Step | Hyp | Ref
| Expression |
1 | | fvexd 6203 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (0g‘𝑃) ∈ V) |
2 | | ovexd 6680 |
. 2
⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑅 Σg
(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) · (𝑘 ↑ 𝑋)) ∈ V) |
3 | | ply1mulgsum.p |
. . . 4
⊢ 𝑃 = (Poly1‘𝑅) |
4 | | ply1mulgsum.b |
. . . 4
⊢ 𝐵 = (Base‘𝑃) |
5 | | ply1mulgsum.a |
. . . 4
⊢ 𝐴 = (coe1‘𝐾) |
6 | | ply1mulgsum.c |
. . . 4
⊢ 𝐶 = (coe1‘𝐿) |
7 | | ply1mulgsum.x |
. . . 4
⊢ 𝑋 = (var1‘𝑅) |
8 | | ply1mulgsum.pm |
. . . 4
⊢ × =
(.r‘𝑃) |
9 | | ply1mulgsum.sm |
. . . 4
⊢ · = (
·𝑠 ‘𝑃) |
10 | | ply1mulgsum.rm |
. . . 4
⊢ ∗ =
(.r‘𝑅) |
11 | | ply1mulgsum.m |
. . . 4
⊢ 𝑀 = (mulGrp‘𝑃) |
12 | | ply1mulgsum.e |
. . . 4
⊢ ↑ =
(.g‘𝑀) |
13 | 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | ply1mulgsumlem2 42175 |
. . 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))) |
14 | | vex 3203 |
. . . . . . . . 9
⊢ 𝑛 ∈ V |
15 | | csbov12g 6689 |
. . . . . . . . . 10
⊢ (𝑛 ∈ V →
⦋𝑛 / 𝑘⦌((𝑅 Σg
(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) · (𝑘 ↑ 𝑋)) = (⦋𝑛 / 𝑘⦌(𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) ·
⦋𝑛 / 𝑘⦌(𝑘 ↑ 𝑋))) |
16 | | csbov2g 6691 |
. . . . . . . . . . . 12
⊢ (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)))))) |
17 | | id 22 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ V → 𝑛 ∈ V) |
18 | | oveq2 6658 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑛 → (0...𝑘) = (0...𝑛)) |
19 | | oveq1 6657 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑛 → (𝑘 − 𝑙) = (𝑛 − 𝑙)) |
20 | 19 | fveq2d 6195 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑛 → (𝐶‘(𝑘 − 𝑙)) = (𝐶‘(𝑛 − 𝑙))) |
21 | 20 | oveq2d 6666 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑛 → ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))) = ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙)))) |
22 | 18, 21 | mpteq12dv 4733 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑛 → (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙)))) = (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) |
23 | 22 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ V ∧ 𝑘 = 𝑛) → (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙)))) = (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) |
24 | 17, 23 | csbied 3560 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ V →
⦋𝑛 / 𝑘⦌(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙)))) = (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) |
25 | 24 | oveq2d 6666 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ V → (𝑅 Σg
⦋𝑛 / 𝑘⦌(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) = (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙)))))) |
26 | 16, 25 | eqtrd 2656 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ V →
⦋𝑛 / 𝑘⦌(𝑅 Σg
(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) = (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙)))))) |
27 | | csbov1g 6690 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ V →
⦋𝑛 / 𝑘⦌(𝑘 ↑ 𝑋) = (⦋𝑛 / 𝑘⦌𝑘 ↑ 𝑋)) |
28 | | csbvarg 4003 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ V →
⦋𝑛 / 𝑘⦌𝑘 = 𝑛) |
29 | 28 | oveq1d 6665 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ V →
(⦋𝑛 / 𝑘⦌𝑘 ↑ 𝑋) = (𝑛 ↑ 𝑋)) |
30 | 27, 29 | eqtrd 2656 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ V →
⦋𝑛 / 𝑘⦌(𝑘 ↑ 𝑋) = (𝑛 ↑ 𝑋)) |
31 | 26, 30 | oveq12d 6668 |
. . . . . . . . . 10
⊢ (𝑛 ∈ V →
(⦋𝑛 / 𝑘⦌(𝑅 Σg
(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) ·
⦋𝑛 / 𝑘⦌(𝑘 ↑ 𝑋)) = ((𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) · (𝑛 ↑ 𝑋))) |
32 | 15, 31 | eqtrd 2656 |
. . . . . . . . 9
⊢ (𝑛 ∈ V →
⦋𝑛 / 𝑘⦌((𝑅 Σg
(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) · (𝑘 ↑ 𝑋)) = ((𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) · (𝑛 ↑ 𝑋))) |
33 | 14, 32 | ax-mp 5 |
. . . . . . . 8
⊢
⦋𝑛 /
𝑘⦌((𝑅 Σg
(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) · (𝑘 ↑ 𝑋)) = ((𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) · (𝑛 ↑ 𝑋)) |
34 | | oveq1 6657 |
. . . . . . . . 9
⊢ ((𝑅 Σg
(𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) = (0g‘𝑅) → ((𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) · (𝑛 ↑ 𝑋)) = ((0g‘𝑅) · (𝑛 ↑ 𝑋))) |
35 | 3 | ply1sca 19623 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃)) |
36 | 35 | 3ad2ant1 1082 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → 𝑅 = (Scalar‘𝑃)) |
37 | 36 | ad2antrr 762 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ 𝑅 =
(Scalar‘𝑃)) |
38 | 37 | fveq2d 6195 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ (0g‘𝑅) = (0g‘(Scalar‘𝑃))) |
39 | 38 | oveq1d 6665 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ ((0g‘𝑅) · (𝑛 ↑ 𝑋)) =
((0g‘(Scalar‘𝑃)) · (𝑛 ↑ 𝑋))) |
40 | 3 | ply1lmod 19622 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
41 | 40 | 3ad2ant1 1082 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → 𝑃 ∈ LMod) |
42 | 41 | ad2antrr 762 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ 𝑃 ∈
LMod) |
43 | 3 | ply1ring 19618 |
. . . . . . . . . . . . . . 15
⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
44 | 11 | ringmgp 18553 |
. . . . . . . . . . . . . . 15
⊢ (𝑃 ∈ Ring → 𝑀 ∈ Mnd) |
45 | 43, 44 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd) |
46 | 45 | 3ad2ant1 1082 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → 𝑀 ∈ Mnd) |
47 | 46 | ad2antrr 762 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ 𝑀 ∈
Mnd) |
48 | | simpr 477 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ 𝑛 ∈
ℕ0) |
49 | 7, 3, 4 | vr1cl 19587 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
50 | 49 | 3ad2ant1 1082 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → 𝑋 ∈ 𝐵) |
51 | 50 | ad2antrr 762 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ 𝑋 ∈ 𝐵) |
52 | 11, 4 | mgpbas 18495 |
. . . . . . . . . . . . 13
⊢ 𝐵 = (Base‘𝑀) |
53 | 52, 12 | mulgnn0cl 17558 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ Mnd ∧ 𝑛 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) → (𝑛 ↑ 𝑋) ∈ 𝐵) |
54 | 47, 48, 51, 53 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ (𝑛 ↑ 𝑋) ∈ 𝐵) |
55 | | eqid 2622 |
. . . . . . . . . . . 12
⊢
(Scalar‘𝑃) =
(Scalar‘𝑃) |
56 | | eqid 2622 |
. . . . . . . . . . . 12
⊢
(0g‘(Scalar‘𝑃)) =
(0g‘(Scalar‘𝑃)) |
57 | | eqid 2622 |
. . . . . . . . . . . 12
⊢
(0g‘𝑃) = (0g‘𝑃) |
58 | 4, 55, 9, 56, 57 | lmod0vs 18896 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ LMod ∧ (𝑛 ↑ 𝑋) ∈ 𝐵) →
((0g‘(Scalar‘𝑃)) · (𝑛 ↑ 𝑋)) = (0g‘𝑃)) |
59 | 42, 54, 58 | syl2anc 693 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ ((0g‘(Scalar‘𝑃)) · (𝑛 ↑ 𝑋)) = (0g‘𝑃)) |
60 | 39, 59 | eqtrd 2656 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ ((0g‘𝑅) · (𝑛 ↑ 𝑋)) = (0g‘𝑃)) |
61 | 34, 60 | sylan9eqr 2678 |
. . . . . . . 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))))) · (𝑛 ↑ 𝑋)) = (0g‘𝑃)) |
62 | 33, 61 | syl5eq 2668 |
. . . . . . 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))))) · (𝑘 ↑ 𝑋)) = (0g‘𝑃)) |
63 | 62 | ex 450 |
. . . . . 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))))) · (𝑘 ↑ 𝑋)) = (0g‘𝑃))) |
64 | 63 | imim2d 57 |
. . . . 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))))) · (𝑘 ↑ 𝑋)) = (0g‘𝑃)))) |
65 | 64 | ralimdva 2962 |
. . . 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))))) · (𝑘 ↑ 𝑋)) = (0g‘𝑃)))) |
66 | 65 | reximdva 3017 |
. . 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))))) · (𝑘 ↑ 𝑋)) = (0g‘𝑃)))) |
67 | 13, 66 | 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))))) · (𝑘 ↑ 𝑋)) = (0g‘𝑃))) |
68 | 1, 2, 67 | mptnn0fsupp 12797 |
1
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑅 Σg
(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) · (𝑘 ↑ 𝑋))) finSupp (0g‘𝑃)) |