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Metamath Proof Explorer

Theorem deg1pw

Description: Exact degree of a variable power over a nontrivial ring. (Contributed by Stefan O'Rear, 1-Apr-2015)

Ref Expression
Hypotheses deg1pw.d 
|- D = ( deg1 ` R )
deg1pw.p 
|- P = ( Poly1 ` R )
deg1pw.x 
|- X = ( var1 ` R )
deg1pw.n 
|- N = ( mulGrp ` P )
deg1pw.e 
|- .^ = ( .g ` N )
Assertion deg1pw 
|- ( ( R e. NzRing /\ F e. NN0 ) -> ( D ` ( F .^ X ) ) = F )

Proof

Step Hyp Ref Expression
1 deg1pw.d 
 |-  D = ( deg1 ` R )
2 deg1pw.p 
 |-  P = ( Poly1 ` R )
3 deg1pw.x 
 |-  X = ( var1 ` R )
4 deg1pw.n 
 |-  N = ( mulGrp ` P )
5 deg1pw.e 
 |-  .^ = ( .g ` N )
6 2 ply1sca 
 |-  ( R e. NzRing -> R = ( Scalar ` P ) )
7 6 adantr 
 |-  ( ( R e. NzRing /\ F e. NN0 ) -> R = ( Scalar ` P ) )
8 7 fveq2d 
 |-  ( ( R e. NzRing /\ F e. NN0 ) -> ( 1r ` R ) = ( 1r ` ( Scalar ` P ) ) )
9 8 oveq1d 
 |-  ( ( R e. NzRing /\ F e. NN0 ) -> ( ( 1r ` R ) ( .s ` P ) ( F .^ X ) ) = ( ( 1r ` ( Scalar ` P ) ) ( .s ` P ) ( F .^ X ) ) )
10 nzrring 
 |-  ( R e. NzRing -> R e. Ring )
11 10 adantr 
 |-  ( ( R e. NzRing /\ F e. NN0 ) -> R e. Ring )
12 2 ply1lmod 
 |-  ( R e. Ring -> P e. LMod )
13 11 12 syl 
 |-  ( ( R e. NzRing /\ F e. NN0 ) -> P e. LMod )
14 eqid 
 |-  ( Base ` P ) = ( Base ` P )
15 4 14 mgpbas 
 |-  ( Base ` P ) = ( Base ` N )
16 2 ply1ring 
 |-  ( R e. Ring -> P e. Ring )
17 4 ringmgp 
 |-  ( P e. Ring -> N e. Mnd )
18 11 16 17 3syl 
 |-  ( ( R e. NzRing /\ F e. NN0 ) -> N e. Mnd )
19 simpr 
 |-  ( ( R e. NzRing /\ F e. NN0 ) -> F e. NN0 )
20 3 2 14 vr1cl 
 |-  ( R e. Ring -> X e. ( Base ` P ) )
21 11 20 syl 
 |-  ( ( R e. NzRing /\ F e. NN0 ) -> X e. ( Base ` P ) )
22 15 5 18 19 21 mulgnn0cld 
 |-  ( ( R e. NzRing /\ F e. NN0 ) -> ( F .^ X ) e. ( Base ` P ) )
23 eqid 
 |-  ( Scalar ` P ) = ( Scalar ` P )
24 eqid 
 |-  ( .s ` P ) = ( .s ` P )
25 eqid 
 |-  ( 1r ` ( Scalar ` P ) ) = ( 1r ` ( Scalar ` P ) )
26 14 23 24 25 lmodvs1 
 |-  ( ( P e. LMod /\ ( F .^ X ) e. ( Base ` P ) ) -> ( ( 1r ` ( Scalar ` P ) ) ( .s ` P ) ( F .^ X ) ) = ( F .^ X ) )
27 13 22 26 syl2anc 
 |-  ( ( R e. NzRing /\ F e. NN0 ) -> ( ( 1r ` ( Scalar ` P ) ) ( .s ` P ) ( F .^ X ) ) = ( F .^ X ) )
28 9 27 eqtrd 
 |-  ( ( R e. NzRing /\ F e. NN0 ) -> ( ( 1r ` R ) ( .s ` P ) ( F .^ X ) ) = ( F .^ X ) )
29 28 fveq2d 
 |-  ( ( R e. NzRing /\ F e. NN0 ) -> ( D ` ( ( 1r ` R ) ( .s ` P ) ( F .^ X ) ) ) = ( D ` ( F .^ X ) ) )
30 eqid 
 |-  ( Base ` R ) = ( Base ` R )
31 eqid 
 |-  ( 1r ` R ) = ( 1r ` R )
32 30 31 ringidcl 
 |-  ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) )
33 11 32 syl 
 |-  ( ( R e. NzRing /\ F e. NN0 ) -> ( 1r ` R ) e. ( Base ` R ) )
34 eqid 
 |-  ( 0g ` R ) = ( 0g ` R )
35 31 34 nzrnz 
 |-  ( R e. NzRing -> ( 1r ` R ) =/= ( 0g ` R ) )
36 35 adantr 
 |-  ( ( R e. NzRing /\ F e. NN0 ) -> ( 1r ` R ) =/= ( 0g ` R ) )
37 1 30 2 3 24 4 5 34 deg1tm 
 |-  ( ( R e. Ring /\ ( ( 1r ` R ) e. ( Base ` R ) /\ ( 1r ` R ) =/= ( 0g ` R ) ) /\ F e. NN0 ) -> ( D ` ( ( 1r ` R ) ( .s ` P ) ( F .^ X ) ) ) = F )
38 11 33 36 19 37 syl121anc 
 |-  ( ( R e. NzRing /\ F e. NN0 ) -> ( D ` ( ( 1r ` R ) ( .s ` P ) ( F .^ X ) ) ) = F )
39 29 38 eqtr3d 
 |-  ( ( R e. NzRing /\ F e. NN0 ) -> ( D ` ( F .^ X ) ) = F )