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Metamath Proof Explorer

Theorem deg1invg

Description: The degree of the negated polynomial is the same as the original. (Contributed by Stefan O'Rear, 28-Mar-2015)

Ref Expression
Hypotheses deg1addle.y 
|- Y = ( Poly1 ` R )
deg1addle.d 
|- D = ( deg1 ` R )
deg1addle.r 
|- ( ph -> R e. Ring )
deg1invg.b 
|- B = ( Base ` Y )
deg1invg.n 
|- N = ( invg ` Y )
deg1invg.f 
|- ( ph -> F e. B )
Assertion deg1invg 
|- ( ph -> ( D ` ( N ` F ) ) = ( D ` F ) )

Proof

Step Hyp Ref Expression
1 deg1addle.y 
 |-  Y = ( Poly1 ` R )
2 deg1addle.d 
 |-  D = ( deg1 ` R )
3 deg1addle.r 
 |-  ( ph -> R e. Ring )
4 deg1invg.b 
 |-  B = ( Base ` Y )
5 deg1invg.n 
 |-  N = ( invg ` Y )
6 deg1invg.f 
 |-  ( ph -> F e. B )
7 1 ply1lmod 
 |-  ( R e. Ring -> Y e. LMod )
8 3 7 syl 
 |-  ( ph -> Y e. LMod )
9 1 ply1sca2 
 |-  ( _I ` R ) = ( Scalar ` Y )
10 eqid 
 |-  ( .s ` Y ) = ( .s ` Y )
11 eqid 
 |-  ( 1r ` ( _I ` R ) ) = ( 1r ` ( _I ` R ) )
12 eqid 
 |-  ( invg ` R ) = ( invg ` R )
13 12 grpinvfvi 
 |-  ( invg ` R ) = ( invg ` ( _I ` R ) )
14 4 5 9 10 11 13 lmodvneg1 
 |-  ( ( Y e. LMod /\ F e. B ) -> ( ( ( invg ` R ) ` ( 1r ` ( _I ` R ) ) ) ( .s ` Y ) F ) = ( N ` F ) )
15 8 6 14 syl2anc 
 |-  ( ph -> ( ( ( invg ` R ) ` ( 1r ` ( _I ` R ) ) ) ( .s ` Y ) F ) = ( N ` F ) )
16 15 fveq2d 
 |-  ( ph -> ( D ` ( ( ( invg ` R ) ` ( 1r ` ( _I ` R ) ) ) ( .s ` Y ) F ) ) = ( D ` ( N ` F ) ) )
17 eqid 
 |-  ( RLReg ` R ) = ( RLReg ` R )
18 fvi 
 |-  ( R e. Ring -> ( _I ` R ) = R )
19 3 18 syl 
 |-  ( ph -> ( _I ` R ) = R )
20 19 fveq2d 
 |-  ( ph -> ( 1r ` ( _I ` R ) ) = ( 1r ` R ) )
21 20 fveq2d 
 |-  ( ph -> ( ( invg ` R ) ` ( 1r ` ( _I ` R ) ) ) = ( ( invg ` R ) ` ( 1r ` R ) ) )
22 eqid 
 |-  ( Unit ` R ) = ( Unit ` R )
23 17 22 unitrrg 
 |-  ( R e. Ring -> ( Unit ` R ) C_ ( RLReg ` R ) )
24 3 23 syl 
 |-  ( ph -> ( Unit ` R ) C_ ( RLReg ` R ) )
25 eqid 
 |-  ( 1r ` R ) = ( 1r ` R )
26 22 25 1unit 
 |-  ( R e. Ring -> ( 1r ` R ) e. ( Unit ` R ) )
27 22 12 unitnegcl 
 |-  ( ( R e. Ring /\ ( 1r ` R ) e. ( Unit ` R ) ) -> ( ( invg ` R ) ` ( 1r ` R ) ) e. ( Unit ` R ) )
28 3 26 27 syl2anc2 
 |-  ( ph -> ( ( invg ` R ) ` ( 1r ` R ) ) e. ( Unit ` R ) )
29 24 28 sseldd 
 |-  ( ph -> ( ( invg ` R ) ` ( 1r ` R ) ) e. ( RLReg ` R ) )
30 21 29 eqeltrd 
 |-  ( ph -> ( ( invg ` R ) ` ( 1r ` ( _I ` R ) ) ) e. ( RLReg ` R ) )
31 1 2 3 4 17 10 30 6 deg1vsca 
 |-  ( ph -> ( D ` ( ( ( invg ` R ) ` ( 1r ` ( _I ` R ) ) ) ( .s ` Y ) F ) ) = ( D ` F ) )
32 16 31 eqtr3d 
 |-  ( ph -> ( D ` ( N ` F ) ) = ( D ` F ) )