Step | Hyp | Ref
| Expression |
1 | | eldifi 4086 |
. . . . 5
β’ (πΉ β ((Polyββ)
β {0π}) β πΉ β
(Polyββ)) |
2 | | ax-resscn 11108 |
. . . . . . 7
β’ β
β β |
3 | | 1re 11155 |
. . . . . . 7
β’ 1 β
β |
4 | | plyid 25570 |
. . . . . . 7
β’ ((β
β β β§ 1 β β) β Xp β
(Polyββ)) |
5 | 2, 3, 4 | mp2an 690 |
. . . . . 6
β’
Xp β (Polyββ) |
6 | 5 | a1i 11 |
. . . . 5
β’ (πΉ β ((Polyββ)
β {0π}) β Xp β
(Polyββ)) |
7 | | simprl 769 |
. . . . . 6
β’ ((πΉ β ((Polyββ)
β {0π}) β§ (π₯ β β β§ π¦ β β)) β π₯ β β) |
8 | | simprr 771 |
. . . . . 6
β’ ((πΉ β ((Polyββ)
β {0π}) β§ (π₯ β β β§ π¦ β β)) β π¦ β β) |
9 | 7, 8 | readdcld 11184 |
. . . . 5
β’ ((πΉ β ((Polyββ)
β {0π}) β§ (π₯ β β β§ π¦ β β)) β (π₯ + π¦) β β) |
10 | 7, 8 | remulcld 11185 |
. . . . 5
β’ ((πΉ β ((Polyββ)
β {0π}) β§ (π₯ β β β§ π¦ β β)) β (π₯ Β· π¦) β β) |
11 | 1, 6, 9, 10 | plymul 25579 |
. . . 4
β’ (πΉ β ((Polyββ)
β {0π}) β (πΉ βf Β·
Xp) β (Polyββ)) |
12 | | 0re 11157 |
. . . 4
β’ 0 β
β |
13 | | eqid 2736 |
. . . . 5
β’
(coeffβ(πΉ
βf Β· Xp)) = (coeffβ(πΉ βf Β·
Xp)) |
14 | 13 | coef2 25592 |
. . . 4
β’ (((πΉ βf Β·
Xp) β (Polyββ) β§ 0 β β)
β (coeffβ(πΉ
βf Β·
Xp)):β0βΆβ) |
15 | 11, 12, 14 | sylancl 586 |
. . 3
β’ (πΉ β ((Polyββ)
β {0π}) β (coeffβ(πΉ βf Β·
Xp)):β0βΆβ) |
16 | 15 | feqmptd 6910 |
. 2
β’ (πΉ β ((Polyββ)
β {0π}) β (coeffβ(πΉ βf Β·
Xp)) = (π
β β0 β¦ ((coeffβ(πΉ βf Β·
Xp))βπ))) |
17 | | cnex 11132 |
. . . . . . . . 9
β’ β
β V |
18 | 17 | a1i 11 |
. . . . . . . 8
β’ (πΉ β ((Polyββ)
β {0π}) β β β V) |
19 | | plyf 25559 |
. . . . . . . . 9
β’ (πΉ β (Polyββ)
β πΉ:ββΆβ) |
20 | 1, 19 | syl 17 |
. . . . . . . 8
β’ (πΉ β ((Polyββ)
β {0π}) β πΉ:ββΆβ) |
21 | | plyf 25559 |
. . . . . . . . . 10
β’
(Xp β (Polyββ) β
Xp:ββΆβ) |
22 | 5, 21 | ax-mp 5 |
. . . . . . . . 9
β’
Xp:ββΆβ |
23 | 22 | a1i 11 |
. . . . . . . 8
β’ (πΉ β ((Polyββ)
β {0π}) β
Xp:ββΆβ) |
24 | | simprl 769 |
. . . . . . . . 9
β’ ((πΉ β ((Polyββ)
β {0π}) β§ (π₯ β β β§ π¦ β β)) β π₯ β β) |
25 | | simprr 771 |
. . . . . . . . 9
β’ ((πΉ β ((Polyββ)
β {0π}) β§ (π₯ β β β§ π¦ β β)) β π¦ β β) |
26 | 24, 25 | mulcomd 11176 |
. . . . . . . 8
β’ ((πΉ β ((Polyββ)
β {0π}) β§ (π₯ β β β§ π¦ β β)) β (π₯ Β· π¦) = (π¦ Β· π₯)) |
27 | 18, 20, 23, 26 | caofcom 7652 |
. . . . . . 7
β’ (πΉ β ((Polyββ)
β {0π}) β (πΉ βf Β·
Xp) = (Xp βf Β·
πΉ)) |
28 | 27 | fveq2d 6846 |
. . . . . 6
β’ (πΉ β ((Polyββ)
β {0π}) β (coeffβ(πΉ βf Β·
Xp)) = (coeffβ(Xp βf
Β· πΉ))) |
29 | 28 | fveq1d 6844 |
. . . . 5
β’ (πΉ β ((Polyββ)
β {0π}) β ((coeffβ(πΉ βf Β·
Xp))βπ) = ((coeffβ(Xp
βf Β· πΉ))βπ)) |
30 | 29 | adantr 481 |
. . . 4
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β
((coeffβ(πΉ
βf Β· Xp))βπ) = ((coeffβ(Xp
βf Β· πΉ))βπ)) |
31 | 5 | a1i 11 |
. . . . . 6
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β
Xp β (Polyββ)) |
32 | 1 | adantr 481 |
. . . . . 6
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β πΉ β
(Polyββ)) |
33 | | simpr 485 |
. . . . . 6
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β π β
β0) |
34 | | eqid 2736 |
. . . . . . 7
β’
(coeffβXp) =
(coeffβXp) |
35 | | eqid 2736 |
. . . . . . 7
β’
(coeffβπΉ) =
(coeffβπΉ) |
36 | 34, 35 | coemul 25613 |
. . . . . 6
β’
((Xp β (Polyββ) β§ πΉ β (Polyββ)
β§ π β
β0) β ((coeffβ(Xp
βf Β· πΉ))βπ) = Ξ£π β (0...π)(((coeffβXp)βπ) Β· ((coeffβπΉ)β(π β π)))) |
37 | 31, 32, 33, 36 | syl3anc 1371 |
. . . . 5
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β
((coeffβ(Xp βf Β· πΉ))βπ) = Ξ£π β (0...π)(((coeffβXp)βπ) Β· ((coeffβπΉ)β(π β π)))) |
38 | | elfznn0 13534 |
. . . . . . . . . 10
β’ (π β (0...π) β π β β0) |
39 | | coeidp 25624 |
. . . . . . . . . 10
β’ (π β β0
β ((coeffβXp)βπ) = if(π = 1, 1, 0)) |
40 | 38, 39 | syl 17 |
. . . . . . . . 9
β’ (π β (0...π) β
((coeffβXp)βπ) = if(π = 1, 1, 0)) |
41 | 40 | oveq1d 7372 |
. . . . . . . 8
β’ (π β (0...π) β
(((coeffβXp)βπ) Β· ((coeffβπΉ)β(π β π))) = (if(π = 1, 1, 0) Β· ((coeffβπΉ)β(π β π)))) |
42 | | ovif 7454 |
. . . . . . . 8
β’ (if(π = 1, 1, 0) Β·
((coeffβπΉ)β(π β π))) = if(π = 1, (1 Β· ((coeffβπΉ)β(π β π))), (0 Β· ((coeffβπΉ)β(π β π)))) |
43 | 41, 42 | eqtrdi 2792 |
. . . . . . 7
β’ (π β (0...π) β
(((coeffβXp)βπ) Β· ((coeffβπΉ)β(π β π))) = if(π = 1, (1 Β· ((coeffβπΉ)β(π β π))), (0 Β· ((coeffβπΉ)β(π β π))))) |
44 | 43 | adantl 482 |
. . . . . 6
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β§ π β (0...π)) β
(((coeffβXp)βπ) Β· ((coeffβπΉ)β(π β π))) = if(π = 1, (1 Β· ((coeffβπΉ)β(π β π))), (0 Β· ((coeffβπΉ)β(π β π))))) |
45 | 44 | sumeq2dv 15588 |
. . . . 5
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β
Ξ£π β (0...π)(((coeffβXp)βπ) Β· ((coeffβπΉ)β(π β π))) = Ξ£π β (0...π)if(π = 1, (1 Β· ((coeffβπΉ)β(π β π))), (0 Β· ((coeffβπΉ)β(π β π))))) |
46 | | velsn 4602 |
. . . . . . . . . 10
β’ (π β {1} β π = 1) |
47 | 46 | bicomi 223 |
. . . . . . . . 9
β’ (π = 1 β π β {1}) |
48 | 47 | a1i 11 |
. . . . . . . 8
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β§ π β (0...π)) β (π = 1 β π β {1})) |
49 | 35 | coef2 25592 |
. . . . . . . . . . . . 13
β’ ((πΉ β (Polyββ)
β§ 0 β β) β (coeffβπΉ):β0βΆβ) |
50 | 1, 12, 49 | sylancl 586 |
. . . . . . . . . . . 12
β’ (πΉ β ((Polyββ)
β {0π}) β (coeffβπΉ):β0βΆβ) |
51 | 50 | ad2antrr 724 |
. . . . . . . . . . 11
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β§ π β (0...π)) β (coeffβπΉ):β0βΆβ) |
52 | | fznn0sub 13473 |
. . . . . . . . . . . 12
β’ (π β (0...π) β (π β π) β
β0) |
53 | 52 | adantl 482 |
. . . . . . . . . . 11
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β§ π β (0...π)) β (π β π) β
β0) |
54 | 51, 53 | ffvelcdmd 7036 |
. . . . . . . . . 10
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β§ π β (0...π)) β ((coeffβπΉ)β(π β π)) β β) |
55 | 54 | recnd 11183 |
. . . . . . . . 9
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β§ π β (0...π)) β ((coeffβπΉ)β(π β π)) β β) |
56 | 55 | mulid2d 11173 |
. . . . . . . 8
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β§ π β (0...π)) β (1 Β· ((coeffβπΉ)β(π β π))) = ((coeffβπΉ)β(π β π))) |
57 | 55 | mul02d 11353 |
. . . . . . . 8
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β§ π β (0...π)) β (0 Β· ((coeffβπΉ)β(π β π))) = 0) |
58 | 48, 56, 57 | ifbieq12d 4514 |
. . . . . . 7
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β§ π β (0...π)) β if(π = 1, (1 Β· ((coeffβπΉ)β(π β π))), (0 Β· ((coeffβπΉ)β(π β π)))) = if(π β {1}, ((coeffβπΉ)β(π β π)), 0)) |
59 | 58 | sumeq2dv 15588 |
. . . . . 6
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β
Ξ£π β (0...π)if(π = 1, (1 Β· ((coeffβπΉ)β(π β π))), (0 Β· ((coeffβπΉ)β(π β π)))) = Ξ£π β (0...π)if(π β {1}, ((coeffβπΉ)β(π β π)), 0)) |
60 | | eqeq2 2748 |
. . . . . . 7
β’ (0 =
if(π = 0, 0,
((coeffβπΉ)β(π β 1))) β (Ξ£π β (0...π)if(π β {1}, ((coeffβπΉ)β(π β π)), 0) = 0 β Ξ£π β (0...π)if(π β {1}, ((coeffβπΉ)β(π β π)), 0) = if(π = 0, 0, ((coeffβπΉ)β(π β 1))))) |
61 | | eqeq2 2748 |
. . . . . . 7
β’
(((coeffβπΉ)β(π β 1)) = if(π = 0, 0, ((coeffβπΉ)β(π β 1))) β (Ξ£π β (0...π)if(π β {1}, ((coeffβπΉ)β(π β π)), 0) = ((coeffβπΉ)β(π β 1)) β Ξ£π β (0...π)if(π β {1}, ((coeffβπΉ)β(π β π)), 0) = if(π = 0, 0, ((coeffβπΉ)β(π β 1))))) |
62 | | oveq2 7365 |
. . . . . . . . . . 11
β’ (π = 0 β (0...π) = (0...0)) |
63 | | 0z 12510 |
. . . . . . . . . . . 12
β’ 0 β
β€ |
64 | | fzsn 13483 |
. . . . . . . . . . . 12
β’ (0 β
β€ β (0...0) = {0}) |
65 | 63, 64 | ax-mp 5 |
. . . . . . . . . . 11
β’ (0...0) =
{0} |
66 | 62, 65 | eqtrdi 2792 |
. . . . . . . . . 10
β’ (π = 0 β (0...π) = {0}) |
67 | | elsni 4603 |
. . . . . . . . . . . . 13
β’ (π β {0} β π = 0) |
68 | 67 | adantl 482 |
. . . . . . . . . . . 12
β’ ((π = 0 β§ π β {0}) β π = 0) |
69 | | ax-1ne0 11120 |
. . . . . . . . . . . . . 14
β’ 1 β
0 |
70 | 69 | nesymi 3001 |
. . . . . . . . . . . . 13
β’ Β¬ 0
= 1 |
71 | | eqeq1 2740 |
. . . . . . . . . . . . 13
β’ (π = 0 β (π = 1 β 0 = 1)) |
72 | 70, 71 | mtbiri 326 |
. . . . . . . . . . . 12
β’ (π = 0 β Β¬ π = 1) |
73 | 68, 72 | syl 17 |
. . . . . . . . . . 11
β’ ((π = 0 β§ π β {0}) β Β¬ π = 1) |
74 | 47 | notbii 319 |
. . . . . . . . . . . 12
β’ (Β¬
π = 1 β Β¬ π β {1}) |
75 | 74 | biimpi 215 |
. . . . . . . . . . 11
β’ (Β¬
π = 1 β Β¬ π β {1}) |
76 | | iffalse 4495 |
. . . . . . . . . . 11
β’ (Β¬
π β {1} β
if(π β {1},
((coeffβπΉ)β(π β π)), 0) = 0) |
77 | 73, 75, 76 | 3syl 18 |
. . . . . . . . . 10
β’ ((π = 0 β§ π β {0}) β if(π β {1}, ((coeffβπΉ)β(π β π)), 0) = 0) |
78 | 66, 77 | sumeq12rdv 15592 |
. . . . . . . . 9
β’ (π = 0 β Ξ£π β (0...π)if(π β {1}, ((coeffβπΉ)β(π β π)), 0) = Ξ£π β {0}0) |
79 | | snfi 8988 |
. . . . . . . . . . 11
β’ {0}
β Fin |
80 | 79 | olci 864 |
. . . . . . . . . 10
β’ ({0}
β (β€β₯β0) β¨ {0} β Fin) |
81 | | sumz 15607 |
. . . . . . . . . 10
β’ (({0}
β (β€β₯β0) β¨ {0} β Fin) β
Ξ£π β {0}0 =
0) |
82 | 80, 81 | ax-mp 5 |
. . . . . . . . 9
β’
Ξ£π β {0}0
= 0 |
83 | 78, 82 | eqtrdi 2792 |
. . . . . . . 8
β’ (π = 0 β Ξ£π β (0...π)if(π β {1}, ((coeffβπΉ)β(π β π)), 0) = 0) |
84 | 83 | adantl 482 |
. . . . . . 7
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β§ π = 0) β Ξ£π β (0...π)if(π β {1}, ((coeffβπΉ)β(π β π)), 0) = 0) |
85 | | simpll 765 |
. . . . . . . 8
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β§ Β¬
π = 0) β πΉ β ((Polyββ)
β {0π})) |
86 | 33 | adantr 481 |
. . . . . . . . 9
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β§ Β¬
π = 0) β π β
β0) |
87 | | simpr 485 |
. . . . . . . . . 10
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β§ Β¬
π = 0) β Β¬ π = 0) |
88 | 87 | neqned 2950 |
. . . . . . . . 9
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β§ Β¬
π = 0) β π β 0) |
89 | | elnnne0 12427 |
. . . . . . . . 9
β’ (π β β β (π β β0
β§ π β
0)) |
90 | 86, 88, 89 | sylanbrc 583 |
. . . . . . . 8
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β§ Β¬
π = 0) β π β
β) |
91 | | 1nn0 12429 |
. . . . . . . . . . . . 13
β’ 1 β
β0 |
92 | 91 | a1i 11 |
. . . . . . . . . . . 12
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β) β 1 β
β0) |
93 | | simpr 485 |
. . . . . . . . . . . . 13
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β) β π β β) |
94 | 93 | nnnn0d 12473 |
. . . . . . . . . . . 12
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β) β π β β0) |
95 | 93 | nnge1d 12201 |
. . . . . . . . . . . 12
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β) β 1 β€ π) |
96 | | elfz2nn0 13532 |
. . . . . . . . . . . 12
β’ (1 β
(0...π) β (1 β
β0 β§ π
β β0 β§ 1 β€ π)) |
97 | 92, 94, 95, 96 | syl3anbrc 1343 |
. . . . . . . . . . 11
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β) β 1 β (0...π)) |
98 | 97 | snssd 4769 |
. . . . . . . . . 10
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β) β {1} β
(0...π)) |
99 | 50 | ad2antrr 724 |
. . . . . . . . . . . . 13
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β) β§ π β {1}) β (coeffβπΉ):β0βΆβ) |
100 | | oveq2 7365 |
. . . . . . . . . . . . . . . 16
β’ (π = 1 β (π β π) = (π β 1)) |
101 | 46, 100 | sylbi 216 |
. . . . . . . . . . . . . . 15
β’ (π β {1} β (π β π) = (π β 1)) |
102 | 101 | adantl 482 |
. . . . . . . . . . . . . 14
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β) β§ π β {1}) β (π β π) = (π β 1)) |
103 | | nnm1nn0 12454 |
. . . . . . . . . . . . . . 15
β’ (π β β β (π β 1) β
β0) |
104 | 103 | ad2antlr 725 |
. . . . . . . . . . . . . 14
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β) β§ π β {1}) β (π β 1) β
β0) |
105 | 102, 104 | eqeltrd 2837 |
. . . . . . . . . . . . 13
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β) β§ π β {1}) β (π β π) β
β0) |
106 | 99, 105 | ffvelcdmd 7036 |
. . . . . . . . . . . 12
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β) β§ π β {1}) β ((coeffβπΉ)β(π β π)) β β) |
107 | 106 | recnd 11183 |
. . . . . . . . . . 11
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β) β§ π β {1}) β ((coeffβπΉ)β(π β π)) β β) |
108 | 107 | ralrimiva 3143 |
. . . . . . . . . 10
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β) β βπ β {1} ((coeffβπΉ)β(π β π)) β β) |
109 | | fzfi 13877 |
. . . . . . . . . . . 12
β’
(0...π) β
Fin |
110 | 109 | olci 864 |
. . . . . . . . . . 11
β’
((0...π) β
(β€β₯β0) β¨ (0...π) β Fin) |
111 | 110 | a1i 11 |
. . . . . . . . . 10
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β) β ((0...π) β
(β€β₯β0) β¨ (0...π) β Fin)) |
112 | | sumss2 15611 |
. . . . . . . . . 10
β’ ((({1}
β (0...π) β§
βπ β {1}
((coeffβπΉ)β(π β π)) β β) β§ ((0...π) β
(β€β₯β0) β¨ (0...π) β Fin)) β Ξ£π β {1} ((coeffβπΉ)β(π β π)) = Ξ£π β (0...π)if(π β {1}, ((coeffβπΉ)β(π β π)), 0)) |
113 | 98, 108, 111, 112 | syl21anc 836 |
. . . . . . . . 9
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β) β Ξ£π β {1} ((coeffβπΉ)β(π β π)) = Ξ£π β (0...π)if(π β {1}, ((coeffβπΉ)β(π β π)), 0)) |
114 | 50 | adantr 481 |
. . . . . . . . . . . 12
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β) β (coeffβπΉ):β0βΆβ) |
115 | 103 | adantl 482 |
. . . . . . . . . . . 12
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β) β (π β 1) β
β0) |
116 | 114, 115 | ffvelcdmd 7036 |
. . . . . . . . . . 11
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β) β ((coeffβπΉ)β(π β 1)) β β) |
117 | 116 | recnd 11183 |
. . . . . . . . . 10
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β) β ((coeffβπΉ)β(π β 1)) β β) |
118 | 100 | fveq2d 6846 |
. . . . . . . . . . 11
β’ (π = 1 β ((coeffβπΉ)β(π β π)) = ((coeffβπΉ)β(π β 1))) |
119 | 118 | sumsn 15631 |
. . . . . . . . . 10
β’ ((1
β β β§ ((coeffβπΉ)β(π β 1)) β β) β
Ξ£π β {1}
((coeffβπΉ)β(π β π)) = ((coeffβπΉ)β(π β 1))) |
120 | 3, 117, 119 | sylancr 587 |
. . . . . . . . 9
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β) β Ξ£π β {1} ((coeffβπΉ)β(π β π)) = ((coeffβπΉ)β(π β 1))) |
121 | 113, 120 | eqtr3d 2778 |
. . . . . . . 8
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β) β Ξ£π β (0...π)if(π β {1}, ((coeffβπΉ)β(π β π)), 0) = ((coeffβπΉ)β(π β 1))) |
122 | 85, 90, 121 | syl2anc 584 |
. . . . . . 7
β’ (((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β§ Β¬
π = 0) β Ξ£π β (0...π)if(π β {1}, ((coeffβπΉ)β(π β π)), 0) = ((coeffβπΉ)β(π β 1))) |
123 | 60, 61, 84, 122 | ifbothda 4524 |
. . . . . 6
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β
Ξ£π β (0...π)if(π β {1}, ((coeffβπΉ)β(π β π)), 0) = if(π = 0, 0, ((coeffβπΉ)β(π β 1)))) |
124 | 59, 123 | eqtrd 2776 |
. . . . 5
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β
Ξ£π β (0...π)if(π = 1, (1 Β· ((coeffβπΉ)β(π β π))), (0 Β· ((coeffβπΉ)β(π β π)))) = if(π = 0, 0, ((coeffβπΉ)β(π β 1)))) |
125 | 37, 45, 124 | 3eqtrd 2780 |
. . . 4
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β
((coeffβ(Xp βf Β· πΉ))βπ) = if(π = 0, 0, ((coeffβπΉ)β(π β 1)))) |
126 | 30, 125 | eqtrd 2776 |
. . 3
β’ ((πΉ β ((Polyββ)
β {0π}) β§ π β β0) β
((coeffβ(πΉ
βf Β· Xp))βπ) = if(π = 0, 0, ((coeffβπΉ)β(π β 1)))) |
127 | 126 | mpteq2dva 5205 |
. 2
β’ (πΉ β ((Polyββ)
β {0π}) β (π β β0 β¦
((coeffβ(πΉ
βf Β· Xp))βπ)) = (π β β0 β¦ if(π = 0, 0, ((coeffβπΉ)β(π β 1))))) |
128 | 16, 127 | eqtrd 2776 |
1
β’ (πΉ β ((Polyββ)
β {0π}) β (coeffβ(πΉ βf Β·
Xp)) = (π
β β0 β¦ if(π = 0, 0, ((coeffβπΉ)β(π β 1))))) |