Step | Hyp | Ref
| Expression |
1 | | coe1tmmul.r |
. . 3
β’ (π β π
β Ring) |
2 | | coe1tmmul.c |
. . . 4
β’ (π β πΆ β πΎ) |
3 | | coe1tmmul.d |
. . . 4
β’ (π β π· β
β0) |
4 | | coe1tm.k |
. . . . 5
β’ πΎ = (Baseβπ
) |
5 | | coe1tm.p |
. . . . 5
β’ π = (Poly1βπ
) |
6 | | coe1tm.x |
. . . . 5
β’ π = (var1βπ
) |
7 | | coe1tm.m |
. . . . 5
β’ Β· = (
Β·π βπ) |
8 | | coe1tm.n |
. . . . 5
β’ π = (mulGrpβπ) |
9 | | coe1tm.e |
. . . . 5
β’ β =
(.gβπ) |
10 | | coe1tmmul.b |
. . . . 5
β’ π΅ = (Baseβπ) |
11 | 4, 5, 6, 7, 8, 9, 10 | ply1tmcl 21643 |
. . . 4
β’ ((π
β Ring β§ πΆ β πΎ β§ π· β β0) β (πΆ Β· (π· β π)) β π΅) |
12 | 1, 2, 3, 11 | syl3anc 1371 |
. . 3
β’ (π β (πΆ Β· (π· β π)) β π΅) |
13 | | coe1tmmul.a |
. . 3
β’ (π β π΄ β π΅) |
14 | | coe1tmmul.t |
. . . 4
β’ β =
(.rβπ) |
15 | | coe1tmmul.u |
. . . 4
β’ Γ =
(.rβπ
) |
16 | 5, 14, 15, 10 | coe1mul 21641 |
. . 3
β’ ((π
β Ring β§ (πΆ Β· (π· β π)) β π΅ β§ π΄ β π΅) β (coe1β((πΆ Β· (π· β π)) β π΄)) = (π₯ β β0 β¦ (π
Ξ£g
(π¦ β (0...π₯) β¦
(((coe1β(πΆ
Β·
(π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))))))) |
17 | 1, 12, 13, 16 | syl3anc 1371 |
. 2
β’ (π β
(coe1β((πΆ
Β·
(π· β π)) β π΄)) = (π₯ β β0 β¦ (π
Ξ£g
(π¦ β (0...π₯) β¦
(((coe1β(πΆ
Β·
(π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))))))) |
18 | | eqeq2 2748 |
. . . 4
β’ ((πΆ Γ
((coe1βπ΄)β(π₯ β π·))) = if(π· β€ π₯, (πΆ Γ
((coe1βπ΄)β(π₯ β π·))), 0 ) β ((π
Ξ£g
(π¦ β (0...π₯) β¦
(((coe1β(πΆ
Β·
(π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))))) = (πΆ Γ
((coe1βπ΄)β(π₯ β π·))) β (π
Ξ£g (π¦ β (0...π₯) β¦ (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))))) = if(π· β€ π₯, (πΆ Γ
((coe1βπ΄)β(π₯ β π·))), 0 ))) |
19 | | eqeq2 2748 |
. . . 4
β’ ( 0 = if(π· β€ π₯, (πΆ Γ
((coe1βπ΄)β(π₯ β π·))), 0 ) β ((π
Ξ£g
(π¦ β (0...π₯) β¦
(((coe1β(πΆ
Β·
(π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))))) = 0 β (π
Ξ£g (π¦ β (0...π₯) β¦ (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))))) = if(π· β€ π₯, (πΆ Γ
((coe1βπ΄)β(π₯ β π·))), 0 ))) |
20 | | coe1tm.z |
. . . . . 6
β’ 0 =
(0gβπ
) |
21 | 1 | ad2antrr 724 |
. . . . . . 7
β’ (((π β§ π₯ β β0) β§ π· β€ π₯) β π
β Ring) |
22 | | ringmnd 19974 |
. . . . . . 7
β’ (π
β Ring β π
β Mnd) |
23 | 21, 22 | syl 17 |
. . . . . 6
β’ (((π β§ π₯ β β0) β§ π· β€ π₯) β π
β Mnd) |
24 | | ovexd 7392 |
. . . . . 6
β’ (((π β§ π₯ β β0) β§ π· β€ π₯) β (0...π₯) β V) |
25 | 3 | ad2antrr 724 |
. . . . . . 7
β’ (((π β§ π₯ β β0) β§ π· β€ π₯) β π· β
β0) |
26 | | simpr 485 |
. . . . . . 7
β’ (((π β§ π₯ β β0) β§ π· β€ π₯) β π· β€ π₯) |
27 | | fznn0 13533 |
. . . . . . . 8
β’ (π₯ β β0
β (π· β (0...π₯) β (π· β β0 β§ π· β€ π₯))) |
28 | 27 | ad2antlr 725 |
. . . . . . 7
β’ (((π β§ π₯ β β0) β§ π· β€ π₯) β (π· β (0...π₯) β (π· β β0 β§ π· β€ π₯))) |
29 | 25, 26, 28 | mpbir2and 711 |
. . . . . 6
β’ (((π β§ π₯ β β0) β§ π· β€ π₯) β π· β (0...π₯)) |
30 | 1 | ad2antrr 724 |
. . . . . . . . 9
β’ (((π β§ π₯ β β0) β§ π¦ β (0...π₯)) β π
β Ring) |
31 | | eqid 2736 |
. . . . . . . . . . . . 13
β’
(coe1β(πΆ Β· (π· β π))) = (coe1β(πΆ Β· (π· β π))) |
32 | 31, 10, 5, 4 | coe1f 21582 |
. . . . . . . . . . . 12
β’ ((πΆ Β· (π· β π)) β π΅ β (coe1β(πΆ Β· (π· β π))):β0βΆπΎ) |
33 | 12, 32 | syl 17 |
. . . . . . . . . . 11
β’ (π β
(coe1β(πΆ
Β·
(π· β π))):β0βΆπΎ) |
34 | 33 | adantr 481 |
. . . . . . . . . 10
β’ ((π β§ π₯ β β0) β
(coe1β(πΆ
Β·
(π· β π))):β0βΆπΎ) |
35 | | elfznn0 13534 |
. . . . . . . . . 10
β’ (π¦ β (0...π₯) β π¦ β β0) |
36 | | ffvelcdm 7032 |
. . . . . . . . . 10
β’
(((coe1β(πΆ Β· (π· β π))):β0βΆπΎ β§ π¦ β β0) β
((coe1β(πΆ
Β·
(π· β π)))βπ¦) β πΎ) |
37 | 34, 35, 36 | syl2an 596 |
. . . . . . . . 9
β’ (((π β§ π₯ β β0) β§ π¦ β (0...π₯)) β ((coe1β(πΆ Β· (π· β π)))βπ¦) β πΎ) |
38 | | eqid 2736 |
. . . . . . . . . . . . 13
β’
(coe1βπ΄) = (coe1βπ΄) |
39 | 38, 10, 5, 4 | coe1f 21582 |
. . . . . . . . . . . 12
β’ (π΄ β π΅ β (coe1βπ΄):β0βΆπΎ) |
40 | 13, 39 | syl 17 |
. . . . . . . . . . 11
β’ (π β
(coe1βπ΄):β0βΆπΎ) |
41 | 40 | adantr 481 |
. . . . . . . . . 10
β’ ((π β§ π₯ β β0) β
(coe1βπ΄):β0βΆπΎ) |
42 | | fznn0sub 13473 |
. . . . . . . . . 10
β’ (π¦ β (0...π₯) β (π₯ β π¦) β
β0) |
43 | | ffvelcdm 7032 |
. . . . . . . . . 10
β’
(((coe1βπ΄):β0βΆπΎ β§ (π₯ β π¦) β β0) β
((coe1βπ΄)β(π₯ β π¦)) β πΎ) |
44 | 41, 42, 43 | syl2an 596 |
. . . . . . . . 9
β’ (((π β§ π₯ β β0) β§ π¦ β (0...π₯)) β ((coe1βπ΄)β(π₯ β π¦)) β πΎ) |
45 | 4, 15 | ringcl 19981 |
. . . . . . . . 9
β’ ((π
β Ring β§
((coe1β(πΆ
Β·
(π· β π)))βπ¦) β πΎ β§ ((coe1βπ΄)β(π₯ β π¦)) β πΎ) β (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))) β πΎ) |
46 | 30, 37, 44, 45 | syl3anc 1371 |
. . . . . . . 8
β’ (((π β§ π₯ β β0) β§ π¦ β (0...π₯)) β (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))) β πΎ) |
47 | 46 | fmpttd 7063 |
. . . . . . 7
β’ ((π β§ π₯ β β0) β (π¦ β (0...π₯) β¦ (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦)))):(0...π₯)βΆπΎ) |
48 | 47 | adantr 481 |
. . . . . 6
β’ (((π β§ π₯ β β0) β§ π· β€ π₯) β (π¦ β (0...π₯) β¦ (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦)))):(0...π₯)βΆπΎ) |
49 | 1 | ad3antrrr 728 |
. . . . . . . . . 10
β’ ((((π β§ π₯ β β0) β§ π· β€ π₯) β§ π¦ β ((0...π₯) β {π·})) β π
β Ring) |
50 | 2 | ad3antrrr 728 |
. . . . . . . . . 10
β’ ((((π β§ π₯ β β0) β§ π· β€ π₯) β§ π¦ β ((0...π₯) β {π·})) β πΆ β πΎ) |
51 | 3 | ad3antrrr 728 |
. . . . . . . . . 10
β’ ((((π β§ π₯ β β0) β§ π· β€ π₯) β§ π¦ β ((0...π₯) β {π·})) β π· β
β0) |
52 | | eldifi 4086 |
. . . . . . . . . . . 12
β’ (π¦ β ((0...π₯) β {π·}) β π¦ β (0...π₯)) |
53 | 52, 35 | syl 17 |
. . . . . . . . . . 11
β’ (π¦ β ((0...π₯) β {π·}) β π¦ β β0) |
54 | 53 | adantl 482 |
. . . . . . . . . 10
β’ ((((π β§ π₯ β β0) β§ π· β€ π₯) β§ π¦ β ((0...π₯) β {π·})) β π¦ β β0) |
55 | | eldifsni 4750 |
. . . . . . . . . . . 12
β’ (π¦ β ((0...π₯) β {π·}) β π¦ β π·) |
56 | 55 | necomd 2999 |
. . . . . . . . . . 11
β’ (π¦ β ((0...π₯) β {π·}) β π· β π¦) |
57 | 56 | adantl 482 |
. . . . . . . . . 10
β’ ((((π β§ π₯ β β0) β§ π· β€ π₯) β§ π¦ β ((0...π₯) β {π·})) β π· β π¦) |
58 | 20, 4, 5, 6, 7, 8, 9, 49, 50, 51, 54, 57 | coe1tmfv2 21646 |
. . . . . . . . 9
β’ ((((π β§ π₯ β β0) β§ π· β€ π₯) β§ π¦ β ((0...π₯) β {π·})) β ((coe1β(πΆ Β· (π· β π)))βπ¦) = 0 ) |
59 | 58 | oveq1d 7372 |
. . . . . . . 8
β’ ((((π β§ π₯ β β0) β§ π· β€ π₯) β§ π¦ β ((0...π₯) β {π·})) β (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))) = ( 0 Γ
((coe1βπ΄)β(π₯ β π¦)))) |
60 | 4, 15, 20 | ringlz 20011 |
. . . . . . . . . . 11
β’ ((π
β Ring β§
((coe1βπ΄)β(π₯ β π¦)) β πΎ) β ( 0 Γ
((coe1βπ΄)β(π₯ β π¦))) = 0 ) |
61 | 30, 44, 60 | syl2anc 584 |
. . . . . . . . . 10
β’ (((π β§ π₯ β β0) β§ π¦ β (0...π₯)) β ( 0 Γ
((coe1βπ΄)β(π₯ β π¦))) = 0 ) |
62 | 52, 61 | sylan2 593 |
. . . . . . . . 9
β’ (((π β§ π₯ β β0) β§ π¦ β ((0...π₯) β {π·})) β ( 0 Γ
((coe1βπ΄)β(π₯ β π¦))) = 0 ) |
63 | 62 | adantlr 713 |
. . . . . . . 8
β’ ((((π β§ π₯ β β0) β§ π· β€ π₯) β§ π¦ β ((0...π₯) β {π·})) β ( 0 Γ
((coe1βπ΄)β(π₯ β π¦))) = 0 ) |
64 | 59, 63 | eqtrd 2776 |
. . . . . . 7
β’ ((((π β§ π₯ β β0) β§ π· β€ π₯) β§ π¦ β ((0...π₯) β {π·})) β (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))) = 0 ) |
65 | 64, 24 | suppss2 8131 |
. . . . . 6
β’ (((π β§ π₯ β β0) β§ π· β€ π₯) β ((π¦ β (0...π₯) β¦ (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦)))) supp 0 ) β {π·}) |
66 | 4, 20, 23, 24, 29, 48, 65 | gsumpt 19739 |
. . . . 5
β’ (((π β§ π₯ β β0) β§ π· β€ π₯) β (π
Ξ£g (π¦ β (0...π₯) β¦ (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))))) = ((π¦ β (0...π₯) β¦ (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))))βπ·)) |
67 | | fveq2 6842 |
. . . . . . . . 9
β’ (π¦ = π· β ((coe1β(πΆ Β· (π· β π)))βπ¦) = ((coe1β(πΆ Β· (π· β π)))βπ·)) |
68 | | oveq2 7365 |
. . . . . . . . . 10
β’ (π¦ = π· β (π₯ β π¦) = (π₯ β π·)) |
69 | 68 | fveq2d 6846 |
. . . . . . . . 9
β’ (π¦ = π· β ((coe1βπ΄)β(π₯ β π¦)) = ((coe1βπ΄)β(π₯ β π·))) |
70 | 67, 69 | oveq12d 7375 |
. . . . . . . 8
β’ (π¦ = π· β (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))) = (((coe1β(πΆ Β· (π· β π)))βπ·) Γ
((coe1βπ΄)β(π₯ β π·)))) |
71 | | eqid 2736 |
. . . . . . . 8
β’ (π¦ β (0...π₯) β¦ (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦)))) = (π¦ β (0...π₯) β¦ (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦)))) |
72 | | ovex 7390 |
. . . . . . . 8
β’
(((coe1β(πΆ Β· (π· β π)))βπ·) Γ
((coe1βπ΄)β(π₯ β π·))) β V |
73 | 70, 71, 72 | fvmpt 6948 |
. . . . . . 7
β’ (π· β (0...π₯) β ((π¦ β (0...π₯) β¦ (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))))βπ·) = (((coe1β(πΆ Β· (π· β π)))βπ·) Γ
((coe1βπ΄)β(π₯ β π·)))) |
74 | 29, 73 | syl 17 |
. . . . . 6
β’ (((π β§ π₯ β β0) β§ π· β€ π₯) β ((π¦ β (0...π₯) β¦ (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))))βπ·) = (((coe1β(πΆ Β· (π· β π)))βπ·) Γ
((coe1βπ΄)β(π₯ β π·)))) |
75 | 20, 4, 5, 6, 7, 8, 9 | coe1tmfv1 21645 |
. . . . . . . . 9
β’ ((π
β Ring β§ πΆ β πΎ β§ π· β β0) β
((coe1β(πΆ
Β·
(π· β π)))βπ·) = πΆ) |
76 | 1, 2, 3, 75 | syl3anc 1371 |
. . . . . . . 8
β’ (π β
((coe1β(πΆ
Β·
(π· β π)))βπ·) = πΆ) |
77 | 76 | ad2antrr 724 |
. . . . . . 7
β’ (((π β§ π₯ β β0) β§ π· β€ π₯) β ((coe1β(πΆ Β· (π· β π)))βπ·) = πΆ) |
78 | 77 | oveq1d 7372 |
. . . . . 6
β’ (((π β§ π₯ β β0) β§ π· β€ π₯) β (((coe1β(πΆ Β· (π· β π)))βπ·) Γ
((coe1βπ΄)β(π₯ β π·))) = (πΆ Γ
((coe1βπ΄)β(π₯ β π·)))) |
79 | 74, 78 | eqtrd 2776 |
. . . . 5
β’ (((π β§ π₯ β β0) β§ π· β€ π₯) β ((π¦ β (0...π₯) β¦ (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))))βπ·) = (πΆ Γ
((coe1βπ΄)β(π₯ β π·)))) |
80 | 66, 79 | eqtrd 2776 |
. . . 4
β’ (((π β§ π₯ β β0) β§ π· β€ π₯) β (π
Ξ£g (π¦ β (0...π₯) β¦ (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))))) = (πΆ Γ
((coe1βπ΄)β(π₯ β π·)))) |
81 | 1 | ad3antrrr 728 |
. . . . . . . . . 10
β’ ((((π β§ π₯ β β0) β§ Β¬
π· β€ π₯) β§ π¦ β (0...π₯)) β π
β Ring) |
82 | 2 | ad3antrrr 728 |
. . . . . . . . . 10
β’ ((((π β§ π₯ β β0) β§ Β¬
π· β€ π₯) β§ π¦ β (0...π₯)) β πΆ β πΎ) |
83 | 3 | ad3antrrr 728 |
. . . . . . . . . 10
β’ ((((π β§ π₯ β β0) β§ Β¬
π· β€ π₯) β§ π¦ β (0...π₯)) β π· β
β0) |
84 | 35 | adantl 482 |
. . . . . . . . . 10
β’ ((((π β§ π₯ β β0) β§ Β¬
π· β€ π₯) β§ π¦ β (0...π₯)) β π¦ β β0) |
85 | | elfzle2 13445 |
. . . . . . . . . . . . . . 15
β’ (π¦ β (0...π₯) β π¦ β€ π₯) |
86 | 85 | adantl 482 |
. . . . . . . . . . . . . 14
β’ (((π β§ π₯ β β0) β§ π¦ β (0...π₯)) β π¦ β€ π₯) |
87 | | breq1 5108 |
. . . . . . . . . . . . . 14
β’ (π· = π¦ β (π· β€ π₯ β π¦ β€ π₯)) |
88 | 86, 87 | syl5ibrcom 246 |
. . . . . . . . . . . . 13
β’ (((π β§ π₯ β β0) β§ π¦ β (0...π₯)) β (π· = π¦ β π· β€ π₯)) |
89 | 88 | necon3bd 2957 |
. . . . . . . . . . . 12
β’ (((π β§ π₯ β β0) β§ π¦ β (0...π₯)) β (Β¬ π· β€ π₯ β π· β π¦)) |
90 | 89 | imp 407 |
. . . . . . . . . . 11
β’ ((((π β§ π₯ β β0) β§ π¦ β (0...π₯)) β§ Β¬ π· β€ π₯) β π· β π¦) |
91 | 90 | an32s 650 |
. . . . . . . . . 10
β’ ((((π β§ π₯ β β0) β§ Β¬
π· β€ π₯) β§ π¦ β (0...π₯)) β π· β π¦) |
92 | 20, 4, 5, 6, 7, 8, 9, 81, 82, 83, 84, 91 | coe1tmfv2 21646 |
. . . . . . . . 9
β’ ((((π β§ π₯ β β0) β§ Β¬
π· β€ π₯) β§ π¦ β (0...π₯)) β ((coe1β(πΆ Β· (π· β π)))βπ¦) = 0 ) |
93 | 92 | oveq1d 7372 |
. . . . . . . 8
β’ ((((π β§ π₯ β β0) β§ Β¬
π· β€ π₯) β§ π¦ β (0...π₯)) β (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))) = ( 0 Γ
((coe1βπ΄)β(π₯ β π¦)))) |
94 | 61 | adantlr 713 |
. . . . . . . 8
β’ ((((π β§ π₯ β β0) β§ Β¬
π· β€ π₯) β§ π¦ β (0...π₯)) β ( 0 Γ
((coe1βπ΄)β(π₯ β π¦))) = 0 ) |
95 | 93, 94 | eqtrd 2776 |
. . . . . . 7
β’ ((((π β§ π₯ β β0) β§ Β¬
π· β€ π₯) β§ π¦ β (0...π₯)) β (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))) = 0 ) |
96 | 95 | mpteq2dva 5205 |
. . . . . 6
β’ (((π β§ π₯ β β0) β§ Β¬
π· β€ π₯) β (π¦ β (0...π₯) β¦ (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦)))) = (π¦ β (0...π₯) β¦ 0 )) |
97 | 96 | oveq2d 7373 |
. . . . 5
β’ (((π β§ π₯ β β0) β§ Β¬
π· β€ π₯) β (π
Ξ£g (π¦ β (0...π₯) β¦ (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))))) = (π
Ξ£g (π¦ β (0...π₯) β¦ 0 ))) |
98 | 1, 22 | syl 17 |
. . . . . . 7
β’ (π β π
β Mnd) |
99 | 98 | ad2antrr 724 |
. . . . . 6
β’ (((π β§ π₯ β β0) β§ Β¬
π· β€ π₯) β π
β Mnd) |
100 | | ovexd 7392 |
. . . . . 6
β’ (((π β§ π₯ β β0) β§ Β¬
π· β€ π₯) β (0...π₯) β V) |
101 | 20 | gsumz 18646 |
. . . . . 6
β’ ((π
β Mnd β§ (0...π₯) β V) β (π
Ξ£g
(π¦ β (0...π₯) β¦ 0 )) = 0 ) |
102 | 99, 100, 101 | syl2anc 584 |
. . . . 5
β’ (((π β§ π₯ β β0) β§ Β¬
π· β€ π₯) β (π
Ξ£g (π¦ β (0...π₯) β¦ 0 )) = 0 ) |
103 | 97, 102 | eqtrd 2776 |
. . . 4
β’ (((π β§ π₯ β β0) β§ Β¬
π· β€ π₯) β (π
Ξ£g (π¦ β (0...π₯) β¦ (((coe1β(πΆ Β· (π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))))) = 0 ) |
104 | 18, 19, 80, 103 | ifbothda 4524 |
. . 3
β’ ((π β§ π₯ β β0) β (π
Ξ£g
(π¦ β (0...π₯) β¦
(((coe1β(πΆ
Β·
(π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦))))) = if(π· β€ π₯, (πΆ Γ
((coe1βπ΄)β(π₯ β π·))), 0 )) |
105 | 104 | mpteq2dva 5205 |
. 2
β’ (π β (π₯ β β0 β¦ (π
Ξ£g
(π¦ β (0...π₯) β¦
(((coe1β(πΆ
Β·
(π· β π)))βπ¦) Γ
((coe1βπ΄)β(π₯ β π¦)))))) = (π₯ β β0 β¦ if(π· β€ π₯, (πΆ Γ
((coe1βπ΄)β(π₯ β π·))), 0 ))) |
106 | 17, 105 | eqtrd 2776 |
1
β’ (π β
(coe1β((πΆ
Β·
(π· β π)) β π΄)) = (π₯ β β0 β¦ if(π· β€ π₯, (πΆ Γ
((coe1βπ΄)β(π₯ β π·))), 0 ))) |