Metamath Proof Explorer


Theorem evl1scvarpwval

Description: Value of a univariate polynomial evaluation mapping a multiple of an exponentiation of a variable to the multiple of the exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019)

Ref Expression
Hypotheses evl1varpw.q Q = eval 1 R
evl1varpw.w W = Poly 1 R
evl1varpw.g G = mulGrp W
evl1varpw.x X = var 1 R
evl1varpw.b B = Base R
evl1varpw.e × ˙ = G
evl1varpw.r φ R CRing
evl1varpw.n φ N 0
evl1scvarpw.t1 × ˙ = W
evl1scvarpw.a φ A B
evl1scvarpwval.c φ C B
evl1scvarpwval.h H = mulGrp R
evl1scvarpwval.e E = H
evl1scvarpwval.t · ˙ = R
Assertion evl1scvarpwval φ Q A × ˙ N × ˙ X C = A · ˙ N E C

Proof

Step Hyp Ref Expression
1 evl1varpw.q Q = eval 1 R
2 evl1varpw.w W = Poly 1 R
3 evl1varpw.g G = mulGrp W
4 evl1varpw.x X = var 1 R
5 evl1varpw.b B = Base R
6 evl1varpw.e × ˙ = G
7 evl1varpw.r φ R CRing
8 evl1varpw.n φ N 0
9 evl1scvarpw.t1 × ˙ = W
10 evl1scvarpw.a φ A B
11 evl1scvarpwval.c φ C B
12 evl1scvarpwval.h H = mulGrp R
13 evl1scvarpwval.e E = H
14 evl1scvarpwval.t · ˙ = R
15 eqid Base W = Base W
16 3 15 mgpbas Base W = Base G
17 crngring R CRing R Ring
18 7 17 syl φ R Ring
19 2 ply1ring R Ring W Ring
20 18 19 syl φ W Ring
21 3 ringmgp W Ring G Mnd
22 20 21 syl φ G Mnd
23 4 2 15 vr1cl R Ring X Base W
24 18 23 syl φ X Base W
25 16 6 22 8 24 mulgnn0cld φ N × ˙ X Base W
26 1 2 3 4 5 6 7 8 11 12 13 evl1varpwval φ Q N × ˙ X C = N E C
27 25 26 jca φ N × ˙ X Base W Q N × ˙ X C = N E C
28 1 2 5 15 7 11 27 10 9 14 evl1vsd φ A × ˙ N × ˙ X Base W Q A × ˙ N × ˙ X C = A · ˙ N E C
29 28 simprd φ Q A × ˙ N × ˙ X C = A · ˙ N E C