Proof of Theorem telgsumfzslem
Step | Hyp | Ref
| Expression |
1 | | telgsumfzs.b |
. . . . 5
⊢ 𝐵 = (Base‘𝐺) |
2 | | eqid 2622 |
. . . . 5
⊢
(+g‘𝐺) = (+g‘𝐺) |
3 | | telgsumfzs.g |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ Abel) |
4 | 3 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ Abel) |
5 | | ablcmn 18199 |
. . . . . . 7
⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) |
6 | 4, 5 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ CMnd) |
7 | 6 | adantl 482 |
. . . . 5
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → 𝐺 ∈ CMnd) |
8 | | fzfid 12772 |
. . . . 5
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (𝑀...(𝑦 + 1)) ∈ Fin) |
9 | | ablgrp 18198 |
. . . . . . . . 9
⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) |
10 | 3, 9 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ Grp) |
11 | 10 | ad2antrl 764 |
. . . . . . 7
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → 𝐺 ∈ Grp) |
12 | 11 | adantr 481 |
. . . . . 6
⊢ (((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) ∧ 𝑖 ∈ (𝑀...(𝑦 + 1))) → 𝐺 ∈ Grp) |
13 | | fzelp1 12393 |
. . . . . . 7
⊢ (𝑖 ∈ (𝑀...(𝑦 + 1)) → 𝑖 ∈ (𝑀...((𝑦 + 1) + 1))) |
14 | | simpr 477 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) |
15 | 14 | adantl 482 |
. . . . . . 7
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) |
16 | | rspcsbela 4006 |
. . . . . . 7
⊢ ((𝑖 ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋𝑖 / 𝑘⦌𝐶 ∈ 𝐵) |
17 | 13, 15, 16 | syl2anr 495 |
. . . . . 6
⊢ (((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) ∧ 𝑖 ∈ (𝑀...(𝑦 + 1))) → ⦋𝑖 / 𝑘⦌𝐶 ∈ 𝐵) |
18 | | fzp1elp1 12394 |
. . . . . . 7
⊢ (𝑖 ∈ (𝑀...(𝑦 + 1)) → (𝑖 + 1) ∈ (𝑀...((𝑦 + 1) + 1))) |
19 | | rspcsbela 4006 |
. . . . . . 7
⊢ (((𝑖 + 1) ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 ∈ 𝐵) |
20 | 18, 15, 19 | syl2anr 495 |
. . . . . 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 17495 |
. . . . . 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 1326 |
. . . . 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 12399 |
. . . . . 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 12388 |
. . . . . 6
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑀...(𝑦 + 1)) = ((𝑀...𝑦) ∪ {(𝑦 + 1)})) |
27 | 26 | adantr 481 |
. . . . 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 18330 |
. . . 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 481 |
. . 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 477 |
. . . 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 | | grpmnd 17429 |
. . . . . . . 8
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) |
32 | 10, 31 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ Mnd) |
33 | 32 | ad2antrl 764 |
. . . . . 6
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → 𝐺 ∈ Mnd) |
34 | | ovexd 6680 |
. . . . . 6
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (𝑦 + 1) ∈ V) |
35 | | peano2uz 11741 |
. . . . . . . . . 10
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑦 + 1) ∈
(ℤ≥‘𝑀)) |
36 | | eluzfz2 12349 |
. . . . . . . . . 10
⊢ ((𝑦 + 1) ∈
(ℤ≥‘𝑀) → (𝑦 + 1) ∈ (𝑀...(𝑦 + 1))) |
37 | 35, 36 | syl 17 |
. . . . . . . . 9
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑦 + 1) ∈ (𝑀...(𝑦 + 1))) |
38 | | fzelp1 12393 |
. . . . . . . . 9
⊢ ((𝑦 + 1) ∈ (𝑀...(𝑦 + 1)) → (𝑦 + 1) ∈ (𝑀...((𝑦 + 1) + 1))) |
39 | 37, 38 | syl 17 |
. . . . . . . 8
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → (𝑦 + 1) ∈ (𝑀...((𝑦 + 1) + 1))) |
40 | | rspcsbela 4006 |
. . . . . . . 8
⊢ (((𝑦 + 1) ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋(𝑦 + 1) / 𝑘⦌𝐶 ∈ 𝐵) |
41 | 39, 14, 40 | syl2an 494 |
. . . . . . 7
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ⦋(𝑦 + 1) / 𝑘⦌𝐶 ∈ 𝐵) |
42 | | peano2uz 11741 |
. . . . . . . . . 10
⊢ ((𝑦 + 1) ∈
(ℤ≥‘𝑀) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘𝑀)) |
43 | 35, 42 | syl 17 |
. . . . . . . . 9
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘𝑀)) |
44 | | eluzfz2 12349 |
. . . . . . . . 9
⊢ (((𝑦 + 1) + 1) ∈
(ℤ≥‘𝑀) → ((𝑦 + 1) + 1) ∈ (𝑀...((𝑦 + 1) + 1))) |
45 | 43, 44 | syl 17 |
. . . . . . . 8
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → ((𝑦 + 1) + 1) ∈ (𝑀...((𝑦 + 1) + 1))) |
46 | | rspcsbela 4006 |
. . . . . . . 8
⊢ ((((𝑦 + 1) + 1) ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶 ∈ 𝐵) |
47 | 45, 14, 46 | syl2an 494 |
. . . . . . 7
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶 ∈ 𝐵) |
48 | 1, 21 | grpsubcl 17495 |
. . . . . . 7
⊢ ((𝐺 ∈ Grp ∧
⦋(𝑦 + 1) /
𝑘⦌𝐶 ∈ 𝐵 ∧ ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶 ∈ 𝐵) → (⦋(𝑦 + 1) / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶) ∈ 𝐵) |
49 | 11, 41, 47, 48 | syl3anc 1326 |
. . . . . 6
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (⦋(𝑦 + 1) / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶) ∈ 𝐵) |
50 | | csbeq1 3536 |
. . . . . . . 8
⊢ (𝑖 = (𝑦 + 1) → ⦋𝑖 / 𝑘⦌𝐶 = ⦋(𝑦 + 1) / 𝑘⦌𝐶) |
51 | | oveq1 6657 |
. . . . . . . . 9
⊢ (𝑖 = (𝑦 + 1) → (𝑖 + 1) = ((𝑦 + 1) + 1)) |
52 | 51 | csbeq1d 3540 |
. . . . . . . 8
⊢ (𝑖 = (𝑦 + 1) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶) |
53 | 50, 52 | oveq12d 6668 |
. . . . . . 7
⊢ (𝑖 = (𝑦 + 1) → (⦋𝑖 / 𝑘⦌𝐶 −
⦋(𝑖 + 1) /
𝑘⦌𝐶) = (⦋(𝑦 + 1) / 𝑘⦌𝐶 −
⦋((𝑦 + 1) +
1) / 𝑘⦌𝐶)) |
54 | 53 | adantl 482 |
. . . . . 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)) |
55 | 1, 33, 34, 49, 54 | gsumsnd 18352 |
. . . . 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)) |
56 | 55 | adantr 481 |
. . . 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)) |
57 | 30, 56 | oveq12d 6668 |
. . 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))) |
58 | | eluzfz1 12348 |
. . . . . . 7
⊢ (((𝑦 + 1) + 1) ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...((𝑦 + 1) + 1))) |
59 | 43, 58 | syl 17 |
. . . . . 6
⊢ (𝑦 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...((𝑦 + 1) + 1))) |
60 | | rspcsbela 4006 |
. . . . . 6
⊢ ((𝑀 ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵) |
61 | 59, 14, 60 | syl2an 494 |
. . . . 5
⊢ ((𝑦 ∈
(ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵) |
62 | 1, 2, 21 | grpnpncan 17510 |
. . . . 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)) |
63 | 11, 61, 41, 47, 62 | syl13anc 1328 |
. . . 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)) |
64 | 63 | adantr 481 |
. . 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)) |
65 | 29, 57, 64 | 3eqtrd 2660 |
. 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)) |
66 | 65 | ex 450 |
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))) |