Metamath Proof Explorer

Theorem uc1pdeg

Description: Unitic polynomials have nonnegative degrees. (Contributed by Stefan O'Rear, 28-Mar-2015)

Ref Expression
Hypotheses uc1pdeg.d 
|- D = ( deg1 ` R )
uc1pdeg.c 
|- C = ( Unic1p ` R )
Assertion uc1pdeg 
|- ( ( R e. Ring /\ F e. C ) -> ( D ` F ) e. NN0 )

Proof

Step Hyp Ref Expression
1 uc1pdeg.d 
 |-  D = ( deg1 ` R )
2 uc1pdeg.c 
 |-  C = ( Unic1p ` R )
3 simpl 
 |-  ( ( R e. Ring /\ F e. C ) -> R e. Ring )
4 eqid 
 |-  ( Poly1 ` R ) = ( Poly1 ` R )
5 eqid 
 |-  ( Base ` ( Poly1 ` R ) ) = ( Base ` ( Poly1 ` R ) )
6 4 5 2 uc1pcl 
 |-  ( F e. C -> F e. ( Base ` ( Poly1 ` R ) ) )
7 6 adantl 
 |-  ( ( R e. Ring /\ F e. C ) -> F e. ( Base ` ( Poly1 ` R ) ) )
8 eqid 
 |-  ( 0g ` ( Poly1 ` R ) ) = ( 0g ` ( Poly1 ` R ) )
9 4 8 2 uc1pn0 
 |-  ( F e. C -> F =/= ( 0g ` ( Poly1 ` R ) ) )
10 9 adantl 
 |-  ( ( R e. Ring /\ F e. C ) -> F =/= ( 0g ` ( Poly1 ` R ) ) )
11 1 4 8 5 deg1nn0cl 
 |-  ( ( R e. Ring /\ F e. ( Base ` ( Poly1 ` R ) ) /\ F =/= ( 0g ` ( Poly1 ` R ) ) ) -> ( D ` F ) e. NN0 )
12 3 7 10 11 syl3anc 
 |-  ( ( R e. Ring /\ F e. C ) -> ( D ` F ) e. NN0 )