))) |
| 45 | | rspcsbela 4438 |
. . . . . . . 8
⊢ ((((𝑦 + 1) + 1) ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶 ∈ 𝐵) |
| 46 | 44, 14, 45 | syl2an 596 |
. . . . . . 7
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶 ∈ 𝐵) |
| 47 | 1, 21 | grpsubcl 19038 |
. . . . . . 7
⊢ ((𝐺 ∈ Grp ∧
⦋(𝑦 + 1) /
𝑘⦌𝐶 ∈ 𝐵 ∧ ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶 ∈ 𝐵) → (⦋(𝑦 + 1) / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶) ∈ 𝐵) |
| 48 | 11, 40, 46, 47 | syl3anc 1373 |
. . . . . 6
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (⦋(𝑦 + 1) / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶) ∈ 𝐵) |
| 49 | | csbeq1 3902 |
. . . . . . . 8
⊢ (𝑖 = (𝑦 + 1) → ⦋𝑖 / 𝑘⦌𝐶 = ⦋(𝑦 + 1) / 𝑘⦌𝐶) |
| 50 | | oveq1 7438 |
. . . . . . . . 9
⊢ (𝑖 = (𝑦 + 1) → (𝑖 + 1) = ((𝑦 + 1) + 1)) |
| 51 | 50 | csbeq1d 3903 |
. . . . . . . 8
⊢ (𝑖 = (𝑦 + 1) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶) |
| 52 | 49, 51 | oveq12d 7449 |
. . . . . . 7
⊢ (𝑖 = (𝑦 + 1) → (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶) = (⦋(𝑦 + 1) / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶)) |
| 53 | 52 | adantl 481 |
. . . . . 6
⊢ (((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) ∧ 𝑖 = (𝑦 + 1)) → (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶) = (⦋(𝑦 + 1) / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶)) |
| 54 | 1, 32, 33, 48, 53 | gsumsnd 19970 |
. . . . 5
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (𝐺 Σg (𝑖 ∈ {(𝑦 + 1)} ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋(𝑦 + 1) / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶)) |
| 55 | 54 | adantr 480 |
. . . 4
⊢ (((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 ∈ {(𝑦 + 1)} ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶))) = (⦋(𝑦 + 1) / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶)) |
| 56 | 30, 55 | oveq12d 7449 |
. . 3
⊢ (((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) ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶)))(+g‘𝐺)(𝐺 Σg (𝑖 ∈ {(𝑦 + 1)} ↦ (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶)))) = ((⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑦 + 1) /
𝑘⦌𝐶)(+g‘𝐺)(⦋(𝑦 + 1) / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶))) |
| 57 | | eluzfz1 13571 |
. . . . . . 7
⊢ (((𝑦 + 1) + 1) ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...((𝑦 + 1) + 1))) |
| 58 | 42, 57 | syl 17 |
. . . . . 6
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...((𝑦 + 1) + 1))) |
| 59 | | rspcsbela 4438 |
. . . . . 6
⊢ ((𝑀 ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵) |
| 60 | 58, 14, 59 | syl2an 596 |
. . . . 5
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵) |
| 61 | 1, 2, 21 | grpnpncan 19053 |
. . . . 5
⊢ ((𝐺 ∈ Grp ∧
(⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵 ∧ ⦋(𝑦 + 1) / 𝑘⦌𝐶 ∈ 𝐵 ∧ ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶 ∈ 𝐵)) → ((⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑦 + 1) /
𝑘⦌𝐶)(+g‘𝐺)(⦋(𝑦 + 1) / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶)) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶)) |
| 62 | 11, 60, 40, 46, 61 | syl13anc 1374 |
. . . 4
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ((⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑦 + 1) /
𝑘⦌𝐶)(+g‘𝐺)(⦋(𝑦 + 1) / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶)) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶)) |
| 63 | 62 | adantr 480 |
. . 3
⊢ (((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)) → ((⦋𝑀 / 𝑘⦌𝐶 −
⦋(𝑦 + 1) /
𝑘⦌𝐶)(+g‘𝐺)(⦋(𝑦 + 1) / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶)) = (⦋𝑀 / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶)) |
| 64 | 29, 56, 63 | 3eqtrd 2781 |
. 2
⊢ (((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)) |
| 65 | 64 | ex 412 |
1
⊢ ((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))) |