Step | Hyp | Ref
| Expression |
1 | | fvex 6113 |
. . 3
⊢
(0g‘𝑃) ∈ V |
2 | 1 | a1i 11 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (0g‘𝑃) ∈ V) |
3 | | ovex 6577 |
. . 3
⊢ ((𝑅 Σg
(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) · (𝑘 ↑ 𝑋)) ∈ V |
4 | 3 | a1i 11 |
. 2
⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑅 Σg
(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) · (𝑘 ↑ 𝑋)) ∈ 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 | | csbov12g 6587 |
. . . . . . . . . 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))) |
18 | | csbov2g 6589 |
. . . . . . . . . . . 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)))))) |
19 | | id 22 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ V → 𝑛 ∈ V) |
20 | | oveq2 6557 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑛 → (0...𝑘) = (0...𝑛)) |
21 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑛 → (𝑘 − 𝑙) = (𝑛 − 𝑙)) |
22 | 21 | fveq2d 6107 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑛 → (𝐶‘(𝑘 − 𝑙)) = (𝐶‘(𝑛 − 𝑙))) |
23 | 22 | oveq2d 6565 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑛 → ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))) = ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙)))) |
24 | 20, 23 | mpteq12dv 4663 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑛 → (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙)))) = (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) |
25 | 24 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ V ∧ 𝑘 = 𝑛) → (𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙)))) = (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) |
26 | 19, 25 | csbied 3526 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ V →
⦋𝑛 / 𝑘⦌(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙)))) = (𝑙 ∈ (0...𝑛) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑛 − 𝑙))))) |
27 | 26 | oveq2d 6565 |
. . . . . . . . . . . 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)))))) |
28 | 18, 27 | eqtrd 2644 |
. . . . . . . . . . 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)))))) |
29 | | csbov1g 6588 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ V →
⦋𝑛 / 𝑘⦌(𝑘 ↑ 𝑋) = (⦋𝑛 / 𝑘⦌𝑘 ↑ 𝑋)) |
30 | | csbvarg 3955 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ V →
⦋𝑛 / 𝑘⦌𝑘 = 𝑛) |
31 | 30 | oveq1d 6564 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ V →
(⦋𝑛 / 𝑘⦌𝑘 ↑ 𝑋) = (𝑛 ↑ 𝑋)) |
32 | 29, 31 | eqtrd 2644 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ V →
⦋𝑛 / 𝑘⦌(𝑘 ↑ 𝑋) = (𝑛 ↑ 𝑋)) |
33 | 28, 32 | oveq12d 6567 |
. . . . . . . . . 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))) |
34 | 17, 33 | eqtrd 2644 |
. . . . . . . . 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))) |
35 | 16, 34 | 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)) |
36 | | oveq1 6556 |
. . . . . . . . 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))) |
37 | 5 | ply1sca 19444 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃)) |
38 | 37 | 3ad2ant1 1075 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → 𝑅 = (Scalar‘𝑃)) |
39 | 38 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ 𝑅 =
(Scalar‘𝑃)) |
40 | 39 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ (0g‘𝑅) = (0g‘(Scalar‘𝑃))) |
41 | 40 | oveq1d 6564 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ ((0g‘𝑅) · (𝑛 ↑ 𝑋)) =
((0g‘(Scalar‘𝑃)) · (𝑛 ↑ 𝑋))) |
42 | 5 | ply1lmod 19443 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
43 | 42 | 3ad2ant1 1075 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → 𝑃 ∈ LMod) |
44 | 43 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ 𝑃 ∈
LMod) |
45 | 5 | ply1ring 19439 |
. . . . . . . . . . . . . . 15
⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
46 | 13 | ringmgp 18376 |
. . . . . . . . . . . . . . 15
⊢ (𝑃 ∈ Ring → 𝑀 ∈ Mnd) |
47 | 45, 46 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd) |
48 | 47 | 3ad2ant1 1075 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → 𝑀 ∈ Mnd) |
49 | 48 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ 𝑀 ∈
Mnd) |
50 | | simpr 476 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ 𝑛 ∈
ℕ0) |
51 | 9, 5, 6 | vr1cl 19408 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
52 | 51 | 3ad2ant1 1075 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → 𝑋 ∈ 𝐵) |
53 | 52 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ 𝑋 ∈ 𝐵) |
54 | 13, 6 | mgpbas 18318 |
. . . . . . . . . . . . 13
⊢ 𝐵 = (Base‘𝑀) |
55 | 54, 14 | mulgnn0cl 17381 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ Mnd ∧ 𝑛 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) → (𝑛 ↑ 𝑋) ∈ 𝐵) |
56 | 49, 50, 53, 55 | syl3anc 1318 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ (𝑛 ↑ 𝑋) ∈ 𝐵) |
57 | | eqid 2610 |
. . . . . . . . . . . 12
⊢
(Scalar‘𝑃) =
(Scalar‘𝑃) |
58 | | eqid 2610 |
. . . . . . . . . . . 12
⊢
(0g‘(Scalar‘𝑃)) =
(0g‘(Scalar‘𝑃)) |
59 | | eqid 2610 |
. . . . . . . . . . . 12
⊢
(0g‘𝑃) = (0g‘𝑃) |
60 | 6, 57, 11, 58, 59 | lmod0vs 18719 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ LMod ∧ (𝑛 ↑ 𝑋) ∈ 𝐵) →
((0g‘(Scalar‘𝑃)) · (𝑛 ↑ 𝑋)) = (0g‘𝑃)) |
61 | 44, 56, 60 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ ((0g‘(Scalar‘𝑃)) · (𝑛 ↑ 𝑋)) = (0g‘𝑃)) |
62 | 41, 61 | eqtrd 2644 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝑠 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0)
→ ((0g‘𝑅) · (𝑛 ↑ 𝑋)) = (0g‘𝑃)) |
63 | 36, 62 | sylan9eqr 2666 |
. . . . . . . 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)) |
64 | 35, 63 | 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))))) · (𝑘 ↑ 𝑋)) = (0g‘𝑃)) |
65 | 64 | 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))))) · (𝑘 ↑ 𝑋)) = (0g‘𝑃))) |
66 | 65 | 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))))) · (𝑘 ↑ 𝑋)) = (0g‘𝑃)))) |
67 | 66 | 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))))) · (𝑘 ↑ 𝑋)) = (0g‘𝑃)))) |
68 | 67 | 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))))) · (𝑘 ↑ 𝑋)) = (0g‘𝑃)))) |
69 | 15, 68 | 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))) |
70 | 2, 4, 69 | mptnn0fsupp 12659 |
1
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑅 Σg
(𝑙 ∈ (0...𝑘) ↦ ((𝐴‘𝑙) ∗ (𝐶‘(𝑘 − 𝑙))))) · (𝑘 ↑ 𝑋))) finSupp (0g‘𝑃)) |