Step | Hyp | Ref
| Expression |
1 | | chfacfisf.0 |
. . . 4
⊢ 0 =
(0g‘𝑌) |
2 | | fvex 6113 |
. . . 4
⊢
(0g‘𝑌) ∈ V |
3 | 1, 2 | eqeltri 2684 |
. . 3
⊢ 0 ∈
V |
4 | 3 | a1i 11 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 0 ∈ V) |
5 | | ovex 6577 |
. . 3
⊢ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ V |
6 | 5 | a1i 11 |
. 2
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) ∈ V) |
7 | | nnnn0 11176 |
. . . . 5
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
ℕ0) |
8 | | peano2nn0 11210 |
. . . . 5
⊢ (𝑠 ∈ ℕ0
→ (𝑠 + 1) ∈
ℕ0) |
9 | 7, 8 | syl 17 |
. . . 4
⊢ (𝑠 ∈ ℕ → (𝑠 + 1) ∈
ℕ0) |
10 | 9 | ad2antrl 760 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → (𝑠 + 1) ∈
ℕ0) |
11 | | vex 3176 |
. . . . . . 7
⊢ 𝑘 ∈ V |
12 | | csbov12g 6587 |
. . . . . . . 8
⊢ (𝑘 ∈ V →
⦋𝑘 / 𝑖⦌((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = (⦋𝑘 / 𝑖⦌(𝑖 ↑ 𝑋) ·
⦋𝑘 / 𝑖⦌(𝐺‘𝑖))) |
13 | | csbov1g 6588 |
. . . . . . . . . 10
⊢ (𝑘 ∈ V →
⦋𝑘 / 𝑖⦌(𝑖 ↑ 𝑋) = (⦋𝑘 / 𝑖⦌𝑖 ↑ 𝑋)) |
14 | | csbvarg 3955 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ V →
⦋𝑘 / 𝑖⦌𝑖 = 𝑘) |
15 | 14 | oveq1d 6564 |
. . . . . . . . . 10
⊢ (𝑘 ∈ V →
(⦋𝑘 / 𝑖⦌𝑖 ↑ 𝑋) = (𝑘 ↑ 𝑋)) |
16 | 13, 15 | eqtrd 2644 |
. . . . . . . . 9
⊢ (𝑘 ∈ V →
⦋𝑘 / 𝑖⦌(𝑖 ↑ 𝑋) = (𝑘 ↑ 𝑋)) |
17 | | csbfv 6143 |
. . . . . . . . . 10
⊢
⦋𝑘 /
𝑖⦌(𝐺‘𝑖) = (𝐺‘𝑘) |
18 | 17 | a1i 11 |
. . . . . . . . 9
⊢ (𝑘 ∈ V →
⦋𝑘 / 𝑖⦌(𝐺‘𝑖) = (𝐺‘𝑘)) |
19 | 16, 18 | oveq12d 6567 |
. . . . . . . 8
⊢ (𝑘 ∈ V →
(⦋𝑘 / 𝑖⦌(𝑖 ↑ 𝑋) ·
⦋𝑘 / 𝑖⦌(𝐺‘𝑖)) = ((𝑘 ↑ 𝑋) · (𝐺‘𝑘))) |
20 | 12, 19 | eqtrd 2644 |
. . . . . . 7
⊢ (𝑘 ∈ V →
⦋𝑘 / 𝑖⦌((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = ((𝑘 ↑ 𝑋) · (𝐺‘𝑘))) |
21 | 11, 20 | mp1i 13 |
. . . . . 6
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → ⦋𝑘 / 𝑖⦌((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = ((𝑘 ↑ 𝑋) · (𝐺‘𝑘))) |
22 | | simplll 794 |
. . . . . . 7
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵)) |
23 | | simpllr 795 |
. . . . . . 7
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) |
24 | 7 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝑠 ∈ ℕ0) |
25 | 24 | ad2antlr 759 |
. . . . . . . . . . 11
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) → 𝑠 ∈
ℕ0) |
26 | 25 | nn0zd 11356 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) → 𝑠 ∈
ℤ) |
27 | 26 | adantr 480 |
. . . . . . . . 9
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 𝑠 ∈ ℤ) |
28 | | 2z 11286 |
. . . . . . . . . 10
⊢ 2 ∈
ℤ |
29 | 28 | a1i 11 |
. . . . . . . . 9
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 2 ∈ ℤ) |
30 | 27, 29 | zaddcld 11362 |
. . . . . . . 8
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → (𝑠 + 2) ∈ ℤ) |
31 | | simplr 788 |
. . . . . . . . 9
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 𝑘 ∈ ℕ0) |
32 | 31 | nn0zd 11356 |
. . . . . . . 8
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 𝑘 ∈ ℤ) |
33 | 10 | nn0zd 11356 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → (𝑠 + 1) ∈ ℤ) |
34 | | nn0z 11277 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℤ) |
35 | | zltp1le 11304 |
. . . . . . . . . . 11
⊢ (((𝑠 + 1) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑠 + 1) < 𝑘 ↔ ((𝑠 + 1) + 1) ≤ 𝑘)) |
36 | 33, 34, 35 | syl2an 493 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) → ((𝑠 + 1) < 𝑘 ↔ ((𝑠 + 1) + 1) ≤ 𝑘)) |
37 | 36 | biimpa 500 |
. . . . . . . . 9
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → ((𝑠 + 1) + 1) ≤ 𝑘) |
38 | | nncn 10905 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
ℂ) |
39 | | add1p1 11160 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 ∈ ℂ → ((𝑠 + 1) + 1) = (𝑠 + 2)) |
40 | 38, 39 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑠 ∈ ℕ → ((𝑠 + 1) + 1) = (𝑠 + 2)) |
41 | 40 | breq1d 4593 |
. . . . . . . . . . . . 13
⊢ (𝑠 ∈ ℕ → (((𝑠 + 1) + 1) ≤ 𝑘 ↔ (𝑠 + 2) ≤ 𝑘)) |
42 | 41 | bicomd 212 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ ℕ → ((𝑠 + 2) ≤ 𝑘 ↔ ((𝑠 + 1) + 1) ≤ 𝑘)) |
43 | 42 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → ((𝑠 + 2) ≤ 𝑘 ↔ ((𝑠 + 1) + 1) ≤ 𝑘)) |
44 | 43 | ad2antlr 759 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) → ((𝑠 + 2) ≤ 𝑘 ↔ ((𝑠 + 1) + 1) ≤ 𝑘)) |
45 | 44 | adantr 480 |
. . . . . . . . 9
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → ((𝑠 + 2) ≤ 𝑘 ↔ ((𝑠 + 1) + 1) ≤ 𝑘)) |
46 | 37, 45 | mpbird 246 |
. . . . . . . 8
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → (𝑠 + 2) ≤ 𝑘) |
47 | | eluz2 11569 |
. . . . . . . 8
⊢ (𝑘 ∈
(ℤ≥‘(𝑠 + 2)) ↔ ((𝑠 + 2) ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ (𝑠 + 2) ≤ 𝑘)) |
48 | 30, 32, 46, 47 | syl3anbrc 1239 |
. . . . . . 7
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → 𝑘 ∈ (ℤ≥‘(𝑠 + 2))) |
49 | | chfacfisf.a |
. . . . . . . 8
⊢ 𝐴 = (𝑁 Mat 𝑅) |
50 | | chfacfisf.b |
. . . . . . . 8
⊢ 𝐵 = (Base‘𝐴) |
51 | | chfacfisf.p |
. . . . . . . 8
⊢ 𝑃 = (Poly1‘𝑅) |
52 | | chfacfisf.y |
. . . . . . . 8
⊢ 𝑌 = (𝑁 Mat 𝑃) |
53 | | chfacfisf.r |
. . . . . . . 8
⊢ × =
(.r‘𝑌) |
54 | | chfacfisf.s |
. . . . . . . 8
⊢ − =
(-g‘𝑌) |
55 | | chfacfisf.t |
. . . . . . . 8
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
56 | | chfacfisf.g |
. . . . . . . 8
⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) |
57 | | chfacfscmulcl.x |
. . . . . . . 8
⊢ 𝑋 = (var1‘𝑅) |
58 | | chfacfscmulcl.m |
. . . . . . . 8
⊢ · = (
·𝑠 ‘𝑌) |
59 | | chfacfscmulcl.e |
. . . . . . . 8
⊢ ↑ =
(.g‘(mulGrp‘𝑃)) |
60 | 49, 50, 51, 52, 53, 54, 1, 55, 56, 57, 58, 59 | chfacfscmul0 20482 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑘 ∈ (ℤ≥‘(𝑠 + 2))) → ((𝑘 ↑ 𝑋) · (𝐺‘𝑘)) = 0 ) |
61 | 22, 23, 48, 60 | syl3anc 1318 |
. . . . . 6
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → ((𝑘 ↑ 𝑋) · (𝐺‘𝑘)) = 0 ) |
62 | 21, 61 | eqtrd 2644 |
. . . . 5
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) ∧ (𝑠 + 1) < 𝑘) → ⦋𝑘 / 𝑖⦌((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = 0 ) |
63 | 62 | ex 449 |
. . . 4
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑘 ∈ ℕ0) → ((𝑠 + 1) < 𝑘 → ⦋𝑘 / 𝑖⦌((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = 0 )) |
64 | 63 | ralrimiva 2949 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → ∀𝑘 ∈ ℕ0
((𝑠 + 1) < 𝑘 → ⦋𝑘 / 𝑖⦌((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = 0 )) |
65 | | breq1 4586 |
. . . . . 6
⊢ (𝑧 = (𝑠 + 1) → (𝑧 < 𝑘 ↔ (𝑠 + 1) < 𝑘)) |
66 | 65 | imbi1d 330 |
. . . . 5
⊢ (𝑧 = (𝑠 + 1) → ((𝑧 < 𝑘 → ⦋𝑘 / 𝑖⦌((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = 0 ) ↔ ((𝑠 + 1) < 𝑘 → ⦋𝑘 / 𝑖⦌((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = 0 ))) |
67 | 66 | ralbidv 2969 |
. . . 4
⊢ (𝑧 = (𝑠 + 1) → (∀𝑘 ∈ ℕ0 (𝑧 < 𝑘 → ⦋𝑘 / 𝑖⦌((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = 0 ) ↔ ∀𝑘 ∈ ℕ0
((𝑠 + 1) < 𝑘 → ⦋𝑘 / 𝑖⦌((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = 0 ))) |
68 | 67 | rspcev 3282 |
. . 3
⊢ (((𝑠 + 1) ∈ ℕ0
∧ ∀𝑘 ∈
ℕ0 ((𝑠 +
1) < 𝑘 →
⦋𝑘 / 𝑖⦌((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = 0 )) → ∃𝑧 ∈ ℕ0
∀𝑘 ∈
ℕ0 (𝑧 <
𝑘 →
⦋𝑘 / 𝑖⦌((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = 0 )) |
69 | 10, 64, 68 | syl2anc 691 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → ∃𝑧 ∈ ℕ0
∀𝑘 ∈
ℕ0 (𝑧 <
𝑘 →
⦋𝑘 / 𝑖⦌((𝑖 ↑ 𝑋) · (𝐺‘𝑖)) = 0 )) |
70 | 4, 6, 69 | mptnn0fsupp 12659 |
1
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))) finSupp 0 ) |