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Definition df-ig1p 25652
Description: Define a choice function for generators of ideals over a division ring; this is the unique monic polynomial of minimal degree in the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Sep-2020.)
Assertion
Ref Expression
df-ig1p idlGen1p = (π‘Ÿ ∈ V ↦ (𝑖 ∈ (LIdealβ€˜(Poly1β€˜π‘Ÿ)) ↦ if(𝑖 = {(0gβ€˜(Poly1β€˜π‘Ÿ))}, (0gβ€˜(Poly1β€˜π‘Ÿ)), (℩𝑔 ∈ (𝑖 ∩ (Monic1pβ€˜π‘Ÿ))(( deg1 β€˜π‘Ÿ)β€˜π‘”) = inf((( deg1 β€˜π‘Ÿ) β€œ (𝑖 βˆ– {(0gβ€˜(Poly1β€˜π‘Ÿ))})), ℝ, < )))))
Distinct variable group:   𝑔,π‘Ÿ,𝑖

Detailed syntax breakdown of Definition df-ig1p
StepHypRef Expression
1 cig1p 25647 . 2 class idlGen1p
2 vr . . 3 setvar π‘Ÿ
3 cvv 3475 . . 3 class V
4 vi . . . 4 setvar 𝑖
52cv 1541 . . . . . 6 class π‘Ÿ
6 cpl1 21701 . . . . . 6 class Poly1
75, 6cfv 6544 . . . . 5 class (Poly1β€˜π‘Ÿ)
8 clidl 20783 . . . . 5 class LIdeal
97, 8cfv 6544 . . . 4 class (LIdealβ€˜(Poly1β€˜π‘Ÿ))
104cv 1541 . . . . . 6 class 𝑖
11 c0g 17385 . . . . . . . 8 class 0g
127, 11cfv 6544 . . . . . . 7 class (0gβ€˜(Poly1β€˜π‘Ÿ))
1312csn 4629 . . . . . 6 class {(0gβ€˜(Poly1β€˜π‘Ÿ))}
1410, 13wceq 1542 . . . . 5 wff 𝑖 = {(0gβ€˜(Poly1β€˜π‘Ÿ))}
15 vg . . . . . . . . 9 setvar 𝑔
1615cv 1541 . . . . . . . 8 class 𝑔
17 cdg1 25569 . . . . . . . . 9 class deg1
185, 17cfv 6544 . . . . . . . 8 class ( deg1 β€˜π‘Ÿ)
1916, 18cfv 6544 . . . . . . 7 class (( deg1 β€˜π‘Ÿ)β€˜π‘”)
2010, 13cdif 3946 . . . . . . . . 9 class (𝑖 βˆ– {(0gβ€˜(Poly1β€˜π‘Ÿ))})
2118, 20cima 5680 . . . . . . . 8 class (( deg1 β€˜π‘Ÿ) β€œ (𝑖 βˆ– {(0gβ€˜(Poly1β€˜π‘Ÿ))}))
22 cr 11109 . . . . . . . 8 class ℝ
23 clt 11248 . . . . . . . 8 class <
2421, 22, 23cinf 9436 . . . . . . 7 class inf((( deg1 β€˜π‘Ÿ) β€œ (𝑖 βˆ– {(0gβ€˜(Poly1β€˜π‘Ÿ))})), ℝ, < )
2519, 24wceq 1542 . . . . . 6 wff (( deg1 β€˜π‘Ÿ)β€˜π‘”) = inf((( deg1 β€˜π‘Ÿ) β€œ (𝑖 βˆ– {(0gβ€˜(Poly1β€˜π‘Ÿ))})), ℝ, < )
26 cmn1 25643 . . . . . . . 8 class Monic1p
275, 26cfv 6544 . . . . . . 7 class (Monic1pβ€˜π‘Ÿ)
2810, 27cin 3948 . . . . . 6 class (𝑖 ∩ (Monic1pβ€˜π‘Ÿ))
2925, 15, 28crio 7364 . . . . 5 class (℩𝑔 ∈ (𝑖 ∩ (Monic1pβ€˜π‘Ÿ))(( deg1 β€˜π‘Ÿ)β€˜π‘”) = inf((( deg1 β€˜π‘Ÿ) β€œ (𝑖 βˆ– {(0gβ€˜(Poly1β€˜π‘Ÿ))})), ℝ, < ))
3014, 12, 29cif 4529 . . . 4 class if(𝑖 = {(0gβ€˜(Poly1β€˜π‘Ÿ))}, (0gβ€˜(Poly1β€˜π‘Ÿ)), (℩𝑔 ∈ (𝑖 ∩ (Monic1pβ€˜π‘Ÿ))(( deg1 β€˜π‘Ÿ)β€˜π‘”) = inf((( deg1 β€˜π‘Ÿ) β€œ (𝑖 βˆ– {(0gβ€˜(Poly1β€˜π‘Ÿ))})), ℝ, < )))
314, 9, 30cmpt 5232 . . 3 class (𝑖 ∈ (LIdealβ€˜(Poly1β€˜π‘Ÿ)) ↦ if(𝑖 = {(0gβ€˜(Poly1β€˜π‘Ÿ))}, (0gβ€˜(Poly1β€˜π‘Ÿ)), (℩𝑔 ∈ (𝑖 ∩ (Monic1pβ€˜π‘Ÿ))(( deg1 β€˜π‘Ÿ)β€˜π‘”) = inf((( deg1 β€˜π‘Ÿ) β€œ (𝑖 βˆ– {(0gβ€˜(Poly1β€˜π‘Ÿ))})), ℝ, < ))))
322, 3, 31cmpt 5232 . 2 class (π‘Ÿ ∈ V ↦ (𝑖 ∈ (LIdealβ€˜(Poly1β€˜π‘Ÿ)) ↦ if(𝑖 = {(0gβ€˜(Poly1β€˜π‘Ÿ))}, (0gβ€˜(Poly1β€˜π‘Ÿ)), (℩𝑔 ∈ (𝑖 ∩ (Monic1pβ€˜π‘Ÿ))(( deg1 β€˜π‘Ÿ)β€˜π‘”) = inf((( deg1 β€˜π‘Ÿ) β€œ (𝑖 βˆ– {(0gβ€˜(Poly1β€˜π‘Ÿ))})), ℝ, < )))))
331, 32wceq 1542 1 wff idlGen1p = (π‘Ÿ ∈ V ↦ (𝑖 ∈ (LIdealβ€˜(Poly1β€˜π‘Ÿ)) ↦ if(𝑖 = {(0gβ€˜(Poly1β€˜π‘Ÿ))}, (0gβ€˜(Poly1β€˜π‘Ÿ)), (℩𝑔 ∈ (𝑖 ∩ (Monic1pβ€˜π‘Ÿ))(( deg1 β€˜π‘Ÿ)β€˜π‘”) = inf((( deg1 β€˜π‘Ÿ) β€œ (𝑖 βˆ– {(0gβ€˜(Poly1β€˜π‘Ÿ))})), ℝ, < )))))
Colors of variables: wff setvar class
This definition is referenced by:  ig1pval  25690
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