Step | Hyp | Ref
| Expression |
1 | | fvex 6113 |
. . 3
⊢
(0g‘𝑅) ∈ V |
2 | 1 | a1i 11 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (0g‘𝑅) ∈ V) |
3 | | ovex 6577 |
. . 3
⊢ (𝑅 Σg
(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) ∈ V |
4 | 3 | a1i 11 |
. 2
⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑅 Σg
(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) ∈ V) |
5 | | ply1mulgsum.p |
. . . 4
⊢ 𝑃 = (Poly1‘𝑅) |
6 | | ply1mulgsum.b |
. . . 4
⊢ 𝐵 = (Base‘𝑃) |
7 | | ply1mulgsum.a |
. . . 4
⊢ 𝐴 = (coe1‘𝐾) |
8 | | ply1mulgsum.c |
. . . 4
⊢ 𝐶 = (coe1‘𝐿) |
9 | | ply1mulgsum.x |
. . . 4
⊢ 𝑋 = (var1‘𝑅) |
10 | | ply1mulgsum.pm |
. . . 4
⊢ × =
(.r‘𝑃) |
11 | | ply1mulgsum.sm |
. . . 4
⊢ · = (
·𝑠 ‘𝑃) |
12 | | ply1mulgsum.rm |
. . . 4
⊢ ∗ =
(.r‘𝑅) |
13 | | ply1mulgsum.m |
. . . 4
⊢ 𝑀 = (mulGrp‘𝑃) |
14 | | ply1mulgsum.e |
. . . 4
⊢ ↑ =
(.g‘𝑀) |
15 | 5, 6, 7, 8, 9, 10,
11, 12, 13, 14 | ply1mulgsumlem2 41969 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅))) |
16 | | vex 3176 |
. . . . . . . . 9
⊢ 𝑛 ∈ V |
17 | | csbov2g 6589 |
. . . . . . . . . 10
⊢ (𝑛 ∈ V →
⦋𝑛 / 𝑘⦌(𝑅 Σg
(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) = (𝑅 Σg
⦋𝑛 / 𝑘⦌(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙)))))) |
18 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ V → 𝑛 ∈ V) |
19 | | oveq2 6557 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑛 → (0...𝑘) = (0...𝑛)) |
20 | | oveq1 6556 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑛 → (𝑘 − 𝑙) = (𝑛 − 𝑙)) |
21 | 20 | fveq2d 6107 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑛 → (𝐶‘(𝑘 − 𝑙)) = (𝐶‘(𝑛 − 𝑙))) |
22 | 21 | oveq2d 6565 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑛 → ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))) = ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙)))) |
23 | 19, 22 | mpteq12dv 4663 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑛 → (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙)))) = (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) |
24 | 23 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈ V ∧ 𝑘 = 𝑛) → (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙)))) = (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) |
25 | 18, 24 | csbied 3526 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ V →
⦋𝑛 / 𝑘⦌(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙)))) = (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) |
26 | 25 | oveq2d 6565 |
. . . . . . . . . 10
⊢ (𝑛 ∈ V → (𝑅 Σg
⦋𝑛 / 𝑘⦌(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) = (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙)))))) |
27 | 17, 26 | eqtrd 2644 |
. . . . . . . . 9
⊢ (𝑛 ∈ V →
⦋𝑛 / 𝑘⦌(𝑅 Σg
(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) = (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙)))))) |
28 | 16, 27 | ax-mp 5 |
. . . . . . . 8
⊢
⦋𝑛 /
𝑘⦌(𝑅 Σg
(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) = (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) |
29 | | simpr 476 |
. . . . . . . 8
⊢
(((((𝑅 ∈ Ring
∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
∧ (𝑅
Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅)) → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅)) |
30 | 28, 29 | syl5eq 2656 |
. . . . . . 7
⊢
(((((𝑅 ∈ Ring
∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
∧ (𝑅
Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅)) → ⦋𝑛 / 𝑘⦌(𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) = (0g‘𝑅)) |
31 | 30 | ex 449 |
. . . . . 6
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ ((𝑅
Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅) → ⦋𝑛 / 𝑘⦌(𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) = (0g‘𝑅))) |
32 | 31 | imim2d 55 |
. . . . 5
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ ((𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅)) → (𝑠 < 𝑛 → ⦋𝑛 / 𝑘⦌(𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) = (0g‘𝑅)))) |
33 | 32 | ralimdva 2945 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) →
(∀𝑛 ∈
ℕ0 (𝑠 <
𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅)) → ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ⦋𝑛 / 𝑘⦌(𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) = (0g‘𝑅)))) |
34 | 33 | reximdva 3000 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅)) → ∃𝑠 ∈ ℕ0
∀𝑛 ∈
ℕ0 (𝑠 <
𝑛 →
⦋𝑛 / 𝑘⦌(𝑅 Σg
(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) = (0g‘𝑅)))) |
35 | 15, 34 | mpd 15 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ⦋𝑛 / 𝑘⦌(𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) = (0g‘𝑅))) |
36 | 2, 4, 35 | mptnn0fsupp 12659 |
1
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑅 Σg
(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙)))))) finSupp (0g‘𝑅)) |