Step | Hyp | Ref
| Expression |
1 | | id 22 |
. . 3
⊢ ((𝐴 Σg
(𝑛 ∈
ℕ0 ↦ (((coe1‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0
↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))) → (𝐴 Σg (𝑛 ∈ ℕ0
↦ (((coe1‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0
↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))))) |
2 | | simp1 1054 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin) |
3 | 2 | ad2antrr 758 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝑁 ∈ Fin) |
4 | | crngring 18381 |
. . . . . . 7
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) |
5 | 4 | 3ad2ant2 1076 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring) |
6 | 5 | ad2antrr 758 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝑅 ∈ Ring) |
7 | | chcoeffeq.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝐴) |
8 | | eqid 2610 |
. . . . . 6
⊢
(0g‘𝐴) = (0g‘𝐴) |
9 | | chcoeffeq.a |
. . . . . . . . . . 11
⊢ 𝐴 = (𝑁 Mat 𝑅) |
10 | 9 | matring 20068 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
11 | 4, 10 | sylan2 490 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring) |
12 | | ringcmn 18404 |
. . . . . . . . 9
⊢ (𝐴 ∈ Ring → 𝐴 ∈ CMnd) |
13 | 11, 12 | syl 17 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ CMnd) |
14 | 13 | 3adant3 1074 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ CMnd) |
15 | 14 | ad2antrr 758 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝐴 ∈ CMnd) |
16 | | nn0ex 11175 |
. . . . . . 7
⊢
ℕ0 ∈ V |
17 | 16 | a1i 11 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → ℕ0
∈ V) |
18 | 3, 6, 10 | syl2anc 691 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝐴 ∈ Ring) |
19 | 18 | adantr 480 |
. . . . . . . 8
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring) |
20 | 2, 5, 10 | syl2anc 691 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ Ring) |
21 | | eqid 2610 |
. . . . . . . . . . . 12
⊢
(mulGrp‘𝐴) =
(mulGrp‘𝐴) |
22 | 21 | ringmgp 18376 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ Ring →
(mulGrp‘𝐴) ∈
Mnd) |
23 | 20, 22 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝐴) ∈ Mnd) |
24 | 23 | ad3antrrr 762 |
. . . . . . . . 9
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) →
(mulGrp‘𝐴) ∈
Mnd) |
25 | | simpr 476 |
. . . . . . . . 9
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈
ℕ0) |
26 | | simpll3 1095 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝑀 ∈ 𝐵) |
27 | 26 | adantr 480 |
. . . . . . . . 9
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ 𝐵) |
28 | 21, 7 | mgpbas 18318 |
. . . . . . . . . 10
⊢ 𝐵 =
(Base‘(mulGrp‘𝐴)) |
29 | | cayhamlem.e1 |
. . . . . . . . . 10
⊢ ↑ =
(.g‘(mulGrp‘𝐴)) |
30 | 28, 29 | mulgnn0cl 17381 |
. . . . . . . . 9
⊢
(((mulGrp‘𝐴)
∈ Mnd ∧ 𝑛 ∈
ℕ0 ∧ 𝑀
∈ 𝐵) → (𝑛 ↑ 𝑀) ∈ 𝐵) |
31 | 24, 25, 27, 30 | syl3anc 1318 |
. . . . . . . 8
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑀) ∈ 𝐵) |
32 | | eqid 2610 |
. . . . . . . . . . . 12
⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅) |
33 | | chcoeffeq.u |
. . . . . . . . . . . 12
⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅) |
34 | 9, 7, 32, 33 | cpm2mf 20376 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵) |
35 | 2, 5, 34 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵) |
36 | 35 | ad3antrrr 762 |
. . . . . . . . 9
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵) |
37 | | simplr 788 |
. . . . . . . . . . 11
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝑠 ∈ ℕ) |
38 | | simpr 476 |
. . . . . . . . . . 11
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) |
39 | | chcoeffeq.p |
. . . . . . . . . . . 12
⊢ 𝑃 = (Poly1‘𝑅) |
40 | | chcoeffeq.y |
. . . . . . . . . . . 12
⊢ 𝑌 = (𝑁 Mat 𝑃) |
41 | | chcoeffeq.r |
. . . . . . . . . . . 12
⊢ × =
(.r‘𝑌) |
42 | | chcoeffeq.s |
. . . . . . . . . . . 12
⊢ − =
(-g‘𝑌) |
43 | | chcoeffeq.0 |
. . . . . . . . . . . 12
⊢ 0 =
(0g‘𝑌) |
44 | | chcoeffeq.t |
. . . . . . . . . . . 12
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
45 | | chcoeffeq.g |
. . . . . . . . . . . 12
⊢ 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)))))))) |
46 | 9, 7, 39, 40, 41, 42, 43, 44, 45, 32 | chfacfisfcpmat 20479 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅)) |
47 | 3, 6, 26, 37, 38, 46 | syl32anc 1326 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅)) |
48 | 47 | ffvelrnda 6267 |
. . . . . . . . 9
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝐺‘𝑛) ∈ (𝑁 ConstPolyMat 𝑅)) |
49 | 36, 48 | ffvelrnd 6268 |
. . . . . . . 8
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺‘𝑛)) ∈ 𝐵) |
50 | | eqid 2610 |
. . . . . . . . 9
⊢
(.r‘𝐴) = (.r‘𝐴) |
51 | 7, 50 | ringcl 18384 |
. . . . . . . 8
⊢ ((𝐴 ∈ Ring ∧ (𝑛 ↑ 𝑀) ∈ 𝐵 ∧ (𝑈‘(𝐺‘𝑛)) ∈ 𝐵) → ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) ∈ 𝐵) |
52 | 19, 31, 49, 51 | syl3anc 1318 |
. . . . . . 7
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) ∈ 𝐵) |
53 | | eqid 2610 |
. . . . . . 7
⊢ (𝑛 ∈ ℕ0
↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))) = (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))) |
54 | 52, 53 | fmptd 6292 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))):ℕ0⟶𝐵) |
55 | | fvex 6113 |
. . . . . . . 8
⊢
(0g‘𝐴) ∈ V |
56 | 55 | a1i 11 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) →
(0g‘𝐴)
∈ V) |
57 | | ovex 6577 |
. . . . . . . 8
⊢ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) ∈ V |
58 | 57 | a1i 11 |
. . . . . . 7
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) ∈ V) |
59 | 9, 7, 39, 40, 41, 42, 43, 44, 45 | chfacffsupp 20480 |
. . . . . . . . 9
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝐺 finSupp (0g‘𝑌)) |
60 | 59 | anassrs 678 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝐺 finSupp (0g‘𝑌)) |
61 | | ovex 6577 |
. . . . . . . . . . . . 13
⊢ (𝑁 ConstPolyMat 𝑅) ∈ V |
62 | 61, 16 | pm3.2i 470 |
. . . . . . . . . . . 12
⊢ ((𝑁 ConstPolyMat 𝑅) ∈ V ∧ ℕ0 ∈
V) |
63 | | elmapg 7757 |
. . . . . . . . . . . 12
⊢ (((𝑁 ConstPolyMat 𝑅) ∈ V ∧ ℕ0 ∈
V) → (𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑𝑚
ℕ0) ↔ 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))) |
64 | 62, 63 | mp1i 13 |
. . . . . . . . . . 11
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → (𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑𝑚
ℕ0) ↔ 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))) |
65 | 47, 64 | mpbird 246 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑𝑚
ℕ0)) |
66 | | fvex 6113 |
. . . . . . . . . 10
⊢
(0g‘𝑌) ∈ V |
67 | | fsuppmapnn0ub 12657 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ ((𝑁 ConstPolyMat 𝑅) ↑𝑚
ℕ0) ∧ (0g‘𝑌) ∈ V) → (𝐺 finSupp (0g‘𝑌) → ∃𝑤 ∈ ℕ0
∀𝑧 ∈
ℕ0 (𝑤 <
𝑧 → (𝐺‘𝑧) = (0g‘𝑌)))) |
68 | 65, 66, 67 | sylancl 693 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → (𝐺 finSupp (0g‘𝑌) → ∃𝑤 ∈ ℕ0
∀𝑧 ∈
ℕ0 (𝑤 <
𝑧 → (𝐺‘𝑧) = (0g‘𝑌)))) |
69 | | csbov12g 6587 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 ∈ ℕ0
→ ⦋𝑧 /
𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (⦋𝑧 / 𝑛⦌(𝑛 ↑ 𝑀)(.r‘𝐴)⦋𝑧 / 𝑛⦌(𝑈‘(𝐺‘𝑛)))) |
70 | | csbov1g 6588 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ ℕ0
→ ⦋𝑧 /
𝑛⦌(𝑛 ↑ 𝑀) = (⦋𝑧 / 𝑛⦌𝑛 ↑ 𝑀)) |
71 | | csbvarg 3955 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ ℕ0
→ ⦋𝑧 /
𝑛⦌𝑛 = 𝑧) |
72 | 71 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ ℕ0
→ (⦋𝑧 /
𝑛⦌𝑛 ↑ 𝑀) = (𝑧 ↑ 𝑀)) |
73 | 70, 72 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ ℕ0
→ ⦋𝑧 /
𝑛⦌(𝑛 ↑ 𝑀) = (𝑧 ↑ 𝑀)) |
74 | | csbfv2g 6142 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ ℕ0
→ ⦋𝑧 /
𝑛⦌(𝑈‘(𝐺‘𝑛)) = (𝑈‘⦋𝑧 / 𝑛⦌(𝐺‘𝑛))) |
75 | | csbfv 6143 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
⦋𝑧 /
𝑛⦌(𝐺‘𝑛) = (𝐺‘𝑧) |
76 | 75 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ ℕ0
→ ⦋𝑧 /
𝑛⦌(𝐺‘𝑛) = (𝐺‘𝑧)) |
77 | 76 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ ℕ0
→ (𝑈‘⦋𝑧 / 𝑛⦌(𝐺‘𝑛)) = (𝑈‘(𝐺‘𝑧))) |
78 | 74, 77 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ ℕ0
→ ⦋𝑧 /
𝑛⦌(𝑈‘(𝐺‘𝑛)) = (𝑈‘(𝐺‘𝑧))) |
79 | 73, 78 | oveq12d 6567 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 ∈ ℕ0
→ (⦋𝑧 /
𝑛⦌(𝑛 ↑ 𝑀)(.r‘𝐴)⦋𝑧 / 𝑛⦌(𝑈‘(𝐺‘𝑛))) = ((𝑧 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑧)))) |
80 | 69, 79 | eqtrd 2644 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 ∈ ℕ0
→ ⦋𝑧 /
𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = ((𝑧 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑧)))) |
81 | 80 | ad2antlr 759 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺‘𝑧) = (0g‘𝑌)) → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = ((𝑧 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑧)))) |
82 | | fveq2 6103 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐺‘𝑧) = (0g‘𝑌) → (𝑈‘(𝐺‘𝑧)) = (𝑈‘(0g‘𝑌))) |
83 | 2, 5 | jca 553 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
84 | 83 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
85 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(0g‘𝑌) = (0g‘𝑌) |
86 | 9, 33, 39, 40, 8, 85 | m2cpminv0 20385 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑈‘(0g‘𝑌)) = (0g‘𝐴)) |
87 | 84, 86 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) → (𝑈‘(0g‘𝑌)) = (0g‘𝐴)) |
88 | 87 | ad2antrr 758 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝑈‘(0g‘𝑌)) = (0g‘𝐴)) |
89 | 82, 88 | sylan9eqr 2666 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺‘𝑧) = (0g‘𝑌)) → (𝑈‘(𝐺‘𝑧)) = (0g‘𝐴)) |
90 | 89 | oveq2d 6565 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺‘𝑧) = (0g‘𝑌)) → ((𝑧 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑧))) = ((𝑧 ↑ 𝑀)(.r‘𝐴)(0g‘𝐴))) |
91 | 18 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝐴 ∈ Ring) |
92 | 23 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) →
(mulGrp‘𝐴) ∈
Mnd) |
93 | | simpr 476 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝑧 ∈
ℕ0) |
94 | 26 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → 𝑀 ∈ 𝐵) |
95 | 28, 29 | mulgnn0cl 17381 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((mulGrp‘𝐴)
∈ Mnd ∧ 𝑧 ∈
ℕ0 ∧ 𝑀
∈ 𝐵) → (𝑧 ↑ 𝑀) ∈ 𝐵) |
96 | 92, 93, 94, 95 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝑧 ↑ 𝑀) ∈ 𝐵) |
97 | 91, 96 | jca 553 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → (𝐴 ∈ Ring ∧ (𝑧 ↑ 𝑀) ∈ 𝐵)) |
98 | 97 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺‘𝑧) = (0g‘𝑌)) → (𝐴 ∈ Ring ∧ (𝑧 ↑ 𝑀) ∈ 𝐵)) |
99 | 7, 50, 8 | ringrz 18411 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∈ Ring ∧ (𝑧 ↑ 𝑀) ∈ 𝐵) → ((𝑧 ↑ 𝑀)(.r‘𝐴)(0g‘𝐴)) = (0g‘𝐴)) |
100 | 98, 99 | syl 17 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺‘𝑧) = (0g‘𝑌)) → ((𝑧 ↑ 𝑀)(.r‘𝐴)(0g‘𝐴)) = (0g‘𝐴)) |
101 | 81, 90, 100 | 3eqtrd 2648 |
. . . . . . . . . . . . . 14
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) ∧ (𝐺‘𝑧) = (0g‘𝑌)) → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0g‘𝐴)) |
102 | 101 | ex 449 |
. . . . . . . . . . . . 13
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑧 ∈ ℕ0) → ((𝐺‘𝑧) = (0g‘𝑌) → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0g‘𝐴))) |
103 | 102 | adantlr 747 |
. . . . . . . . . . . 12
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑤 ∈ ℕ0) ∧ 𝑧 ∈ ℕ0)
→ ((𝐺‘𝑧) = (0g‘𝑌) → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0g‘𝐴))) |
104 | 103 | imim2d 55 |
. . . . . . . . . . 11
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑤 ∈ ℕ0) ∧ 𝑧 ∈ ℕ0)
→ ((𝑤 < 𝑧 → (𝐺‘𝑧) = (0g‘𝑌)) → (𝑤 < 𝑧 → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0g‘𝐴)))) |
105 | 104 | ralimdva 2945 |
. . . . . . . . . 10
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑤 ∈ ℕ0) →
(∀𝑧 ∈
ℕ0 (𝑤 <
𝑧 → (𝐺‘𝑧) = (0g‘𝑌)) → ∀𝑧 ∈ ℕ0 (𝑤 < 𝑧 → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0g‘𝐴)))) |
106 | 105 | reximdva 3000 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → (∃𝑤 ∈ ℕ0
∀𝑧 ∈
ℕ0 (𝑤 <
𝑧 → (𝐺‘𝑧) = (0g‘𝑌)) → ∃𝑤 ∈ ℕ0 ∀𝑧 ∈ ℕ0
(𝑤 < 𝑧 → ⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0g‘𝐴)))) |
107 | 68, 106 | syld 46 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → (𝐺 finSupp (0g‘𝑌) → ∃𝑤 ∈ ℕ0
∀𝑧 ∈
ℕ0 (𝑤 <
𝑧 →
⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0g‘𝐴)))) |
108 | 60, 107 | mpd 15 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → ∃𝑤 ∈ ℕ0
∀𝑧 ∈
ℕ0 (𝑤 <
𝑧 →
⦋𝑧 / 𝑛⦌((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) = (0g‘𝐴))) |
109 | 56, 58, 108 | mptnn0fsupp 12659 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))) finSupp (0g‘𝐴)) |
110 | 7, 8, 15, 17, 54, 109 | gsumcl 18139 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → (𝐴 Σg (𝑛 ∈ ℕ0
↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))) ∈ 𝐵) |
111 | 33, 9, 7, 44 | m2cpminvid 20377 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐴 Σg
(𝑛 ∈
ℕ0 ↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))) ∈ 𝐵) → (𝑈‘(𝑇‘(𝐴 Σg (𝑛 ∈ ℕ0
↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))))) = (𝐴 Σg (𝑛 ∈ ℕ0
↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))))) |
112 | 3, 6, 110, 111 | syl3anc 1318 |
. . . 4
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → (𝑈‘(𝑇‘(𝐴 Σg (𝑛 ∈ ℕ0
↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))))) = (𝐴 Σg (𝑛 ∈ ℕ0
↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))))) |
113 | 39, 40 | pmatring 20317 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring) |
114 | 2, 5, 113 | syl2anc 691 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring) |
115 | | ringmnd 18379 |
. . . . . . . . 9
⊢ (𝑌 ∈ Ring → 𝑌 ∈ Mnd) |
116 | 114, 115 | syl 17 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Mnd) |
117 | 116 | ad2antrr 758 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝑌 ∈ Mnd) |
118 | | chcoeffeq.w |
. . . . . . . . . 10
⊢ 𝑊 = (Base‘𝑌) |
119 | 44, 9, 7, 39, 40, 118 | mat2pmatghm 20354 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝑌)) |
120 | 3, 6, 119 | syl2anc 691 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝑇 ∈ (𝐴 GrpHom 𝑌)) |
121 | | ghmmhm 17493 |
. . . . . . . 8
⊢ (𝑇 ∈ (𝐴 GrpHom 𝑌) → 𝑇 ∈ (𝐴 MndHom 𝑌)) |
122 | 120, 121 | syl 17 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝑇 ∈ (𝐴 MndHom 𝑌)) |
123 | 20 | ad3antrrr 762 |
. . . . . . . 8
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring) |
124 | 4, 34 | sylan2 490 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵) |
125 | 124 | 3adant3 1074 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵) |
126 | 125 | ad3antrrr 762 |
. . . . . . . . 9
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵) |
127 | 126, 48 | ffvelrnd 6268 |
. . . . . . . 8
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑈‘(𝐺‘𝑛)) ∈ 𝐵) |
128 | 123, 31, 127, 51 | syl3anc 1318 |
. . . . . . 7
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))) ∈ 𝐵) |
129 | 7, 8, 15, 117, 17, 122, 128, 109 | gsummptmhm 18163 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → (𝑌 Σg (𝑛 ∈ ℕ0
↦ (𝑇‘((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))))) = (𝑇‘(𝐴 Σg (𝑛 ∈ ℕ0
↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))))) |
130 | 44, 9, 7, 39, 40, 118 | mat2pmatrhm 20358 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑌)) |
131 | 130 | 3adant3 1074 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑇 ∈ (𝐴 RingHom 𝑌)) |
132 | 131 | ad3antrrr 762 |
. . . . . . . . . 10
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑇 ∈ (𝐴 RingHom 𝑌)) |
133 | 7, 50, 41 | rhmmul 18550 |
. . . . . . . . . 10
⊢ ((𝑇 ∈ (𝐴 RingHom 𝑌) ∧ (𝑛 ↑ 𝑀) ∈ 𝐵 ∧ (𝑈‘(𝐺‘𝑛)) ∈ 𝐵) → (𝑇‘((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))) = ((𝑇‘(𝑛 ↑ 𝑀)) × (𝑇‘(𝑈‘(𝐺‘𝑛))))) |
134 | 132, 31, 127, 133 | syl3anc 1318 |
. . . . . . . . 9
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))) = ((𝑇‘(𝑛 ↑ 𝑀)) × (𝑇‘(𝑈‘(𝐺‘𝑛))))) |
135 | 44, 9, 7, 39, 40, 118 | mat2pmatmhm 20357 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌))) |
136 | 135 | 3adant3 1074 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌))) |
137 | 136 | ad3antrrr 762 |
. . . . . . . . . . 11
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌))) |
138 | | cayhamlem.e2 |
. . . . . . . . . . . 12
⊢ 𝐸 =
(.g‘(mulGrp‘𝑌)) |
139 | 28, 29, 138 | mhmmulg 17406 |
. . . . . . . . . . 11
⊢ ((𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑌)) ∧ 𝑛 ∈ ℕ0 ∧ 𝑀 ∈ 𝐵) → (𝑇‘(𝑛 ↑ 𝑀)) = (𝑛𝐸(𝑇‘𝑀))) |
140 | 137, 25, 27, 139 | syl3anc 1318 |
. . . . . . . . . 10
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑛 ↑ 𝑀)) = (𝑛𝐸(𝑇‘𝑀))) |
141 | 2 | ad3antrrr 762 |
. . . . . . . . . . 11
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin) |
142 | 5 | ad3antrrr 762 |
. . . . . . . . . . 11
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring) |
143 | 32, 33, 44 | m2cpminvid2 20379 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐺‘𝑛) ∈ (𝑁 ConstPolyMat 𝑅)) → (𝑇‘(𝑈‘(𝐺‘𝑛))) = (𝐺‘𝑛)) |
144 | 141, 142,
48, 143 | syl3anc 1318 |
. . . . . . . . . 10
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑈‘(𝐺‘𝑛))) = (𝐺‘𝑛)) |
145 | 140, 144 | oveq12d 6567 |
. . . . . . . . 9
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → ((𝑇‘(𝑛 ↑ 𝑀)) × (𝑇‘(𝑈‘(𝐺‘𝑛)))) = ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))) |
146 | 134, 145 | eqtrd 2644 |
. . . . . . . 8
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))) = ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))) |
147 | 146 | mpteq2dva 4672 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → (𝑛 ∈ ℕ0 ↦ (𝑇‘((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))) = (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))) |
148 | 147 | oveq2d 6565 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → (𝑌 Σg (𝑛 ∈ ℕ0
↦ (𝑇‘((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))))) = (𝑌 Σg (𝑛 ∈ ℕ0
↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))) |
149 | 129, 148 | eqtr3d 2646 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → (𝑇‘(𝐴 Σg (𝑛 ∈ ℕ0
↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))))) = (𝑌 Σg (𝑛 ∈ ℕ0
↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))) |
150 | 149 | fveq2d 6107 |
. . . 4
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → (𝑈‘(𝑇‘(𝐴 Σg (𝑛 ∈ ℕ0
↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0
↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))))) |
151 | 112, 150 | eqtr3d 2646 |
. . 3
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → (𝐴 Σg (𝑛 ∈ ℕ0
↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛))))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0
↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))))) |
152 | 1, 151 | sylan9eqr 2666 |
. 2
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ CRing ∧
𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) ∧ (𝐴 Σg (𝑛 ∈ ℕ0
↦ (((coe1‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0
↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))))) → (𝐴 Σg (𝑛 ∈ ℕ0
↦ (((coe1‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0
↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))))) |
153 | | chcoeffeq.c |
. . 3
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) |
154 | | chcoeffeq.k |
. . 3
⊢ 𝐾 = (𝐶‘𝑀) |
155 | | chcoeffeq.1 |
. . 3
⊢ 1 =
(1r‘𝐴) |
156 | | chcoeffeq.m |
. . 3
⊢ ∗ = (
·𝑠 ‘𝐴) |
157 | 9, 7, 39, 40, 41, 42, 43, 44, 153, 154, 45, 118, 155, 156, 33, 29, 50 | cayhamlem3 20511 |
. 2
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0
↦ (((coe1‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0
↦ ((𝑛 ↑ 𝑀)(.r‘𝐴)(𝑈‘(𝐺‘𝑛)))))) |
158 | 152, 157 | reximddv2 3002 |
1
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0
↦ (((coe1‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0
↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))))) |