This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem deg1scl

Description: A nonzero scalar polynomial has zero degree. (Contributed by Stefan O'Rear, 29-Mar-2015)

Ref Expression
Hypotheses deg1sclle.d 
|- D = ( deg1 ` R )
deg1sclle.p 
|- P = ( Poly1 ` R )
deg1sclle.k 
|- K = ( Base ` R )
deg1sclle.a 
|- A = ( algSc ` P )
deg1scl.z 
|- .0. = ( 0g ` R )
Assertion deg1scl 
|- ( ( R e. Ring /\ F e. K /\ F =/= .0. ) -> ( D ` ( A ` F ) ) = 0 )

Proof

Step Hyp Ref Expression
1 deg1sclle.d 
 |-  D = ( deg1 ` R )
2 deg1sclle.p 
 |-  P = ( Poly1 ` R )
3 deg1sclle.k 
 |-  K = ( Base ` R )
4 deg1sclle.a 
 |-  A = ( algSc ` P )
5 deg1scl.z 
 |-  .0. = ( 0g ` R )
6 1 2 3 4 deg1sclle 
 |-  ( ( R e. Ring /\ F e. K ) -> ( D ` ( A ` F ) ) <_ 0 )
7 6 3adant3 
 |-  ( ( R e. Ring /\ F e. K /\ F =/= .0. ) -> ( D ` ( A ` F ) ) <_ 0 )
8 simp1 
 |-  ( ( R e. Ring /\ F e. K /\ F =/= .0. ) -> R e. Ring )
9 eqid 
 |-  ( Base ` P ) = ( Base ` P )
10 2 4 3 9 ply1sclcl 
 |-  ( ( R e. Ring /\ F e. K ) -> ( A ` F ) e. ( Base ` P ) )
11 10 3adant3 
 |-  ( ( R e. Ring /\ F e. K /\ F =/= .0. ) -> ( A ` F ) e. ( Base ` P ) )
12 eqid 
 |-  ( 0g ` P ) = ( 0g ` P )
13 2 4 5 12 3 ply1scln0 
 |-  ( ( R e. Ring /\ F e. K /\ F =/= .0. ) -> ( A ` F ) =/= ( 0g ` P ) )
14 1 2 12 9 deg1nn0cl 
 |-  ( ( R e. Ring /\ ( A ` F ) e. ( Base ` P ) /\ ( A ` F ) =/= ( 0g ` P ) ) -> ( D ` ( A ` F ) ) e. NN0 )
15 8 11 13 14 syl3anc 
 |-  ( ( R e. Ring /\ F e. K /\ F =/= .0. ) -> ( D ` ( A ` F ) ) e. NN0 )
16 nn0le0eq0 
 |-  ( ( D ` ( A ` F ) ) e. NN0 -> ( ( D ` ( A ` F ) ) <_ 0 <-> ( D ` ( A ` F ) ) = 0 ) )
17 15 16 syl 
 |-  ( ( R e. Ring /\ F e. K /\ F =/= .0. ) -> ( ( D ` ( A ` F ) ) <_ 0 <-> ( D ` ( A ` F ) ) = 0 ) )
18 7 17 mpbid 
 |-  ( ( R e. Ring /\ F e. K /\ F =/= .0. ) -> ( D ` ( A ` F ) ) = 0 )