Definition df-ig1p 23894

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 = ( r e. _V |-> ( i e. (LIdeal ` (Poly1 ` r ) ) |-> if ( i = { ( 0g ` (Poly1 ` r ) ) } , ( 0g ` (Poly1 ` r ) ) , ( iota_ g e. ( i i^i (Monic1p ` r ) ) ( ( deg1 ` r ) ` g ) = inf ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , < ) ) ) ) )

Distinct variable group:   g, r, i

Detailed syntax breakdown of Definition df-ig1p

Step	Hyp	Ref	Expression
1		cig1p 23889	. 2 class idlGen1p
2		vr	. . 3 setvar r
3		cvv 3200	. . 3 class _V
4		vi	. . . 4 setvar i
5	2	cv 1482	. . . . . 6 class r
6		cpl1 19547	. . . . . 6 class Poly1
7	5, 6	cfv 5888	. . . . 5 class (Poly1 ` r )
8		clidl 19170	. . . . 5 class LIdeal
9	7, 8	cfv 5888	. . . 4 class (LIdeal ` (Poly1 ` r ) )
10	4	cv 1482	. . . . . 6 class i
11		c0g 16100	. . . . . . . 8 class 0g
12	7, 11	cfv 5888	. . . . . . 7 class ( 0g ` (Poly1 ` r ) )
13	12	csn 4177	. . . . . 6 class { ( 0g ` (Poly1 ` r ) ) }
14	10, 13	wceq 1483	. . . . 5 wff i = { ( 0g ` (Poly1 ` r ) ) }
15		vg	. . . . . . . . 9 setvar g
16	15	cv 1482	. . . . . . . 8 class g
17		cdg1 23814	. . . . . . . . 9 class deg1
18	5, 17	cfv 5888	. . . . . . . 8 class ( deg1 ` r )
19	16, 18	cfv 5888	. . . . . . 7 class ( ( deg1 ` r ) ` g )
20	10, 13	cdif 3571	. . . . . . . . 9 class ( i \ { ( 0g ` (Poly1 ` r ) ) } )
21	18, 20	cima 5117	. . . . . . . 8 class ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) )
22		cr 9935	. . . . . . . 8 class RR
23		clt 10074	. . . . . . . 8 class <
24	21, 22, 23	cinf 8347	. . . . . . 7 class inf ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , < )
25	19, 24	wceq 1483	. . . . . 6 wff ( ( deg1 ` r ) ` g ) = inf ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , < )
26		cmn1 23885	. . . . . . . 8 class Monic1p
27	5, 26	cfv 5888	. . . . . . 7 class (Monic1p ` r )
28	10, 27	cin 3573	. . . . . 6 class ( i i^i (Monic1p ` r ) )
29	25, 15, 28	crio 6610	. . . . 5 class ( iota_ g e. ( i i^i (Monic1p ` r ) ) ( ( deg1 ` r ) ` g ) = inf ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , < ) )
30	14, 12, 29	cif 4086	. . . 4 class if ( i = { ( 0g ` (Poly1 ` r ) ) } , ( 0g ` (Poly1 ` r ) ) , ( iota_ g e. ( i i^i (Monic1p ` r ) ) ( ( deg1 ` r ) ` g ) = inf ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , < ) ) )
31	4, 9, 30	cmpt 4729	. . 3 class ( i e. (LIdeal ` (Poly1 ` r ) ) |-> if ( i = { ( 0g ` (Poly1 ` r ) ) } , ( 0g ` (Poly1 ` r ) ) , ( iota_ g e. ( i i^i (Monic1p ` r ) ) ( ( deg1 ` r ) ` g ) = inf ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , < ) ) ) )
32	2, 3, 31	cmpt 4729	. 2 class ( r e. _V |-> ( i e. (LIdeal ` (Poly1 ` r ) ) |-> if ( i = { ( 0g ` (Poly1 ` r ) ) } , ( 0g ` (Poly1 ` r ) ) , ( iota_ g e. ( i i^i (Monic1p ` r ) ) ( ( deg1 ` r ) ` g ) = inf ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , < ) ) ) ) )
33	1, 32	wceq 1483	1 wff idlGen1p = ( r e. _V |-> ( i e. (LIdeal ` (Poly1 ` r ) ) |-> if ( i = { ( 0g ` (Poly1 ` r ) ) } , ( 0g ` (Poly1 ` r ) ) , ( iota_ g e. ( i i^i (Monic1p ` r ) ) ( ( deg1 ` r ) ` g ) = inf ( ( ( deg1 ` r ) " ( i \ { ( 0g ` (Poly1 ` r ) ) } ) ) , RR , < ) ) ) ) )

This definition is referenced by:  ig1pval  23932