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 = deg 1 R
deg1pw.p P = Poly 1 R
deg1pw.x X = var 1 R
deg1pw.n N = mulGrp P
deg1pw.e × ˙ = N
Assertion deg1pw R NzRing F 0 D F × ˙ X = F

Proof

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