Metamath Proof Explorer


Theorem ply1coe

Description: Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015) (Revised by AV, 7-Oct-2019)

Ref Expression
Hypotheses ply1coe.p P = Poly 1 R
ply1coe.x X = var 1 R
ply1coe.b B = Base P
ply1coe.n · ˙ = P
ply1coe.m M = mulGrp P
ply1coe.e × ˙ = M
ply1coe.a A = coe 1 K
Assertion ply1coe R Ring K B K = P k 0 A k · ˙ k × ˙ X

Proof

Step Hyp Ref Expression
1 ply1coe.p P = Poly 1 R
2 ply1coe.x X = var 1 R
3 ply1coe.b B = Base P
4 ply1coe.n · ˙ = P
5 ply1coe.m M = mulGrp P
6 ply1coe.e × ˙ = M
7 ply1coe.a A = coe 1 K
8 eqid 1 𝑜 mPoly R = 1 𝑜 mPoly R
9 psr1baslem 0 1 𝑜 = d 0 1 𝑜 | d -1 Fin
10 eqid 0 R = 0 R
11 eqid 1 R = 1 R
12 1onn 1 𝑜 ω
13 12 a1i R Ring K B 1 𝑜 ω
14 eqid PwSer 1 R = PwSer 1 R
15 1 14 3 ply1bas B = Base 1 𝑜 mPoly R
16 1 8 4 ply1vsca · ˙ = 1 𝑜 mPoly R
17 simpl R Ring K B R Ring
18 simpr R Ring K B K B
19 8 9 10 11 13 15 16 17 18 mplcoe1 R Ring K B K = 1 𝑜 mPoly R a 0 1 𝑜 K a · ˙ b 0 1 𝑜 if b = a 1 R 0 R
20 7 fvcoe1 K B a 0 1 𝑜 K a = A a
21 20 adantll R Ring K B a 0 1 𝑜 K a = A a
22 12 a1i R Ring K B a 0 1 𝑜 1 𝑜 ω
23 eqid mulGrp 1 𝑜 mPoly R = mulGrp 1 𝑜 mPoly R
24 eqid mulGrp 1 𝑜 mPoly R = mulGrp 1 𝑜 mPoly R
25 eqid 1 𝑜 mVar R = 1 𝑜 mVar R
26 simpll R Ring K B a 0 1 𝑜 R Ring
27 simpr R Ring K B a 0 1 𝑜 a 0 1 𝑜
28 eqidd R Ring K B a 0 1 𝑜 1 𝑜 mVar R + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R = 1 𝑜 mVar R + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R
29 0ex V
30 fveq2 b = 1 𝑜 mVar R b = 1 𝑜 mVar R
31 30 oveq1d b = 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R = 1 𝑜 mVar R + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R
32 30 oveq2d b = 1 𝑜 mVar R + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b = 1 𝑜 mVar R + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R
33 31 32 eqeq12d b = 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R = 1 𝑜 mVar R + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b 1 𝑜 mVar R + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R = 1 𝑜 mVar R + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R
34 29 33 ralsn b 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R = 1 𝑜 mVar R + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b 1 𝑜 mVar R + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R = 1 𝑜 mVar R + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R
35 28 34 sylibr R Ring K B a 0 1 𝑜 b 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R = 1 𝑜 mVar R + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b
36 fveq2 x = 1 𝑜 mVar R x = 1 𝑜 mVar R
37 36 oveq2d x = 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R x = 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R
38 36 oveq1d x = 1 𝑜 mVar R x + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b = 1 𝑜 mVar R + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b
39 37 38 eqeq12d x = 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R x = 1 𝑜 mVar R x + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R = 1 𝑜 mVar R + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b
40 39 ralbidv x = b 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R x = 1 𝑜 mVar R x + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b b 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R = 1 𝑜 mVar R + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b
41 29 40 ralsn x b 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R x = 1 𝑜 mVar R x + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b b 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R = 1 𝑜 mVar R + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b
42 35 41 sylibr R Ring K B a 0 1 𝑜 x b 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R x = 1 𝑜 mVar R x + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b
43 df1o2 1 𝑜 =
44 43 raleqi b 1 𝑜 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R x = 1 𝑜 mVar R x + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b b 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R x = 1 𝑜 mVar R x + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b
45 43 44 raleqbii x 1 𝑜 b 1 𝑜 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R x = 1 𝑜 mVar R x + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b x b 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R x = 1 𝑜 mVar R x + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b
46 42 45 sylibr R Ring K B a 0 1 𝑜 x 1 𝑜 b 1 𝑜 1 𝑜 mVar R b + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R x = 1 𝑜 mVar R x + mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R b
47 8 9 10 11 22 23 24 25 26 27 46 mplcoe5 R Ring K B a 0 1 𝑜 b 0 1 𝑜 if b = a 1 R 0 R = mulGrp 1 𝑜 mPoly R c 1 𝑜 a c mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R c
48 43 mpteq1i c 1 𝑜 a c mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R c = c a c mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R c
49 48 oveq2i mulGrp 1 𝑜 mPoly R c 1 𝑜 a c mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R c = mulGrp 1 𝑜 mPoly R c a c mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R c
50 8 mplring 1 𝑜 ω R Ring 1 𝑜 mPoly R Ring
51 12 50 mpan R Ring 1 𝑜 mPoly R Ring
52 23 ringmgp 1 𝑜 mPoly R Ring mulGrp 1 𝑜 mPoly R Mnd
53 51 52 syl R Ring mulGrp 1 𝑜 mPoly R Mnd
54 53 ad2antrr R Ring K B a 0 1 𝑜 mulGrp 1 𝑜 mPoly R Mnd
55 29 a1i R Ring K B a 0 1 𝑜 V
56 23 15 mgpbas B = Base mulGrp 1 𝑜 mPoly R
57 56 a1i R Ring K B B = Base mulGrp 1 𝑜 mPoly R
58 5 3 mgpbas B = Base M
59 58 a1i R Ring K B B = Base M
60 ssv B V
61 60 a1i R Ring K B B V
62 ovexd R Ring K B a V b V a + mulGrp 1 𝑜 mPoly R b V
63 eqid P = P
64 1 8 63 ply1mulr P = 1 𝑜 mPoly R
65 23 64 mgpplusg P = + mulGrp 1 𝑜 mPoly R
66 5 63 mgpplusg P = + M
67 65 66 eqtr3i + mulGrp 1 𝑜 mPoly R = + M
68 67 oveqi a + mulGrp 1 𝑜 mPoly R b = a + M b
69 68 a1i R Ring K B a V b V a + mulGrp 1 𝑜 mPoly R b = a + M b
70 24 6 57 59 61 62 69 mulgpropd R Ring K B mulGrp 1 𝑜 mPoly R = × ˙
71 70 oveqd R Ring K B a mulGrp 1 𝑜 mPoly R X = a × ˙ X
72 71 adantr R Ring K B a 0 1 𝑜 a mulGrp 1 𝑜 mPoly R X = a × ˙ X
73 1 ply1ring R Ring P Ring
74 5 ringmgp P Ring M Mnd
75 73 74 syl R Ring M Mnd
76 75 ad2antrr R Ring K B a 0 1 𝑜 M Mnd
77 elmapi a 0 1 𝑜 a : 1 𝑜 0
78 0lt1o 1 𝑜
79 ffvelcdm a : 1 𝑜 0 1 𝑜 a 0
80 77 78 79 sylancl a 0 1 𝑜 a 0
81 80 adantl R Ring K B a 0 1 𝑜 a 0
82 2 1 3 vr1cl R Ring X B
83 82 ad2antrr R Ring K B a 0 1 𝑜 X B
84 58 6 76 81 83 mulgnn0cld R Ring K B a 0 1 𝑜 a × ˙ X B
85 72 84 eqeltrd R Ring K B a 0 1 𝑜 a mulGrp 1 𝑜 mPoly R X B
86 fveq2 c = a c = a
87 fveq2 c = 1 𝑜 mVar R c = 1 𝑜 mVar R
88 2 vr1val X = 1 𝑜 mVar R
89 87 88 eqtr4di c = 1 𝑜 mVar R c = X
90 86 89 oveq12d c = a c mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R c = a mulGrp 1 𝑜 mPoly R X
91 56 90 gsumsn mulGrp 1 𝑜 mPoly R Mnd V a mulGrp 1 𝑜 mPoly R X B mulGrp 1 𝑜 mPoly R c a c mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R c = a mulGrp 1 𝑜 mPoly R X
92 54 55 85 91 syl3anc R Ring K B a 0 1 𝑜 mulGrp 1 𝑜 mPoly R c a c mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R c = a mulGrp 1 𝑜 mPoly R X
93 49 92 eqtrid R Ring K B a 0 1 𝑜 mulGrp 1 𝑜 mPoly R c 1 𝑜 a c mulGrp 1 𝑜 mPoly R 1 𝑜 mVar R c = a mulGrp 1 𝑜 mPoly R X
94 47 93 72 3eqtrd R Ring K B a 0 1 𝑜 b 0 1 𝑜 if b = a 1 R 0 R = a × ˙ X
95 21 94 oveq12d R Ring K B a 0 1 𝑜 K a · ˙ b 0 1 𝑜 if b = a 1 R 0 R = A a · ˙ a × ˙ X
96 95 mpteq2dva R Ring K B a 0 1 𝑜 K a · ˙ b 0 1 𝑜 if b = a 1 R 0 R = a 0 1 𝑜 A a · ˙ a × ˙ X
97 96 oveq2d R Ring K B 1 𝑜 mPoly R a 0 1 𝑜 K a · ˙ b 0 1 𝑜 if b = a 1 R 0 R = 1 𝑜 mPoly R a 0 1 𝑜 A a · ˙ a × ˙ X
98 nn0ex 0 V
99 98 mptex k 0 A k · ˙ k × ˙ X V
100 99 a1i R Ring K B k 0 A k · ˙ k × ˙ X V
101 1 fvexi P V
102 101 a1i R Ring K B P V
103 ovexd R Ring K B 1 𝑜 mPoly R V
104 3 15 eqtr3i Base P = Base 1 𝑜 mPoly R
105 104 a1i R Ring K B Base P = Base 1 𝑜 mPoly R
106 eqid + P = + P
107 1 8 106 ply1plusg + P = + 1 𝑜 mPoly R
108 107 a1i R Ring K B + P = + 1 𝑜 mPoly R
109 100 102 103 105 108 gsumpropd R Ring K B P k 0 A k · ˙ k × ˙ X = 1 𝑜 mPoly R k 0 A k · ˙ k × ˙ X
110 eqid 0 P = 0 P
111 8 1 110 ply1mpl0 0 P = 0 1 𝑜 mPoly R
112 8 mpllmod 1 𝑜 ω R Ring 1 𝑜 mPoly R LMod
113 12 17 112 sylancr R Ring K B 1 𝑜 mPoly R LMod
114 lmodcmn 1 𝑜 mPoly R LMod 1 𝑜 mPoly R CMnd
115 113 114 syl R Ring K B 1 𝑜 mPoly R CMnd
116 98 a1i R Ring K B 0 V
117 1 ply1lmod R Ring P LMod
118 117 ad2antrr R Ring K B k 0 P LMod
119 eqid Base R = Base R
120 7 3 1 119 coe1f K B A : 0 Base R
121 120 adantl R Ring K B A : 0 Base R
122 121 ffvelcdmda R Ring K B k 0 A k Base R
123 1 ply1sca R Ring R = Scalar P
124 123 eqcomd R Ring Scalar P = R
125 124 ad2antrr R Ring K B k 0 Scalar P = R
126 125 fveq2d R Ring K B k 0 Base Scalar P = Base R
127 122 126 eleqtrrd R Ring K B k 0 A k Base Scalar P
128 75 ad2antrr R Ring K B k 0 M Mnd
129 simpr R Ring K B k 0 k 0
130 82 ad2antrr R Ring K B k 0 X B
131 58 6 128 129 130 mulgnn0cld R Ring K B k 0 k × ˙ X B
132 eqid Scalar P = Scalar P
133 eqid Base Scalar P = Base Scalar P
134 3 132 4 133 lmodvscl P LMod A k Base Scalar P k × ˙ X B A k · ˙ k × ˙ X B
135 118 127 131 134 syl3anc R Ring K B k 0 A k · ˙ k × ˙ X B
136 135 fmpttd R Ring K B k 0 A k · ˙ k × ˙ X : 0 B
137 1 2 3 4 5 6 7 ply1coefsupp R Ring K B finSupp 0 P k 0 A k · ˙ k × ˙ X
138 eqid a 0 1 𝑜 a = a 0 1 𝑜 a
139 43 98 29 138 mapsnf1o2 a 0 1 𝑜 a : 0 1 𝑜 1-1 onto 0
140 139 a1i R Ring K B a 0 1 𝑜 a : 0 1 𝑜 1-1 onto 0
141 15 111 115 116 136 137 140 gsumf1o R Ring K B 1 𝑜 mPoly R k 0 A k · ˙ k × ˙ X = 1 𝑜 mPoly R k 0 A k · ˙ k × ˙ X a 0 1 𝑜 a
142 eqidd R Ring K B a 0 1 𝑜 a = a 0 1 𝑜 a
143 eqidd R Ring K B k 0 A k · ˙ k × ˙ X = k 0 A k · ˙ k × ˙ X
144 fveq2 k = a A k = A a
145 oveq1 k = a k × ˙ X = a × ˙ X
146 144 145 oveq12d k = a A k · ˙ k × ˙ X = A a · ˙ a × ˙ X
147 81 142 143 146 fmptco R Ring K B k 0 A k · ˙ k × ˙ X a 0 1 𝑜 a = a 0 1 𝑜 A a · ˙ a × ˙ X
148 147 oveq2d R Ring K B 1 𝑜 mPoly R k 0 A k · ˙ k × ˙ X a 0 1 𝑜 a = 1 𝑜 mPoly R a 0 1 𝑜 A a · ˙ a × ˙ X
149 109 141 148 3eqtrrd R Ring K B 1 𝑜 mPoly R a 0 1 𝑜 A a · ˙ a × ˙ X = P k 0 A k · ˙ k × ˙ X
150 19 97 149 3eqtrd R Ring K B K = P k 0 A k · ˙ k × ˙ X