Step | Hyp | Ref
| Expression |
1 | | cig1p 25647 |
. 2
class
idlGen1p |
2 | | vr |
. . 3
setvar π |
3 | | cvv 3475 |
. . 3
class
V |
4 | | vi |
. . . 4
setvar π |
5 | 2 | cv 1541 |
. . . . . 6
class π |
6 | | cpl1 21701 |
. . . . . 6
class
Poly1 |
7 | 5, 6 | cfv 6544 |
. . . . 5
class
(Poly1βπ) |
8 | | clidl 20783 |
. . . . 5
class
LIdeal |
9 | 7, 8 | cfv 6544 |
. . . 4
class
(LIdealβ(Poly1βπ)) |
10 | 4 | cv 1541 |
. . . . . 6
class π |
11 | | c0g 17385 |
. . . . . . . 8
class
0g |
12 | 7, 11 | cfv 6544 |
. . . . . . 7
class
(0gβ(Poly1βπ)) |
13 | 12 | csn 4629 |
. . . . . 6
class
{(0gβ(Poly1βπ))} |
14 | 10, 13 | wceq 1542 |
. . . . 5
wff π =
{(0gβ(Poly1βπ))} |
15 | | vg |
. . . . . . . . 9
setvar π |
16 | 15 | cv 1541 |
. . . . . . . 8
class π |
17 | | cdg1 25569 |
. . . . . . . . 9
class
deg1 |
18 | 5, 17 | cfv 6544 |
. . . . . . . 8
class (
deg1 βπ) |
19 | 16, 18 | cfv 6544 |
. . . . . . 7
class ((
deg1 βπ)βπ) |
20 | 10, 13 | cdif 3946 |
. . . . . . . . 9
class (π β
{(0gβ(Poly1βπ))}) |
21 | 18, 20 | cima 5680 |
. . . . . . . 8
class ((
deg1 βπ)
β (π β
{(0gβ(Poly1βπ))})) |
22 | | cr 11109 |
. . . . . . . 8
class
β |
23 | | clt 11248 |
. . . . . . . 8
class
< |
24 | 21, 22, 23 | cinf 9436 |
. . . . . . 7
class inf(((
deg1 βπ)
β (π β
{(0gβ(Poly1βπ))})), β, < ) |
25 | 19, 24 | wceq 1542 |
. . . . . 6
wff ((
deg1 βπ)βπ) = inf((( deg1 βπ) β (π β
{(0gβ(Poly1βπ))})), β, < ) |
26 | | cmn1 25643 |
. . . . . . . 8
class
Monic1p |
27 | 5, 26 | cfv 6544 |
. . . . . . 7
class
(Monic1pβπ) |
28 | 10, 27 | cin 3948 |
. . . . . 6
class (π β©
(Monic1pβπ)) |
29 | 25, 15, 28 | crio 7364 |
. . . . 5
class
(β©π
β (π β©
(Monic1pβπ))(( deg1 βπ)βπ) = inf((( deg1 βπ) β (π β
{(0gβ(Poly1βπ))})), β, < )) |
30 | 14, 12, 29 | cif 4529 |
. . . 4
class if(π =
{(0gβ(Poly1βπ))},
(0gβ(Poly1βπ)), (β©π β (π β© (Monic1pβπ))(( deg1
βπ)βπ) = inf((( deg1
βπ) β (π β
{(0gβ(Poly1βπ))})), β, < ))) |
31 | 4, 9, 30 | cmpt 5232 |
. . 3
class (π β
(LIdealβ(Poly1βπ)) β¦ if(π =
{(0gβ(Poly1βπ))},
(0gβ(Poly1βπ)), (β©π β (π β© (Monic1pβπ))(( deg1
βπ)βπ) = inf((( deg1
βπ) β (π β
{(0gβ(Poly1βπ))})), β, < )))) |
32 | 2, 3, 31 | cmpt 5232 |
. 2
class (π β V β¦ (π β
(LIdealβ(Poly1βπ)) β¦ if(π =
{(0gβ(Poly1βπ))},
(0gβ(Poly1βπ)), (β©π β (π β© (Monic1pβπ))(( deg1
βπ)βπ) = inf((( deg1
βπ) β (π β
{(0gβ(Poly1βπ))})), β, < ))))) |
33 | 1, 32 | wceq 1542 |
1
wff
idlGen1p = (π β V β¦ (π β
(LIdealβ(Poly1βπ)) β¦ if(π =
{(0gβ(Poly1βπ))},
(0gβ(Poly1βπ)), (β©π β (π β© (Monic1pβπ))(( deg1
βπ)βπ) = inf((( deg1
βπ) β (π β
{(0gβ(Poly1βπ))})), β, < ))))) |