Step |
Hyp |
Ref |
Expression |
1 |
|
chcoeffeq.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
chcoeffeq.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
3 |
|
chcoeffeq.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
4 |
|
chcoeffeq.y |
⊢ 𝑌 = ( 𝑁 Mat 𝑃 ) |
5 |
|
chcoeffeq.r |
⊢ × = ( .r ‘ 𝑌 ) |
6 |
|
chcoeffeq.s |
⊢ − = ( -g ‘ 𝑌 ) |
7 |
|
chcoeffeq.0 |
⊢ 0 = ( 0g ‘ 𝑌 ) |
8 |
|
chcoeffeq.t |
⊢ 𝑇 = ( 𝑁 matToPolyMat 𝑅 ) |
9 |
|
chcoeffeq.c |
⊢ 𝐶 = ( 𝑁 CharPlyMat 𝑅 ) |
10 |
|
chcoeffeq.k |
⊢ 𝐾 = ( 𝐶 ‘ 𝑀 ) |
11 |
|
chcoeffeq.g |
⊢ 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 ) ) ) ) ) ) ) ) |
12 |
|
chcoeffeq.w |
⊢ 𝑊 = ( Base ‘ 𝑌 ) |
13 |
|
chcoeffeq.1 |
⊢ 1 = ( 1r ‘ 𝐴 ) |
14 |
|
chcoeffeq.m |
⊢ ∗ = ( ·𝑠 ‘ 𝐴 ) |
15 |
|
chcoeffeq.u |
⊢ 𝑈 = ( 𝑁 cPolyMatToMat 𝑅 ) |
16 |
|
cayhamlem.e1 |
⊢ ↑ = ( .g ‘ ( mulGrp ‘ 𝐴 ) ) |
17 |
|
cayhamlem.e2 |
⊢ 𝐸 = ( .g ‘ ( mulGrp ‘ 𝑌 ) ) |
18 |
|
id |
⊢ ( ( 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 ) ) ) ) ) ) |
19 |
|
simp1 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑁 ∈ Fin ) |
20 |
19
|
ad2antrr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → 𝑁 ∈ Fin ) |
21 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
22 |
21
|
3ad2ant2 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑅 ∈ Ring ) |
23 |
22
|
ad2antrr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → 𝑅 ∈ Ring ) |
24 |
|
eqid |
⊢ ( 0g ‘ 𝐴 ) = ( 0g ‘ 𝐴 ) |
25 |
1
|
matring |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐴 ∈ Ring ) |
26 |
21 25
|
sylan2 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝐴 ∈ Ring ) |
27 |
|
ringcmn |
⊢ ( 𝐴 ∈ Ring → 𝐴 ∈ CMnd ) |
28 |
26 27
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝐴 ∈ CMnd ) |
29 |
28
|
3adant3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝐴 ∈ CMnd ) |
30 |
29
|
ad2antrr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → 𝐴 ∈ CMnd ) |
31 |
|
nn0ex |
⊢ ℕ0 ∈ V |
32 |
31
|
a1i |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → ℕ0 ∈ V ) |
33 |
20 23 25
|
syl2anc |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → 𝐴 ∈ Ring ) |
34 |
33
|
adantr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → 𝐴 ∈ Ring ) |
35 |
|
eqid |
⊢ ( mulGrp ‘ 𝐴 ) = ( mulGrp ‘ 𝐴 ) |
36 |
35 2
|
mgpbas |
⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝐴 ) ) |
37 |
19 22 25
|
syl2anc |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝐴 ∈ Ring ) |
38 |
35
|
ringmgp |
⊢ ( 𝐴 ∈ Ring → ( mulGrp ‘ 𝐴 ) ∈ Mnd ) |
39 |
37 38
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( mulGrp ‘ 𝐴 ) ∈ Mnd ) |
40 |
39
|
ad3antrrr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → ( mulGrp ‘ 𝐴 ) ∈ Mnd ) |
41 |
|
simpr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → 𝑛 ∈ ℕ0 ) |
42 |
|
simpll3 |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → 𝑀 ∈ 𝐵 ) |
43 |
42
|
adantr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → 𝑀 ∈ 𝐵 ) |
44 |
36 16 40 41 43
|
mulgnn0cld |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝑛 ↑ 𝑀 ) ∈ 𝐵 ) |
45 |
|
eqid |
⊢ ( 𝑁 ConstPolyMat 𝑅 ) = ( 𝑁 ConstPolyMat 𝑅 ) |
46 |
1 2 45 15
|
cpm2mf |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑈 : ( 𝑁 ConstPolyMat 𝑅 ) ⟶ 𝐵 ) |
47 |
19 22 46
|
syl2anc |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑈 : ( 𝑁 ConstPolyMat 𝑅 ) ⟶ 𝐵 ) |
48 |
47
|
ad3antrrr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → 𝑈 : ( 𝑁 ConstPolyMat 𝑅 ) ⟶ 𝐵 ) |
49 |
|
simplr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → 𝑠 ∈ ℕ ) |
50 |
|
simpr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) |
51 |
1 2 3 4 5 6 7 8 11 45
|
chfacfisfcpmat |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝐺 : ℕ0 ⟶ ( 𝑁 ConstPolyMat 𝑅 ) ) |
52 |
20 23 42 49 50 51
|
syl32anc |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → 𝐺 : ℕ0 ⟶ ( 𝑁 ConstPolyMat 𝑅 ) ) |
53 |
52
|
ffvelcdmda |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝐺 ‘ 𝑛 ) ∈ ( 𝑁 ConstPolyMat 𝑅 ) ) |
54 |
48 53
|
ffvelcdmd |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) ∈ 𝐵 ) |
55 |
|
eqid |
⊢ ( .r ‘ 𝐴 ) = ( .r ‘ 𝐴 ) |
56 |
2 55
|
ringcl |
⊢ ( ( 𝐴 ∈ Ring ∧ ( 𝑛 ↑ 𝑀 ) ∈ 𝐵 ∧ ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) ∈ 𝐵 ) → ( ( 𝑛 ↑ 𝑀 ) ( .r ‘ 𝐴 ) ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) ) ∈ 𝐵 ) |
57 |
34 44 54 56
|
syl3anc |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝑛 ↑ 𝑀 ) ( .r ‘ 𝐴 ) ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) ) ∈ 𝐵 ) |
58 |
57
|
fmpttd |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → ( 𝑛 ∈ ℕ0 ↦ ( ( 𝑛 ↑ 𝑀 ) ( .r ‘ 𝐴 ) ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) ) ) : ℕ0 ⟶ 𝐵 ) |
59 |
|
fvexd |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → ( 0g ‘ 𝐴 ) ∈ V ) |
60 |
|
ovexd |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝑛 ↑ 𝑀 ) ( .r ‘ 𝐴 ) ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) ) ∈ V ) |
61 |
1 2 3 4 5 6 7 8 11
|
chfacffsupp |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑠 ∈ ℕ ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ) → 𝐺 finSupp ( 0g ‘ 𝑌 ) ) |
62 |
61
|
anassrs |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → 𝐺 finSupp ( 0g ‘ 𝑌 ) ) |
63 |
|
ovex |
⊢ ( 𝑁 ConstPolyMat 𝑅 ) ∈ V |
64 |
63 31
|
pm3.2i |
⊢ ( ( 𝑁 ConstPolyMat 𝑅 ) ∈ V ∧ ℕ0 ∈ V ) |
65 |
|
elmapg |
⊢ ( ( ( 𝑁 ConstPolyMat 𝑅 ) ∈ V ∧ ℕ0 ∈ V ) → ( 𝐺 ∈ ( ( 𝑁 ConstPolyMat 𝑅 ) ↑m ℕ0 ) ↔ 𝐺 : ℕ0 ⟶ ( 𝑁 ConstPolyMat 𝑅 ) ) ) |
66 |
64 65
|
mp1i |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → ( 𝐺 ∈ ( ( 𝑁 ConstPolyMat 𝑅 ) ↑m ℕ0 ) ↔ 𝐺 : ℕ0 ⟶ ( 𝑁 ConstPolyMat 𝑅 ) ) ) |
67 |
52 66
|
mpbird |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → 𝐺 ∈ ( ( 𝑁 ConstPolyMat 𝑅 ) ↑m ℕ0 ) ) |
68 |
|
fvex |
⊢ ( 0g ‘ 𝑌 ) ∈ V |
69 |
|
fsuppmapnn0ub |
⊢ ( ( 𝐺 ∈ ( ( 𝑁 ConstPolyMat 𝑅 ) ↑m ℕ0 ) ∧ ( 0g ‘ 𝑌 ) ∈ V ) → ( 𝐺 finSupp ( 0g ‘ 𝑌 ) → ∃ 𝑤 ∈ ℕ0 ∀ 𝑧 ∈ ℕ0 ( 𝑤 < 𝑧 → ( 𝐺 ‘ 𝑧 ) = ( 0g ‘ 𝑌 ) ) ) ) |
70 |
67 68 69
|
sylancl |
⊢ ( ( ( ( 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 ) ) ) ) |
71 |
|
csbov12g |
⊢ ( 𝑧 ∈ ℕ0 → ⦋ 𝑧 / 𝑛 ⦌ ( ( 𝑛 ↑ 𝑀 ) ( .r ‘ 𝐴 ) ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) ) = ( ⦋ 𝑧 / 𝑛 ⦌ ( 𝑛 ↑ 𝑀 ) ( .r ‘ 𝐴 ) ⦋ 𝑧 / 𝑛 ⦌ ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) ) ) |
72 |
|
csbov1g |
⊢ ( 𝑧 ∈ ℕ0 → ⦋ 𝑧 / 𝑛 ⦌ ( 𝑛 ↑ 𝑀 ) = ( ⦋ 𝑧 / 𝑛 ⦌ 𝑛 ↑ 𝑀 ) ) |
73 |
|
csbvarg |
⊢ ( 𝑧 ∈ ℕ0 → ⦋ 𝑧 / 𝑛 ⦌ 𝑛 = 𝑧 ) |
74 |
73
|
oveq1d |
⊢ ( 𝑧 ∈ ℕ0 → ( ⦋ 𝑧 / 𝑛 ⦌ 𝑛 ↑ 𝑀 ) = ( 𝑧 ↑ 𝑀 ) ) |
75 |
72 74
|
eqtrd |
⊢ ( 𝑧 ∈ ℕ0 → ⦋ 𝑧 / 𝑛 ⦌ ( 𝑛 ↑ 𝑀 ) = ( 𝑧 ↑ 𝑀 ) ) |
76 |
|
csbfv2g |
⊢ ( 𝑧 ∈ ℕ0 → ⦋ 𝑧 / 𝑛 ⦌ ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) = ( 𝑈 ‘ ⦋ 𝑧 / 𝑛 ⦌ ( 𝐺 ‘ 𝑛 ) ) ) |
77 |
|
csbfv |
⊢ ⦋ 𝑧 / 𝑛 ⦌ ( 𝐺 ‘ 𝑛 ) = ( 𝐺 ‘ 𝑧 ) |
78 |
77
|
a1i |
⊢ ( 𝑧 ∈ ℕ0 → ⦋ 𝑧 / 𝑛 ⦌ ( 𝐺 ‘ 𝑛 ) = ( 𝐺 ‘ 𝑧 ) ) |
79 |
78
|
fveq2d |
⊢ ( 𝑧 ∈ ℕ0 → ( 𝑈 ‘ ⦋ 𝑧 / 𝑛 ⦌ ( 𝐺 ‘ 𝑛 ) ) = ( 𝑈 ‘ ( 𝐺 ‘ 𝑧 ) ) ) |
80 |
76 79
|
eqtrd |
⊢ ( 𝑧 ∈ ℕ0 → ⦋ 𝑧 / 𝑛 ⦌ ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) = ( 𝑈 ‘ ( 𝐺 ‘ 𝑧 ) ) ) |
81 |
75 80
|
oveq12d |
⊢ ( 𝑧 ∈ ℕ0 → ( ⦋ 𝑧 / 𝑛 ⦌ ( 𝑛 ↑ 𝑀 ) ( .r ‘ 𝐴 ) ⦋ 𝑧 / 𝑛 ⦌ ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) ) = ( ( 𝑧 ↑ 𝑀 ) ( .r ‘ 𝐴 ) ( 𝑈 ‘ ( 𝐺 ‘ 𝑧 ) ) ) ) |
82 |
71 81
|
eqtrd |
⊢ ( 𝑧 ∈ ℕ0 → ⦋ 𝑧 / 𝑛 ⦌ ( ( 𝑛 ↑ 𝑀 ) ( .r ‘ 𝐴 ) ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) ) = ( ( 𝑧 ↑ 𝑀 ) ( .r ‘ 𝐴 ) ( 𝑈 ‘ ( 𝐺 ‘ 𝑧 ) ) ) ) |
83 |
82
|
ad2antlr |
⊢ ( ( ( ( ( ( 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 ) ) ) ) |
84 |
|
fveq2 |
⊢ ( ( 𝐺 ‘ 𝑧 ) = ( 0g ‘ 𝑌 ) → ( 𝑈 ‘ ( 𝐺 ‘ 𝑧 ) ) = ( 𝑈 ‘ ( 0g ‘ 𝑌 ) ) ) |
85 |
19 22
|
jca |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ) |
86 |
85
|
adantr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ) |
87 |
|
eqid |
⊢ ( 0g ‘ 𝑌 ) = ( 0g ‘ 𝑌 ) |
88 |
1 15 3 4 24 87
|
m2cpminv0 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝑈 ‘ ( 0g ‘ 𝑌 ) ) = ( 0g ‘ 𝐴 ) ) |
89 |
86 88
|
syl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) → ( 𝑈 ‘ ( 0g ‘ 𝑌 ) ) = ( 0g ‘ 𝐴 ) ) |
90 |
89
|
ad2antrr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑧 ∈ ℕ0 ) → ( 𝑈 ‘ ( 0g ‘ 𝑌 ) ) = ( 0g ‘ 𝐴 ) ) |
91 |
84 90
|
sylan9eqr |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑧 ∈ ℕ0 ) ∧ ( 𝐺 ‘ 𝑧 ) = ( 0g ‘ 𝑌 ) ) → ( 𝑈 ‘ ( 𝐺 ‘ 𝑧 ) ) = ( 0g ‘ 𝐴 ) ) |
92 |
91
|
oveq2d |
⊢ ( ( ( ( ( ( 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 ) ) ) |
93 |
33
|
adantr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑧 ∈ ℕ0 ) → 𝐴 ∈ Ring ) |
94 |
39
|
ad3antrrr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑧 ∈ ℕ0 ) → ( mulGrp ‘ 𝐴 ) ∈ Mnd ) |
95 |
|
simpr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑧 ∈ ℕ0 ) → 𝑧 ∈ ℕ0 ) |
96 |
42
|
adantr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑧 ∈ ℕ0 ) → 𝑀 ∈ 𝐵 ) |
97 |
36 16 94 95 96
|
mulgnn0cld |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑧 ∈ ℕ0 ) → ( 𝑧 ↑ 𝑀 ) ∈ 𝐵 ) |
98 |
93 97
|
jca |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑧 ∈ ℕ0 ) → ( 𝐴 ∈ Ring ∧ ( 𝑧 ↑ 𝑀 ) ∈ 𝐵 ) ) |
99 |
98
|
adantr |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑧 ∈ ℕ0 ) ∧ ( 𝐺 ‘ 𝑧 ) = ( 0g ‘ 𝑌 ) ) → ( 𝐴 ∈ Ring ∧ ( 𝑧 ↑ 𝑀 ) ∈ 𝐵 ) ) |
100 |
2 55 24
|
ringrz |
⊢ ( ( 𝐴 ∈ Ring ∧ ( 𝑧 ↑ 𝑀 ) ∈ 𝐵 ) → ( ( 𝑧 ↑ 𝑀 ) ( .r ‘ 𝐴 ) ( 0g ‘ 𝐴 ) ) = ( 0g ‘ 𝐴 ) ) |
101 |
99 100
|
syl |
⊢ ( ( ( ( ( ( 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 ) ) |
102 |
83 92 101
|
3eqtrd |
⊢ ( ( ( ( ( ( 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
|
ex |
⊢ ( ( ( ( ( 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 ) ) ) |
104 |
103
|
adantlr |
⊢ ( ( ( ( ( ( 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 ) ) ) |
105 |
104
|
imim2d |
⊢ ( ( ( ( ( ( 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 ) ) ) ) |
106 |
105
|
ralimdva |
⊢ ( ( ( ( ( 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 ) ) ) ) |
107 |
106
|
reximdva |
⊢ ( ( ( ( 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 ) ) ) ) |
108 |
70 107
|
syld |
⊢ ( ( ( ( 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 ) ) ) ) |
109 |
62 108
|
mpd |
⊢ ( ( ( ( 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 ) ) ) |
110 |
59 60 109
|
mptnn0fsupp |
⊢ ( ( ( ( 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 ) ) |
111 |
2 24 30 32 58 110
|
gsumcl |
⊢ ( ( ( ( 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 ) |
112 |
15 1 2 8
|
m2cpminvid |
⊢ ( ( 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 ) ) ) ) ) ) |
113 |
20 23 111 112
|
syl3anc |
⊢ ( ( ( ( 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 ) ) ) ) ) ) |
114 |
3 4
|
pmatring |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑌 ∈ Ring ) |
115 |
19 22 114
|
syl2anc |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑌 ∈ Ring ) |
116 |
|
ringmnd |
⊢ ( 𝑌 ∈ Ring → 𝑌 ∈ Mnd ) |
117 |
115 116
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑌 ∈ Mnd ) |
118 |
117
|
ad2antrr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → 𝑌 ∈ Mnd ) |
119 |
8 1 2 3 4 12
|
mat2pmatghm |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑇 ∈ ( 𝐴 GrpHom 𝑌 ) ) |
120 |
20 23 119
|
syl2anc |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → 𝑇 ∈ ( 𝐴 GrpHom 𝑌 ) ) |
121 |
|
ghmmhm |
⊢ ( 𝑇 ∈ ( 𝐴 GrpHom 𝑌 ) → 𝑇 ∈ ( 𝐴 MndHom 𝑌 ) ) |
122 |
120 121
|
syl |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) → 𝑇 ∈ ( 𝐴 MndHom 𝑌 ) ) |
123 |
37
|
ad3antrrr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → 𝐴 ∈ Ring ) |
124 |
21 46
|
sylan2 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑈 : ( 𝑁 ConstPolyMat 𝑅 ) ⟶ 𝐵 ) |
125 |
124
|
3adant3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑈 : ( 𝑁 ConstPolyMat 𝑅 ) ⟶ 𝐵 ) |
126 |
125
|
ad3antrrr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → 𝑈 : ( 𝑁 ConstPolyMat 𝑅 ) ⟶ 𝐵 ) |
127 |
126 53
|
ffvelcdmd |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) ∈ 𝐵 ) |
128 |
123 44 127 56
|
syl3anc |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝑛 ↑ 𝑀 ) ( .r ‘ 𝐴 ) ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) ) ∈ 𝐵 ) |
129 |
2 24 30 118 32 122 128 110
|
gsummptmhm |
⊢ ( ( ( ( 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 |
8 1 2 3 4 12
|
mat2pmatrhm |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑇 ∈ ( 𝐴 RingHom 𝑌 ) ) |
131 |
130
|
3adant3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑇 ∈ ( 𝐴 RingHom 𝑌 ) ) |
132 |
131
|
ad3antrrr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → 𝑇 ∈ ( 𝐴 RingHom 𝑌 ) ) |
133 |
2 55 5
|
rhmmul |
⊢ ( ( 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 44 127 133
|
syl3anc |
⊢ ( ( ( ( ( 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 |
8 1 2 3 4 12
|
mat2pmatmhm |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → 𝑇 ∈ ( ( mulGrp ‘ 𝐴 ) MndHom ( mulGrp ‘ 𝑌 ) ) ) |
136 |
135
|
3adant3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → 𝑇 ∈ ( ( mulGrp ‘ 𝐴 ) MndHom ( mulGrp ‘ 𝑌 ) ) ) |
137 |
136
|
ad3antrrr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → 𝑇 ∈ ( ( mulGrp ‘ 𝐴 ) MndHom ( mulGrp ‘ 𝑌 ) ) ) |
138 |
36 16 17
|
mhmmulg |
⊢ ( ( 𝑇 ∈ ( ( mulGrp ‘ 𝐴 ) MndHom ( mulGrp ‘ 𝑌 ) ) ∧ 𝑛 ∈ ℕ0 ∧ 𝑀 ∈ 𝐵 ) → ( 𝑇 ‘ ( 𝑛 ↑ 𝑀 ) ) = ( 𝑛 𝐸 ( 𝑇 ‘ 𝑀 ) ) ) |
139 |
137 41 43 138
|
syl3anc |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝑇 ‘ ( 𝑛 ↑ 𝑀 ) ) = ( 𝑛 𝐸 ( 𝑇 ‘ 𝑀 ) ) ) |
140 |
19
|
ad3antrrr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → 𝑁 ∈ Fin ) |
141 |
22
|
ad3antrrr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → 𝑅 ∈ Ring ) |
142 |
45 15 8
|
m2cpminvid2 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ ( 𝐺 ‘ 𝑛 ) ∈ ( 𝑁 ConstPolyMat 𝑅 ) ) → ( 𝑇 ‘ ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) ) = ( 𝐺 ‘ 𝑛 ) ) |
143 |
140 141 53 142
|
syl3anc |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝑇 ‘ ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) ) = ( 𝐺 ‘ 𝑛 ) ) |
144 |
139 143
|
oveq12d |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝑇 ‘ ( 𝑛 ↑ 𝑀 ) ) × ( 𝑇 ‘ ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) ) ) = ( ( 𝑛 𝐸 ( 𝑇 ‘ 𝑀 ) ) × ( 𝐺 ‘ 𝑛 ) ) ) |
145 |
134 144
|
eqtrd |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑠 ∈ ℕ ) ∧ 𝑏 ∈ ( 𝐵 ↑m ( 0 ... 𝑠 ) ) ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝑇 ‘ ( ( 𝑛 ↑ 𝑀 ) ( .r ‘ 𝐴 ) ( 𝑈 ‘ ( 𝐺 ‘ 𝑛 ) ) ) ) = ( ( 𝑛 𝐸 ( 𝑇 ‘ 𝑀 ) ) × ( 𝐺 ‘ 𝑛 ) ) ) |
146 |
145
|
mpteq2dva |
⊢ ( ( ( ( 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 ↦ ( ( 𝑛 𝐸 ( 𝑇 ‘ 𝑀 ) ) × ( 𝐺 ‘ 𝑛 ) ) ) ) |
147 |
146
|
oveq2d |
⊢ ( ( ( ( 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 ↦ ( ( 𝑛 𝐸 ( 𝑇 ‘ 𝑀 ) ) × ( 𝐺 ‘ 𝑛 ) ) ) ) ) |
148 |
129 147
|
eqtr3d |
⊢ ( ( ( ( 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 ↦ ( ( 𝑛 𝐸 ( 𝑇 ‘ 𝑀 ) ) × ( 𝐺 ‘ 𝑛 ) ) ) ) ) |
149 |
148
|
fveq2d |
⊢ ( ( ( ( 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 ↦ ( ( 𝑛 𝐸 ( 𝑇 ‘ 𝑀 ) ) × ( 𝐺 ‘ 𝑛 ) ) ) ) ) ) |
150 |
113 149
|
eqtr3d |
⊢ ( ( ( ( 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 ↦ ( ( 𝑛 𝐸 ( 𝑇 ‘ 𝑀 ) ) × ( 𝐺 ‘ 𝑛 ) ) ) ) ) ) |
151 |
18 150
|
sylan9eqr |
⊢ ( ( ( ( ( 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 ↦ ( ( 𝑛 𝐸 ( 𝑇 ‘ 𝑀 ) ) × ( 𝐺 ‘ 𝑛 ) ) ) ) ) ) |
152 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 55
|
cayhamlem3 |
⊢ ( ( 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 ) ) ) ) ) ) |
153 |
151 152
|
reximddv2 |
⊢ ( ( 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 ↦ ( ( 𝑛 𝐸 ( 𝑇 ‘ 𝑀 ) ) × ( 𝐺 ‘ 𝑛 ) ) ) ) ) ) |