Metamath Proof Explorer


Theorem coe1tm

Description: Coefficient vector of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015)

Ref Expression
Hypotheses coe1tm.z 0 ˙ = 0 R
coe1tm.k K = Base R
coe1tm.p P = Poly 1 R
coe1tm.x X = var 1 R
coe1tm.m · ˙ = P
coe1tm.n N = mulGrp P
coe1tm.e × ˙ = N
Assertion coe1tm R Ring C K D 0 coe 1 C · ˙ D × ˙ X = x 0 if x = D C 0 ˙

Proof

Step Hyp Ref Expression
1 coe1tm.z 0 ˙ = 0 R
2 coe1tm.k K = Base R
3 coe1tm.p P = Poly 1 R
4 coe1tm.x X = var 1 R
5 coe1tm.m · ˙ = P
6 coe1tm.n N = mulGrp P
7 coe1tm.e × ˙ = N
8 eqid Base P = Base P
9 2 3 4 5 6 7 8 ply1tmcl R Ring C K D 0 C · ˙ D × ˙ X Base P
10 eqid coe 1 C · ˙ D × ˙ X = coe 1 C · ˙ D × ˙ X
11 eqid x 0 1 𝑜 × x = x 0 1 𝑜 × x
12 10 8 3 11 coe1fval2 C · ˙ D × ˙ X Base P coe 1 C · ˙ D × ˙ X = C · ˙ D × ˙ X x 0 1 𝑜 × x
13 9 12 syl R Ring C K D 0 coe 1 C · ˙ D × ˙ X = C · ˙ D × ˙ X x 0 1 𝑜 × x
14 fconst6g x 0 1 𝑜 × x : 1 𝑜 0
15 nn0ex 0 V
16 1oex 1 𝑜 V
17 15 16 elmap 1 𝑜 × x 0 1 𝑜 1 𝑜 × x : 1 𝑜 0
18 14 17 sylibr x 0 1 𝑜 × x 0 1 𝑜
19 18 adantl R Ring C K D 0 x 0 1 𝑜 × x 0 1 𝑜
20 eqidd R Ring C K D 0 x 0 1 𝑜 × x = x 0 1 𝑜 × x
21 eqid mulGrp 1 𝑜 mPoly R = mulGrp 1 𝑜 mPoly R
22 6 8 mgpbas Base P = Base N
23 22 a1i R Ring Base P = Base N
24 eqid mulGrp 1 𝑜 mPoly R = mulGrp 1 𝑜 mPoly R
25 3 8 ply1bas Base P = Base 1 𝑜 mPoly R
26 24 25 mgpbas Base P = Base mulGrp 1 𝑜 mPoly R
27 26 a1i R Ring Base P = Base mulGrp 1 𝑜 mPoly R
28 ssv Base P V
29 28 a1i R Ring Base P V
30 ovexd R Ring x V y V x + N y V
31 eqid P = P
32 6 31 mgpplusg P = + N
33 eqid 1 𝑜 mPoly R = 1 𝑜 mPoly R
34 3 33 31 ply1mulr P = 1 𝑜 mPoly R
35 24 34 mgpplusg P = + mulGrp 1 𝑜 mPoly R
36 32 35 eqtr3i + N = + mulGrp 1 𝑜 mPoly R
37 36 a1i R Ring + N = + mulGrp 1 𝑜 mPoly R
38 37 oveqdr R Ring x V y V x + N y = x + mulGrp 1 𝑜 mPoly R y
39 7 21 23 27 29 30 38 mulgpropd R Ring × ˙ = mulGrp 1 𝑜 mPoly R
40 39 3ad2ant1 R Ring C K D 0 × ˙ = mulGrp 1 𝑜 mPoly R
41 eqidd R Ring C K D 0 D = D
42 4 vr1val X = 1 𝑜 mVar R
43 42 a1i R Ring C K D 0 X = 1 𝑜 mVar R
44 40 41 43 oveq123d R Ring C K D 0 D × ˙ X = D mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R
45 44 oveq2d R Ring C K D 0 C · ˙ D × ˙ X = C · ˙ D mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R
46 psr1baslem 0 1 𝑜 = a 0 1 𝑜 | a -1 Fin
47 eqid 1 R = 1 R
48 1on 1 𝑜 On
49 48 a1i R Ring C K D 0 1 𝑜 On
50 eqid 1 𝑜 mVar R = 1 𝑜 mVar R
51 simp1 R Ring C K D 0 R Ring
52 0lt1o 1 𝑜
53 52 a1i R Ring C K D 0 1 𝑜
54 simp3 R Ring C K D 0 D 0
55 33 46 1 47 49 24 21 50 51 53 54 mplcoe3 R Ring C K D 0 y 0 1 𝑜 if y = b 1 𝑜 if b = D 0 1 R 0 ˙ = D mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R
56 55 oveq2d R Ring C K D 0 C · ˙ y 0 1 𝑜 if y = b 1 𝑜 if b = D 0 1 R 0 ˙ = C · ˙ D mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R
57 3 33 5 ply1vsca · ˙ = 1 𝑜 mPoly R
58 elsni b b =
59 df1o2 1 𝑜 =
60 58 59 eleq2s b 1 𝑜 b =
61 60 iftrued b 1 𝑜 if b = D 0 = D
62 61 adantl R Ring C K D 0 b 1 𝑜 if b = D 0 = D
63 62 mpteq2dva R Ring C K D 0 b 1 𝑜 if b = D 0 = b 1 𝑜 D
64 fconstmpt 1 𝑜 × D = b 1 𝑜 D
65 63 64 eqtr4di R Ring C K D 0 b 1 𝑜 if b = D 0 = 1 𝑜 × D
66 fconst6g D 0 1 𝑜 × D : 1 𝑜 0
67 15 16 elmap 1 𝑜 × D 0 1 𝑜 1 𝑜 × D : 1 𝑜 0
68 66 67 sylibr D 0 1 𝑜 × D 0 1 𝑜
69 68 3ad2ant3 R Ring C K D 0 1 𝑜 × D 0 1 𝑜
70 65 69 eqeltrd R Ring C K D 0 b 1 𝑜 if b = D 0 0 1 𝑜
71 simp2 R Ring C K D 0 C K
72 33 57 46 47 1 2 49 51 70 71 mplmon2 R Ring C K D 0 C · ˙ y 0 1 𝑜 if y = b 1 𝑜 if b = D 0 1 R 0 ˙ = y 0 1 𝑜 if y = b 1 𝑜 if b = D 0 C 0 ˙
73 45 56 72 3eqtr2d R Ring C K D 0 C · ˙ D × ˙ X = y 0 1 𝑜 if y = b 1 𝑜 if b = D 0 C 0 ˙
74 eqeq1 y = 1 𝑜 × x y = b 1 𝑜 if b = D 0 1 𝑜 × x = b 1 𝑜 if b = D 0
75 74 ifbid y = 1 𝑜 × x if y = b 1 𝑜 if b = D 0 C 0 ˙ = if 1 𝑜 × x = b 1 𝑜 if b = D 0 C 0 ˙
76 19 20 73 75 fmptco R Ring C K D 0 C · ˙ D × ˙ X x 0 1 𝑜 × x = x 0 if 1 𝑜 × x = b 1 𝑜 if b = D 0 C 0 ˙
77 65 adantr R Ring C K D 0 x 0 b 1 𝑜 if b = D 0 = 1 𝑜 × D
78 77 eqeq2d R Ring C K D 0 x 0 1 𝑜 × x = b 1 𝑜 if b = D 0 1 𝑜 × x = 1 𝑜 × D
79 fveq1 1 𝑜 × x = 1 𝑜 × D 1 𝑜 × x = 1 𝑜 × D
80 vex x V
81 80 fvconst2 1 𝑜 1 𝑜 × x = x
82 52 81 mp1i R Ring C K D 0 x 0 1 𝑜 × x = x
83 simpl3 R Ring C K D 0 x 0 D 0
84 fvconst2g D 0 1 𝑜 1 𝑜 × D = D
85 83 52 84 sylancl R Ring C K D 0 x 0 1 𝑜 × D = D
86 82 85 eqeq12d R Ring C K D 0 x 0 1 𝑜 × x = 1 𝑜 × D x = D
87 79 86 imbitrid R Ring C K D 0 x 0 1 𝑜 × x = 1 𝑜 × D x = D
88 sneq x = D x = D
89 88 xpeq2d x = D 1 𝑜 × x = 1 𝑜 × D
90 87 89 impbid1 R Ring C K D 0 x 0 1 𝑜 × x = 1 𝑜 × D x = D
91 78 90 bitrd R Ring C K D 0 x 0 1 𝑜 × x = b 1 𝑜 if b = D 0 x = D
92 91 ifbid R Ring C K D 0 x 0 if 1 𝑜 × x = b 1 𝑜 if b = D 0 C 0 ˙ = if x = D C 0 ˙
93 92 mpteq2dva R Ring C K D 0 x 0 if 1 𝑜 × x = b 1 𝑜 if b = D 0 C 0 ˙ = x 0 if x = D C 0 ˙
94 13 76 93 3eqtrd R Ring C K D 0 coe 1 C · ˙ D × ˙ X = x 0 if x = D C 0 ˙