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Metamath Proof Explorer

Theorem ig1pval3

Description: Characterizing properties of the monic generator of a nonzero ideal of polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015) (Revised by AV, 25-Sep-2020)

Ref Expression
Hypotheses ig1pval.p 
|- P = ( Poly1 ` R )
ig1pval.g 
|- G = ( idlGen1p ` R )
ig1pval3.z 
|- .0. = ( 0g ` P )
ig1pval3.u 
|- U = ( LIdeal ` P )
ig1pval3.d 
|- D = ( deg1 ` R )
ig1pval3.m 
|- M = ( Monic1p ` R )
Assertion ig1pval3 
|- ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( ( G ` I ) e. I /\ ( G ` I ) e. M /\ ( D ` ( G ` I ) ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )

Proof

Step Hyp Ref Expression
1 ig1pval.p 
 |-  P = ( Poly1 ` R )
2 ig1pval.g 
 |-  G = ( idlGen1p ` R )
3 ig1pval3.z 
 |-  .0. = ( 0g ` P )
4 ig1pval3.u 
 |-  U = ( LIdeal ` P )
5 ig1pval3.d 
 |-  D = ( deg1 ` R )
6 ig1pval3.m 
 |-  M = ( Monic1p ` R )
7 1 2 3 4 5 6 ig1pval 
 |-  ( ( R e. DivRing /\ I e. U ) -> ( G ` I ) = if ( I = { .0. } , .0. , ( iota_ g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) ) )
8 7 3adant3 
 |-  ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( G ` I ) = if ( I = { .0. } , .0. , ( iota_ g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) ) )
9 simp3 
 |-  ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> I =/= { .0. } )
10 9 neneqd 
 |-  ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> -. I = { .0. } )
11 10 iffalsed 
 |-  ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> if ( I = { .0. } , .0. , ( iota_ g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) ) = ( iota_ g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
12 8 11 eqtrd 
 |-  ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( G ` I ) = ( iota_ g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
13 1 4 3 6 5 ig1peu 
 |-  ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> E! g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) )
14 riotacl2 
 |-  ( E! g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) -> ( iota_ g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) e. { g e. ( I i^i M ) | ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) } )
15 13 14 syl 
 |-  ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( iota_ g e. ( I i^i M ) ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) e. { g e. ( I i^i M ) | ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) } )
16 12 15 eqeltrd 
 |-  ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( G ` I ) e. { g e. ( I i^i M ) | ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) } )
17 elin 
 |-  ( ( G ` I ) e. ( I i^i M ) <-> ( ( G ` I ) e. I /\ ( G ` I ) e. M ) )
18 17 anbi1i 
 |-  ( ( ( G ` I ) e. ( I i^i M ) /\ ( D ` ( G ` I ) ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) <-> ( ( ( G ` I ) e. I /\ ( G ` I ) e. M ) /\ ( D ` ( G ` I ) ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
19 fveqeq2 
 |-  ( g = ( G ` I ) -> ( ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) <-> ( D ` ( G ` I ) ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
20 19 elrab 
 |-  ( ( G ` I ) e. { g e. ( I i^i M ) | ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) } <-> ( ( G ` I ) e. ( I i^i M ) /\ ( D ` ( G ` I ) ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
21 df-3an 
 |-  ( ( ( G ` I ) e. I /\ ( G ` I ) e. M /\ ( D ` ( G ` I ) ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) <-> ( ( ( G ` I ) e. I /\ ( G ` I ) e. M ) /\ ( D ` ( G ` I ) ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
22 18 20 21 3bitr4i 
 |-  ( ( G ` I ) e. { g e. ( I i^i M ) | ( D ` g ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) } <-> ( ( G ` I ) e. I /\ ( G ` I ) e. M /\ ( D ` ( G ` I ) ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )
23 16 22 sylib 
 |-  ( ( R e. DivRing /\ I e. U /\ I =/= { .0. } ) -> ( ( G ` I ) e. I /\ ( G ` I ) e. M /\ ( D ` ( G ` I ) ) = inf ( ( D " ( I \ { .0. } ) ) , RR , < ) ) )