Step |
Hyp |
Ref |
Expression |
0 |
|
cmpaa |
⊢ minPolyAA |
1 |
|
vx |
⊢ 𝑥 |
2 |
|
caa |
⊢ 𝔸 |
3 |
|
vp |
⊢ 𝑝 |
4 |
|
cply |
⊢ Poly |
5 |
|
cq |
⊢ ℚ |
6 |
5 4
|
cfv |
⊢ ( Poly ‘ ℚ ) |
7 |
|
cdgr |
⊢ deg |
8 |
3
|
cv |
⊢ 𝑝 |
9 |
8 7
|
cfv |
⊢ ( deg ‘ 𝑝 ) |
10 |
|
cdgraa |
⊢ degAA |
11 |
1
|
cv |
⊢ 𝑥 |
12 |
11 10
|
cfv |
⊢ ( degAA ‘ 𝑥 ) |
13 |
9 12
|
wceq |
⊢ ( deg ‘ 𝑝 ) = ( degAA ‘ 𝑥 ) |
14 |
11 8
|
cfv |
⊢ ( 𝑝 ‘ 𝑥 ) |
15 |
|
cc0 |
⊢ 0 |
16 |
14 15
|
wceq |
⊢ ( 𝑝 ‘ 𝑥 ) = 0 |
17 |
|
ccoe |
⊢ coeff |
18 |
8 17
|
cfv |
⊢ ( coeff ‘ 𝑝 ) |
19 |
12 18
|
cfv |
⊢ ( ( coeff ‘ 𝑝 ) ‘ ( degAA ‘ 𝑥 ) ) |
20 |
|
c1 |
⊢ 1 |
21 |
19 20
|
wceq |
⊢ ( ( coeff ‘ 𝑝 ) ‘ ( degAA ‘ 𝑥 ) ) = 1 |
22 |
13 16 21
|
w3a |
⊢ ( ( deg ‘ 𝑝 ) = ( degAA ‘ 𝑥 ) ∧ ( 𝑝 ‘ 𝑥 ) = 0 ∧ ( ( coeff ‘ 𝑝 ) ‘ ( degAA ‘ 𝑥 ) ) = 1 ) |
23 |
22 3 6
|
crio |
⊢ ( ℩ 𝑝 ∈ ( Poly ‘ ℚ ) ( ( deg ‘ 𝑝 ) = ( degAA ‘ 𝑥 ) ∧ ( 𝑝 ‘ 𝑥 ) = 0 ∧ ( ( coeff ‘ 𝑝 ) ‘ ( degAA ‘ 𝑥 ) ) = 1 ) ) |
24 |
1 2 23
|
cmpt |
⊢ ( 𝑥 ∈ 𝔸 ↦ ( ℩ 𝑝 ∈ ( Poly ‘ ℚ ) ( ( deg ‘ 𝑝 ) = ( degAA ‘ 𝑥 ) ∧ ( 𝑝 ‘ 𝑥 ) = 0 ∧ ( ( coeff ‘ 𝑝 ) ‘ ( degAA ‘ 𝑥 ) ) = 1 ) ) ) |
25 |
0 24
|
wceq |
⊢ minPolyAA = ( 𝑥 ∈ 𝔸 ↦ ( ℩ 𝑝 ∈ ( Poly ‘ ℚ ) ( ( deg ‘ 𝑝 ) = ( degAA ‘ 𝑥 ) ∧ ( 𝑝 ‘ 𝑥 ) = 0 ∧ ( ( coeff ‘ 𝑝 ) ‘ ( degAA ‘ 𝑥 ) ) = 1 ) ) ) |