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Metamath Proof Explorer

Theorem deg1ldgdomn

Description: A nonzero univariate polynomial over a domain always has a nonzero-divisor leading coefficient. (Contributed by Stefan O'Rear, 28-Mar-2015)

Ref Expression
Hypotheses deg1z.d 
|- D = ( deg1 ` R )
deg1z.p 
|- P = ( Poly1 ` R )
deg1z.z 
|- .0. = ( 0g ` P )
deg1nn0cl.b 
|- B = ( Base ` P )
deg1ldgdomn.e 
|- E = ( RLReg ` R )
deg1ldgdomn.a 
|- A = ( coe1 ` F )
Assertion deg1ldgdomn 
|- ( ( R e. Domn /\ F e. B /\ F =/= .0. ) -> ( A ` ( D ` F ) ) e. E )

Proof

Step Hyp Ref Expression
1 deg1z.d 
 |-  D = ( deg1 ` R )
2 deg1z.p 
 |-  P = ( Poly1 ` R )
3 deg1z.z 
 |-  .0. = ( 0g ` P )
4 deg1nn0cl.b 
 |-  B = ( Base ` P )
5 deg1ldgdomn.e 
 |-  E = ( RLReg ` R )
6 deg1ldgdomn.a 
 |-  A = ( coe1 ` F )
7 simp1 
 |-  ( ( R e. Domn /\ F e. B /\ F =/= .0. ) -> R e. Domn )
8 eqid 
 |-  ( Base ` R ) = ( Base ` R )
9 6 4 2 8 coe1f 
 |-  ( F e. B -> A : NN0 --> ( Base ` R ) )
10 9 3ad2ant2 
 |-  ( ( R e. Domn /\ F e. B /\ F =/= .0. ) -> A : NN0 --> ( Base ` R ) )
11 domnring 
 |-  ( R e. Domn -> R e. Ring )
12 1 2 3 4 deg1nn0cl 
 |-  ( ( R e. Ring /\ F e. B /\ F =/= .0. ) -> ( D ` F ) e. NN0 )
13 11 12 syl3an1 
 |-  ( ( R e. Domn /\ F e. B /\ F =/= .0. ) -> ( D ` F ) e. NN0 )
14 10 13 ffvelcdmd 
 |-  ( ( R e. Domn /\ F e. B /\ F =/= .0. ) -> ( A ` ( D ` F ) ) e. ( Base ` R ) )
15 eqid 
 |-  ( 0g ` R ) = ( 0g ` R )
16 1 2 3 4 15 6 deg1ldg 
 |-  ( ( R e. Ring /\ F e. B /\ F =/= .0. ) -> ( A ` ( D ` F ) ) =/= ( 0g ` R ) )
17 11 16 syl3an1 
 |-  ( ( R e. Domn /\ F e. B /\ F =/= .0. ) -> ( A ` ( D ` F ) ) =/= ( 0g ` R ) )
18 8 5 15 domnrrg 
 |-  ( ( R e. Domn /\ ( A ` ( D ` F ) ) e. ( Base ` R ) /\ ( A ` ( D ` F ) ) =/= ( 0g ` R ) ) -> ( A ` ( D ` F ) ) e. E )
19 7 14 17 18 syl3anc 
 |-  ( ( R e. Domn /\ F e. B /\ F =/= .0. ) -> ( A ` ( D ` F ) ) e. E )