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˙=0R
coe1tm.k K=BaseR
coe1tm.p P=Poly1R
coe1tm.x X=var1R
coe1tm.m ·˙=P
coe1tm.n N=mulGrpP
coe1tm.e ×˙=N
Assertion coe1tm RRingCKD0coe1C·˙D×˙X=x0ifx=DC0˙

Proof

Step Hyp Ref Expression
1 coe1tm.z 0˙=0R
2 coe1tm.k K=BaseR
3 coe1tm.p P=Poly1R
4 coe1tm.x X=var1R
5 coe1tm.m ·˙=P
6 coe1tm.n N=mulGrpP
7 coe1tm.e ×˙=N
8 eqid BaseP=BaseP
9 2 3 4 5 6 7 8 ply1tmcl RRingCKD0C·˙D×˙XBaseP
10 eqid coe1C·˙D×˙X=coe1C·˙D×˙X
11 eqid x01𝑜×x=x01𝑜×x
12 10 8 3 11 coe1fval2 C·˙D×˙XBasePcoe1C·˙D×˙X=C·˙D×˙Xx01𝑜×x
13 9 12 syl RRingCKD0coe1C·˙D×˙X=C·˙D×˙Xx01𝑜×x
14 fconst6g x01𝑜×x:1𝑜0
15 nn0ex 0V
16 1oex 1𝑜V
17 15 16 elmap 1𝑜×x01𝑜1𝑜×x:1𝑜0
18 14 17 sylibr x01𝑜×x01𝑜
19 18 adantl RRingCKD0x01𝑜×x01𝑜
20 eqidd RRingCKD0x01𝑜×x=x01𝑜×x
21 eqid mulGrp1𝑜mPolyR=mulGrp1𝑜mPolyR
22 6 8 mgpbas BaseP=BaseN
23 22 a1i RRingBaseP=BaseN
24 eqid mulGrp1𝑜mPolyR=mulGrp1𝑜mPolyR
25 3 8 ply1bas BaseP=Base1𝑜mPolyR
26 24 25 mgpbas BaseP=BasemulGrp1𝑜mPolyR
27 26 a1i RRingBaseP=BasemulGrp1𝑜mPolyR
28 ssv BasePV
29 28 a1i RRingBasePV
30 ovexd RRingxVyVx+NyV
31 eqid P=P
32 6 31 mgpplusg P=+N
33 eqid 1𝑜mPolyR=1𝑜mPolyR
34 3 33 31 ply1mulr P=1𝑜mPolyR
35 24 34 mgpplusg P=+mulGrp1𝑜mPolyR
36 32 35 eqtr3i +N=+mulGrp1𝑜mPolyR
37 36 a1i RRing+N=+mulGrp1𝑜mPolyR
38 37 oveqdr RRingxVyVx+Ny=x+mulGrp1𝑜mPolyRy
39 7 21 23 27 29 30 38 mulgpropd RRing×˙=mulGrp1𝑜mPolyR
40 39 3ad2ant1 RRingCKD0×˙=mulGrp1𝑜mPolyR
41 eqidd RRingCKD0D=D
42 4 vr1val X=1𝑜mVarR
43 42 a1i RRingCKD0X=1𝑜mVarR
44 40 41 43 oveq123d RRingCKD0D×˙X=DmulGrp1𝑜mPolyR1𝑜mVarR
45 44 oveq2d RRingCKD0C·˙D×˙X=C·˙DmulGrp1𝑜mPolyR1𝑜mVarR
46 psr1baslem 01𝑜=a01𝑜|a-1Fin
47 eqid 1R=1R
48 1on 1𝑜On
49 48 a1i RRingCKD01𝑜On
50 eqid 1𝑜mVarR=1𝑜mVarR
51 simp1 RRingCKD0RRing
52 0lt1o 1𝑜
53 52 a1i RRingCKD01𝑜
54 simp3 RRingCKD0D0
55 33 46 1 47 49 24 21 50 51 53 54 mplcoe3 RRingCKD0y01𝑜ify=b1𝑜ifb=D01R0˙=DmulGrp1𝑜mPolyR1𝑜mVarR
56 55 oveq2d RRingCKD0C·˙y01𝑜ify=b1𝑜ifb=D01R0˙=C·˙DmulGrp1𝑜mPolyR1𝑜mVarR
57 3 33 5 ply1vsca ·˙=1𝑜mPolyR
58 elsni bb=
59 df1o2 1𝑜=
60 58 59 eleq2s b1𝑜b=
61 60 iftrued b1𝑜ifb=D0=D
62 61 adantl RRingCKD0b1𝑜ifb=D0=D
63 62 mpteq2dva RRingCKD0b1𝑜ifb=D0=b1𝑜D
64 fconstmpt 1𝑜×D=b1𝑜D
65 63 64 eqtr4di RRingCKD0b1𝑜ifb=D0=1𝑜×D
66 fconst6g D01𝑜×D:1𝑜0
67 15 16 elmap 1𝑜×D01𝑜1𝑜×D:1𝑜0
68 66 67 sylibr D01𝑜×D01𝑜
69 68 3ad2ant3 RRingCKD01𝑜×D01𝑜
70 65 69 eqeltrd RRingCKD0b1𝑜ifb=D001𝑜
71 simp2 RRingCKD0CK
72 33 57 46 47 1 2 49 51 70 71 mplmon2 RRingCKD0C·˙y01𝑜ify=b1𝑜ifb=D01R0˙=y01𝑜ify=b1𝑜ifb=D0C0˙
73 45 56 72 3eqtr2d RRingCKD0C·˙D×˙X=y01𝑜ify=b1𝑜ifb=D0C0˙
74 eqeq1 y=1𝑜×xy=b1𝑜ifb=D01𝑜×x=b1𝑜ifb=D0
75 74 ifbid y=1𝑜×xify=b1𝑜ifb=D0C0˙=if1𝑜×x=b1𝑜ifb=D0C0˙
76 19 20 73 75 fmptco RRingCKD0C·˙D×˙Xx01𝑜×x=x0if1𝑜×x=b1𝑜ifb=D0C0˙
77 65 adantr RRingCKD0x0b1𝑜ifb=D0=1𝑜×D
78 77 eqeq2d RRingCKD0x01𝑜×x=b1𝑜ifb=D01𝑜×x=1𝑜×D
79 fveq1 1𝑜×x=1𝑜×D1𝑜×x=1𝑜×D
80 vex xV
81 80 fvconst2 1𝑜1𝑜×x=x
82 52 81 mp1i RRingCKD0x01𝑜×x=x
83 simpl3 RRingCKD0x0D0
84 fvconst2g D01𝑜1𝑜×D=D
85 83 52 84 sylancl RRingCKD0x01𝑜×D=D
86 82 85 eqeq12d RRingCKD0x01𝑜×x=1𝑜×Dx=D
87 79 86 imbitrid RRingCKD0x01𝑜×x=1𝑜×Dx=D
88 sneq x=Dx=D
89 88 xpeq2d x=D1𝑜×x=1𝑜×D
90 87 89 impbid1 RRingCKD0x01𝑜×x=1𝑜×Dx=D
91 78 90 bitrd RRingCKD0x01𝑜×x=b1𝑜ifb=D0x=D
92 91 ifbid RRingCKD0x0if1𝑜×x=b1𝑜ifb=D0C0˙=ifx=DC0˙
93 92 mpteq2dva RRingCKD0x0if1𝑜×x=b1𝑜ifb=D0C0˙=x0ifx=DC0˙
94 13 76 93 3eqtrd RRingCKD0coe1C·˙D×˙X=x0ifx=DC0˙