Metamath Proof Explorer


Theorem plyrem

Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout ). If a polynomial F is divided by the linear factor x - A , the remainder is equal to F ( A ) , the evaluation of the polynomial at A (interpreted as a constant polynomial). This is part of Metamath 100 proof #89. (Contributed by Mario Carneiro, 26-Jul-2014)

Ref Expression
Hypotheses plyrem.1 G = X p f × A
plyrem.2 R = F f G × f F quot G
Assertion plyrem F Poly S A R = × F A

Proof

Step Hyp Ref Expression
1 plyrem.1 G = X p f × A
2 plyrem.2 R = F f G × f F quot G
3 plyssc Poly S Poly
4 simpl F Poly S A F Poly S
5 3 4 sselid F Poly S A F Poly
6 1 plyremlem A G Poly deg G = 1 G -1 0 = A
7 6 adantl F Poly S A G Poly deg G = 1 G -1 0 = A
8 7 simp1d F Poly S A G Poly
9 7 simp2d F Poly S A deg G = 1
10 ax-1ne0 1 0
11 10 a1i F Poly S A 1 0
12 9 11 eqnetrd F Poly S A deg G 0
13 fveq2 G = 0 𝑝 deg G = deg 0 𝑝
14 dgr0 deg 0 𝑝 = 0
15 13 14 eqtrdi G = 0 𝑝 deg G = 0
16 15 necon3i deg G 0 G 0 𝑝
17 12 16 syl F Poly S A G 0 𝑝
18 2 quotdgr F Poly G Poly G 0 𝑝 R = 0 𝑝 deg R < deg G
19 5 8 17 18 syl3anc F Poly S A R = 0 𝑝 deg R < deg G
20 0lt1 0 < 1
21 20 9 breqtrrid F Poly S A 0 < deg G
22 fveq2 R = 0 𝑝 deg R = deg 0 𝑝
23 22 14 eqtrdi R = 0 𝑝 deg R = 0
24 23 breq1d R = 0 𝑝 deg R < deg G 0 < deg G
25 21 24 syl5ibrcom F Poly S A R = 0 𝑝 deg R < deg G
26 pm2.62 R = 0 𝑝 deg R < deg G R = 0 𝑝 deg R < deg G deg R < deg G
27 19 25 26 sylc F Poly S A deg R < deg G
28 27 9 breqtrd F Poly S A deg R < 1
29 quotcl2 F Poly G Poly G 0 𝑝 F quot G Poly
30 5 8 17 29 syl3anc F Poly S A F quot G Poly
31 plymulcl G Poly F quot G Poly G × f F quot G Poly
32 8 30 31 syl2anc F Poly S A G × f F quot G Poly
33 plysubcl F Poly G × f F quot G Poly F f G × f F quot G Poly
34 5 32 33 syl2anc F Poly S A F f G × f F quot G Poly
35 2 34 eqeltrid F Poly S A R Poly
36 dgrcl R Poly deg R 0
37 35 36 syl F Poly S A deg R 0
38 nn0lt10b deg R 0 deg R < 1 deg R = 0
39 37 38 syl F Poly S A deg R < 1 deg R = 0
40 28 39 mpbid F Poly S A deg R = 0
41 0dgrb R Poly deg R = 0 R = × R 0
42 35 41 syl F Poly S A deg R = 0 R = × R 0
43 40 42 mpbid F Poly S A R = × R 0
44 43 fveq1d F Poly S A R A = × R 0 A
45 2 fveq1i R A = F f G × f F quot G A
46 plyf F Poly S F :
47 46 adantr F Poly S A F :
48 47 ffnd F Poly S A F Fn
49 plyf G Poly G :
50 8 49 syl F Poly S A G :
51 50 ffnd F Poly S A G Fn
52 plyf F quot G Poly F quot G :
53 30 52 syl F Poly S A F quot G :
54 53 ffnd F Poly S A F quot G Fn
55 cnex V
56 55 a1i F Poly S A V
57 inidm =
58 51 54 56 56 57 offn F Poly S A G × f F quot G Fn
59 eqidd F Poly S A A F A = F A
60 7 simp3d F Poly S A G -1 0 = A
61 ssun1 G -1 0 G -1 0 F quot G -1 0
62 60 61 eqsstrrdi F Poly S A A G -1 0 F quot G -1 0
63 snssg A A G -1 0 F quot G -1 0 A G -1 0 F quot G -1 0
64 63 adantl F Poly S A A G -1 0 F quot G -1 0 A G -1 0 F quot G -1 0
65 62 64 mpbird F Poly S A A G -1 0 F quot G -1 0
66 ofmulrt V G : F quot G : G × f F quot G -1 0 = G -1 0 F quot G -1 0
67 56 50 53 66 syl3anc F Poly S A G × f F quot G -1 0 = G -1 0 F quot G -1 0
68 65 67 eleqtrrd F Poly S A A G × f F quot G -1 0
69 fniniseg G × f F quot G Fn A G × f F quot G -1 0 A G × f F quot G A = 0
70 58 69 syl F Poly S A A G × f F quot G -1 0 A G × f F quot G A = 0
71 68 70 mpbid F Poly S A A G × f F quot G A = 0
72 71 simprd F Poly S A G × f F quot G A = 0
73 72 adantr F Poly S A A G × f F quot G A = 0
74 48 58 56 56 57 59 73 ofval F Poly S A A F f G × f F quot G A = F A 0
75 74 anabss3 F Poly S A F f G × f F quot G A = F A 0
76 45 75 eqtrid F Poly S A R A = F A 0
77 46 ffvelrnda F Poly S A F A
78 77 subid1d F Poly S A F A 0 = F A
79 76 78 eqtrd F Poly S A R A = F A
80 fvex R 0 V
81 80 fvconst2 A × R 0 A = R 0
82 81 adantl F Poly S A × R 0 A = R 0
83 44 79 82 3eqtr3d F Poly S A F A = R 0
84 83 sneqd F Poly S A F A = R 0
85 84 xpeq2d F Poly S A × F A = × R 0
86 43 85 eqtr4d F Poly S A R = × F A