Step | Hyp | Ref
| Expression |
1 | | elfznn0 13541 |
. . 3
β’ (πΎ β (0...(degβπΉ)) β πΎ β
β0) |
2 | | elqaa.6 |
. . . . 5
β’ π
= (seq0( Β· , π)β(degβπΉ)) |
3 | 2 | fveq2i 6850 |
. . . 4
β’ ((π β β β¦ (π mod (πβπΎ)))βπ
) = ((π β β β¦ (π mod (πβπΎ)))β(seq0( Β· , π)β(degβπΉ))) |
4 | | nnmulcl 12184 |
. . . . . 6
β’ ((π β β β§ π β β) β (π Β· π) β β) |
5 | 4 | adantl 483 |
. . . . 5
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β (π Β· π) β β) |
6 | | elfznn0 13541 |
. . . . . 6
β’ (π β (0...(degβπΉ)) β π β β0) |
7 | | elqaa.1 |
. . . . . . . . 9
β’ (π β π΄ β β) |
8 | | elqaa.2 |
. . . . . . . . 9
β’ (π β πΉ β ((Polyββ) β
{0π})) |
9 | | elqaa.3 |
. . . . . . . . 9
β’ (π β (πΉβπ΄) = 0) |
10 | | elqaa.4 |
. . . . . . . . 9
β’ π΅ = (coeffβπΉ) |
11 | | elqaa.5 |
. . . . . . . . 9
β’ π = (π β β0 β¦
inf({π β β
β£ ((π΅βπ) Β· π) β β€}, β, <
)) |
12 | 7, 8, 9, 10, 11, 2 | elqaalem1 25695 |
. . . . . . . 8
β’ ((π β§ π β β0) β ((πβπ) β β β§ ((π΅βπ) Β· (πβπ)) β β€)) |
13 | 12 | simpld 496 |
. . . . . . 7
β’ ((π β§ π β β0) β (πβπ) β β) |
14 | 13 | adantlr 714 |
. . . . . 6
β’ (((π β§ πΎ β β0) β§ π β β0)
β (πβπ) β
β) |
15 | 6, 14 | sylan2 594 |
. . . . 5
β’ (((π β§ πΎ β β0) β§ π β (0...(degβπΉ))) β (πβπ) β β) |
16 | | eldifi 4091 |
. . . . . . . 8
β’ (πΉ β ((Polyββ)
β {0π}) β πΉ β
(Polyββ)) |
17 | | dgrcl 25610 |
. . . . . . . 8
β’ (πΉ β (Polyββ)
β (degβπΉ) β
β0) |
18 | 8, 16, 17 | 3syl 18 |
. . . . . . 7
β’ (π β (degβπΉ) β
β0) |
19 | | nn0uz 12812 |
. . . . . . 7
β’
β0 = (β€β₯β0) |
20 | 18, 19 | eleqtrdi 2848 |
. . . . . 6
β’ (π β (degβπΉ) β
(β€β₯β0)) |
21 | 20 | adantr 482 |
. . . . 5
β’ ((π β§ πΎ β β0) β
(degβπΉ) β
(β€β₯β0)) |
22 | | nnz 12527 |
. . . . . . . . . 10
β’ (π β β β π β
β€) |
23 | 22 | ad2antrl 727 |
. . . . . . . . 9
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β π β
β€) |
24 | 7, 8, 9, 10, 11, 2 | elqaalem1 25695 |
. . . . . . . . . . 11
β’ ((π β§ πΎ β β0) β ((πβπΎ) β β β§ ((π΅βπΎ) Β· (πβπΎ)) β β€)) |
25 | 24 | simpld 496 |
. . . . . . . . . 10
β’ ((π β§ πΎ β β0) β (πβπΎ) β β) |
26 | 25 | adantr 482 |
. . . . . . . . 9
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β (πβπΎ) β β) |
27 | 23, 26 | zmodcld 13804 |
. . . . . . . 8
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β (π mod (πβπΎ)) β
β0) |
28 | 27 | nn0zd 12532 |
. . . . . . 7
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β (π mod (πβπΎ)) β β€) |
29 | | nnz 12527 |
. . . . . . . . . 10
β’ (π β β β π β
β€) |
30 | 29 | ad2antll 728 |
. . . . . . . . 9
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β π β
β€) |
31 | 30, 26 | zmodcld 13804 |
. . . . . . . 8
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β (π mod (πβπΎ)) β
β0) |
32 | 31 | nn0zd 12532 |
. . . . . . 7
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β (π mod (πβπΎ)) β β€) |
33 | 26 | nnrpd 12962 |
. . . . . . 7
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β (πβπΎ) β
β+) |
34 | | nnre 12167 |
. . . . . . . . 9
β’ (π β β β π β
β) |
35 | 34 | ad2antrl 727 |
. . . . . . . 8
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β π β
β) |
36 | | modabs2 13817 |
. . . . . . . 8
β’ ((π β β β§ (πβπΎ) β β+) β ((π mod (πβπΎ)) mod (πβπΎ)) = (π mod (πβπΎ))) |
37 | 35, 33, 36 | syl2anc 585 |
. . . . . . 7
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β ((π mod (πβπΎ)) mod (πβπΎ)) = (π mod (πβπΎ))) |
38 | | nnre 12167 |
. . . . . . . . 9
β’ (π β β β π β
β) |
39 | 38 | ad2antll 728 |
. . . . . . . 8
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β π β
β) |
40 | | modabs2 13817 |
. . . . . . . 8
β’ ((π β β β§ (πβπΎ) β β+) β ((π mod (πβπΎ)) mod (πβπΎ)) = (π mod (πβπΎ))) |
41 | 39, 33, 40 | syl2anc 585 |
. . . . . . 7
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β ((π mod (πβπΎ)) mod (πβπΎ)) = (π mod (πβπΎ))) |
42 | 28, 23, 32, 30, 33, 37, 41 | modmul12d 13837 |
. . . . . 6
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β
(((π mod (πβπΎ)) Β· (π mod (πβπΎ))) mod (πβπΎ)) = ((π Β· π) mod (πβπΎ))) |
43 | | oveq1 7369 |
. . . . . . . . . 10
β’ (π = π β (π mod (πβπΎ)) = (π mod (πβπΎ))) |
44 | | eqid 2737 |
. . . . . . . . . 10
β’ (π β β β¦ (π mod (πβπΎ))) = (π β β β¦ (π mod (πβπΎ))) |
45 | | ovex 7395 |
. . . . . . . . . 10
β’ (π mod (πβπΎ)) β V |
46 | 43, 44, 45 | fvmpt 6953 |
. . . . . . . . 9
β’ (π β β β ((π β β β¦ (π mod (πβπΎ)))βπ) = (π mod (πβπΎ))) |
47 | 46 | ad2antrl 727 |
. . . . . . . 8
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β ((π β β β¦ (π mod (πβπΎ)))βπ) = (π mod (πβπΎ))) |
48 | | oveq1 7369 |
. . . . . . . . . 10
β’ (π = π β (π mod (πβπΎ)) = (π mod (πβπΎ))) |
49 | | ovex 7395 |
. . . . . . . . . 10
β’ (π mod (πβπΎ)) β V |
50 | 48, 44, 49 | fvmpt 6953 |
. . . . . . . . 9
β’ (π β β β ((π β β β¦ (π mod (πβπΎ)))βπ) = (π mod (πβπΎ))) |
51 | 50 | ad2antll 728 |
. . . . . . . 8
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β ((π β β β¦ (π mod (πβπΎ)))βπ) = (π mod (πβπΎ))) |
52 | 47, 51 | oveq12d 7380 |
. . . . . . 7
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β
(((π β β β¦
(π mod (πβπΎ)))βπ)π((π β β β¦ (π mod (πβπΎ)))βπ)) = ((π mod (πβπΎ))π(π mod (πβπΎ)))) |
53 | | oveq12 7371 |
. . . . . . . . . 10
β’ ((π₯ = (π mod (πβπΎ)) β§ π¦ = (π mod (πβπΎ))) β (π₯ Β· π¦) = ((π mod (πβπΎ)) Β· (π mod (πβπΎ)))) |
54 | 53 | oveq1d 7377 |
. . . . . . . . 9
β’ ((π₯ = (π mod (πβπΎ)) β§ π¦ = (π mod (πβπΎ))) β ((π₯ Β· π¦) mod (πβπΎ)) = (((π mod (πβπΎ)) Β· (π mod (πβπΎ))) mod (πβπΎ))) |
55 | | elqaa.7 |
. . . . . . . . 9
β’ π = (π₯ β V, π¦ β V β¦ ((π₯ Β· π¦) mod (πβπΎ))) |
56 | | ovex 7395 |
. . . . . . . . 9
β’ (((π mod (πβπΎ)) Β· (π mod (πβπΎ))) mod (πβπΎ)) β V |
57 | 54, 55, 56 | ovmpoa 7515 |
. . . . . . . 8
β’ (((π mod (πβπΎ)) β V β§ (π mod (πβπΎ)) β V) β ((π mod (πβπΎ))π(π mod (πβπΎ))) = (((π mod (πβπΎ)) Β· (π mod (πβπΎ))) mod (πβπΎ))) |
58 | 45, 49, 57 | mp2an 691 |
. . . . . . 7
β’ ((π mod (πβπΎ))π(π mod (πβπΎ))) = (((π mod (πβπΎ)) Β· (π mod (πβπΎ))) mod (πβπΎ)) |
59 | 52, 58 | eqtrdi 2793 |
. . . . . 6
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β
(((π β β β¦
(π mod (πβπΎ)))βπ)π((π β β β¦ (π mod (πβπΎ)))βπ)) = (((π mod (πβπΎ)) Β· (π mod (πβπΎ))) mod (πβπΎ))) |
60 | | oveq1 7369 |
. . . . . . . 8
β’ (π = (π Β· π) β (π mod (πβπΎ)) = ((π Β· π) mod (πβπΎ))) |
61 | | ovex 7395 |
. . . . . . . 8
β’ ((π Β· π) mod (πβπΎ)) β V |
62 | 60, 44, 61 | fvmpt 6953 |
. . . . . . 7
β’ ((π Β· π) β β β ((π β β β¦ (π mod (πβπΎ)))β(π Β· π)) = ((π Β· π) mod (πβπΎ))) |
63 | 5, 62 | syl 17 |
. . . . . 6
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β ((π β β β¦ (π mod (πβπΎ)))β(π Β· π)) = ((π Β· π) mod (πβπΎ))) |
64 | 42, 59, 63 | 3eqtr4rd 2788 |
. . . . 5
β’ (((π β§ πΎ β β0) β§ (π β β β§ π β β)) β ((π β β β¦ (π mod (πβπΎ)))β(π Β· π)) = (((π β β β¦ (π mod (πβπΎ)))βπ)π((π β β β¦ (π mod (πβπΎ)))βπ))) |
65 | | oveq1 7369 |
. . . . . . . . 9
β’ (π = (πβπ) β (π mod (πβπΎ)) = ((πβπ) mod (πβπΎ))) |
66 | | ovex 7395 |
. . . . . . . . 9
β’ ((πβπ) mod (πβπΎ)) β V |
67 | 65, 44, 66 | fvmpt 6953 |
. . . . . . . 8
β’ ((πβπ) β β β ((π β β β¦ (π mod (πβπΎ)))β(πβπ)) = ((πβπ) mod (πβπΎ))) |
68 | 14, 67 | syl 17 |
. . . . . . 7
β’ (((π β§ πΎ β β0) β§ π β β0)
β ((π β β
β¦ (π mod (πβπΎ)))β(πβπ)) = ((πβπ) mod (πβπΎ))) |
69 | | fveq2 6847 |
. . . . . . . . . 10
β’ (π = π β (πβπ) = (πβπ)) |
70 | 69 | oveq1d 7377 |
. . . . . . . . 9
β’ (π = π β ((πβπ) mod (πβπΎ)) = ((πβπ) mod (πβπΎ))) |
71 | | eqid 2737 |
. . . . . . . . 9
β’ (π β β0
β¦ ((πβπ) mod (πβπΎ))) = (π β β0 β¦ ((πβπ) mod (πβπΎ))) |
72 | 70, 71, 66 | fvmpt 6953 |
. . . . . . . 8
β’ (π β β0
β ((π β
β0 β¦ ((πβπ) mod (πβπΎ)))βπ) = ((πβπ) mod (πβπΎ))) |
73 | 72 | adantl 483 |
. . . . . . 7
β’ (((π β§ πΎ β β0) β§ π β β0)
β ((π β
β0 β¦ ((πβπ) mod (πβπΎ)))βπ) = ((πβπ) mod (πβπΎ))) |
74 | 68, 73 | eqtr4d 2780 |
. . . . . 6
β’ (((π β§ πΎ β β0) β§ π β β0)
β ((π β β
β¦ (π mod (πβπΎ)))β(πβπ)) = ((π β β0 β¦ ((πβπ) mod (πβπΎ)))βπ)) |
75 | 6, 74 | sylan2 594 |
. . . . 5
β’ (((π β§ πΎ β β0) β§ π β (0...(degβπΉ))) β ((π β β β¦ (π mod (πβπΎ)))β(πβπ)) = ((π β β0 β¦ ((πβπ) mod (πβπΎ)))βπ)) |
76 | 5, 15, 21, 64, 75 | seqhomo 13962 |
. . . 4
β’ ((π β§ πΎ β β0) β ((π β β β¦ (π mod (πβπΎ)))β(seq0( Β· , π)β(degβπΉ))) = (seq0(π, (π β β0 β¦ ((πβπ) mod (πβπΎ))))β(degβπΉ))) |
77 | 3, 76 | eqtrid 2789 |
. . 3
β’ ((π β§ πΎ β β0) β ((π β β β¦ (π mod (πβπΎ)))βπ
) = (seq0(π, (π β β0 β¦ ((πβπ) mod (πβπΎ))))β(degβπΉ))) |
78 | 1, 77 | sylan2 594 |
. 2
β’ ((π β§ πΎ β (0...(degβπΉ))) β ((π β β β¦ (π mod (πβπΎ)))βπ
) = (seq0(π, (π β β0 β¦ ((πβπ) mod (πβπΎ))))β(degβπΉ))) |
79 | | 0zd 12518 |
. . . . . . . 8
β’ (π β 0 β
β€) |
80 | 4 | adantl 483 |
. . . . . . . 8
β’ ((π β§ (π β β β§ π β β)) β (π Β· π) β β) |
81 | 19, 79, 13, 80 | seqf 13936 |
. . . . . . 7
β’ (π β seq0( Β· , π):β0βΆβ) |
82 | 81, 18 | ffvelcdmd 7041 |
. . . . . 6
β’ (π β (seq0( Β· , π)β(degβπΉ)) β
β) |
83 | 2, 82 | eqeltrid 2842 |
. . . . 5
β’ (π β π
β β) |
84 | 83 | adantr 482 |
. . . 4
β’ ((π β§ πΎ β β0) β π
β
β) |
85 | | oveq1 7369 |
. . . . 5
β’ (π = π
β (π mod (πβπΎ)) = (π
mod (πβπΎ))) |
86 | | ovex 7395 |
. . . . 5
β’ (π
mod (πβπΎ)) β V |
87 | 85, 44, 86 | fvmpt 6953 |
. . . 4
β’ (π
β β β ((π β β β¦ (π mod (πβπΎ)))βπ
) = (π
mod (πβπΎ))) |
88 | 84, 87 | syl 17 |
. . 3
β’ ((π β§ πΎ β β0) β ((π β β β¦ (π mod (πβπΎ)))βπ
) = (π
mod (πβπΎ))) |
89 | 1, 88 | sylan2 594 |
. 2
β’ ((π β§ πΎ β (0...(degβπΉ))) β ((π β β β¦ (π mod (πβπΎ)))βπ
) = (π
mod (πβπΎ))) |
90 | | oveq12 7371 |
. . . . . . 7
β’ ((π₯ = π β§ π¦ = π) β (π₯ Β· π¦) = (π Β· π)) |
91 | 90 | oveq1d 7377 |
. . . . . 6
β’ ((π₯ = π β§ π¦ = π) β ((π₯ Β· π¦) mod (πβπΎ)) = ((π Β· π) mod (πβπΎ))) |
92 | 91, 55, 61 | ovmpoa 7515 |
. . . . 5
β’ ((π β V β§ π β V) β (πππ) = ((π Β· π) mod (πβπΎ))) |
93 | 92 | el2v 3456 |
. . . 4
β’ (πππ) = ((π Β· π) mod (πβπΎ)) |
94 | | nn0mulcl 12456 |
. . . . . 6
β’ ((π β β0
β§ π β
β0) β (π Β· π) β
β0) |
95 | 94 | nn0zd 12532 |
. . . . 5
β’ ((π β β0
β§ π β
β0) β (π Β· π) β β€) |
96 | 1, 25 | sylan2 594 |
. . . . 5
β’ ((π β§ πΎ β (0...(degβπΉ))) β (πβπΎ) β β) |
97 | | zmodcl 13803 |
. . . . 5
β’ (((π Β· π) β β€ β§ (πβπΎ) β β) β ((π Β· π) mod (πβπΎ)) β
β0) |
98 | 95, 96, 97 | syl2anr 598 |
. . . 4
β’ (((π β§ πΎ β (0...(degβπΉ))) β§ (π β β0 β§ π β β0))
β ((π Β· π) mod (πβπΎ)) β
β0) |
99 | 93, 98 | eqeltrid 2842 |
. . 3
β’ (((π β§ πΎ β (0...(degβπΉ))) β§ (π β β0 β§ π β β0))
β (πππ) β
β0) |
100 | | fveq2 6847 |
. . . . . . . . . . . . . . . . 17
β’ (π = π β (π΅βπ) = (π΅βπ)) |
101 | 100 | oveq1d 7377 |
. . . . . . . . . . . . . . . 16
β’ (π = π β ((π΅βπ) Β· π) = ((π΅βπ) Β· π)) |
102 | 101 | eleq1d 2823 |
. . . . . . . . . . . . . . 15
β’ (π = π β (((π΅βπ) Β· π) β β€ β ((π΅βπ) Β· π) β β€)) |
103 | 102 | rabbidv 3418 |
. . . . . . . . . . . . . 14
β’ (π = π β {π β β β£ ((π΅βπ) Β· π) β β€} = {π β β β£ ((π΅βπ) Β· π) β β€}) |
104 | 103 | infeq1d 9420 |
. . . . . . . . . . . . 13
β’ (π = π β inf({π β β β£ ((π΅βπ) Β· π) β β€}, β, < ) =
inf({π β β
β£ ((π΅βπ) Β· π) β β€}, β, <
)) |
105 | 104 | cbvmptv 5223 |
. . . . . . . . . . . 12
β’ (π β β0
β¦ inf({π β
β β£ ((π΅βπ) Β· π) β β€}, β, < )) = (π β β0
β¦ inf({π β
β β£ ((π΅βπ) Β· π) β β€}, β, <
)) |
106 | 11, 105 | eqtri 2765 |
. . . . . . . . . . 11
β’ π = (π β β0 β¦
inf({π β β
β£ ((π΅βπ) Β· π) β β€}, β, <
)) |
107 | 7, 8, 9, 10, 106, 2 | elqaalem1 25695 |
. . . . . . . . . 10
β’ ((π β§ π β β0) β ((πβπ) β β β§ ((π΅βπ) Β· (πβπ)) β β€)) |
108 | 107 | simpld 496 |
. . . . . . . . 9
β’ ((π β§ π β β0) β (πβπ) β β) |
109 | 108 | adantlr 714 |
. . . . . . . 8
β’ (((π β§ πΎ β β0) β§ π β β0)
β (πβπ) β
β) |
110 | 109 | nnzd 12533 |
. . . . . . 7
β’ (((π β§ πΎ β β0) β§ π β β0)
β (πβπ) β
β€) |
111 | 25 | adantr 482 |
. . . . . . 7
β’ (((π β§ πΎ β β0) β§ π β β0)
β (πβπΎ) β
β) |
112 | 110, 111 | zmodcld 13804 |
. . . . . 6
β’ (((π β§ πΎ β β0) β§ π β β0)
β ((πβπ) mod (πβπΎ)) β
β0) |
113 | 112 | fmpttd 7068 |
. . . . 5
β’ ((π β§ πΎ β β0) β (π β β0
β¦ ((πβπ) mod (πβπΎ))):β0βΆβ0) |
114 | 1, 113 | sylan2 594 |
. . . 4
β’ ((π β§ πΎ β (0...(degβπΉ))) β (π β β0 β¦ ((πβπ) mod (πβπΎ))):β0βΆβ0) |
115 | | ffvelcdm 7037 |
. . . 4
β’ (((π β β0
β¦ ((πβπ) mod (πβπΎ))):β0βΆβ0
β§ π β
β0) β ((π
β β0 β¦ ((πβπ) mod (πβπΎ)))βπ) β β0) |
116 | 114, 6, 115 | syl2an 597 |
. . 3
β’ (((π β§ πΎ β (0...(degβπΉ))) β§ π β (0...(degβπΉ))) β ((π β β0 β¦ ((πβπ) mod (πβπΎ)))βπ) β
β0) |
117 | | c0ex 11156 |
. . . . 5
β’ 0 β
V |
118 | | vex 3452 |
. . . . 5
β’ π β V |
119 | | oveq12 7371 |
. . . . . . 7
β’ ((π₯ = 0 β§ π¦ = π) β (π₯ Β· π¦) = (0 Β· π)) |
120 | 119 | oveq1d 7377 |
. . . . . 6
β’ ((π₯ = 0 β§ π¦ = π) β ((π₯ Β· π¦) mod (πβπΎ)) = ((0 Β· π) mod (πβπΎ))) |
121 | | ovex 7395 |
. . . . . 6
β’ ((0
Β· π) mod (πβπΎ)) β V |
122 | 120, 55, 121 | ovmpoa 7515 |
. . . . 5
β’ ((0
β V β§ π β V)
β (0ππ) = ((0 Β· π) mod (πβπΎ))) |
123 | 117, 118,
122 | mp2an 691 |
. . . 4
β’ (0ππ) = ((0 Β· π) mod (πβπΎ)) |
124 | | nn0cn 12430 |
. . . . . . 7
β’ (π β β0
β π β
β) |
125 | 124 | mul02d 11360 |
. . . . . 6
β’ (π β β0
β (0 Β· π) =
0) |
126 | 125 | oveq1d 7377 |
. . . . 5
β’ (π β β0
β ((0 Β· π) mod
(πβπΎ)) = (0 mod (πβπΎ))) |
127 | 96 | nnrpd 12962 |
. . . . . 6
β’ ((π β§ πΎ β (0...(degβπΉ))) β (πβπΎ) β
β+) |
128 | | 0mod 13814 |
. . . . . 6
β’ ((πβπΎ) β β+ β (0 mod
(πβπΎ)) = 0) |
129 | 127, 128 | syl 17 |
. . . . 5
β’ ((π β§ πΎ β (0...(degβπΉ))) β (0 mod (πβπΎ)) = 0) |
130 | 126, 129 | sylan9eqr 2799 |
. . . 4
β’ (((π β§ πΎ β (0...(degβπΉ))) β§ π β β0) β ((0
Β· π) mod (πβπΎ)) = 0) |
131 | 123, 130 | eqtrid 2789 |
. . 3
β’ (((π β§ πΎ β (0...(degβπΉ))) β§ π β β0) β (0ππ) = 0) |
132 | | oveq12 7371 |
. . . . . . 7
β’ ((π₯ = π β§ π¦ = 0) β (π₯ Β· π¦) = (π Β· 0)) |
133 | 132 | oveq1d 7377 |
. . . . . 6
β’ ((π₯ = π β§ π¦ = 0) β ((π₯ Β· π¦) mod (πβπΎ)) = ((π Β· 0) mod (πβπΎ))) |
134 | | ovex 7395 |
. . . . . 6
β’ ((π Β· 0) mod (πβπΎ)) β V |
135 | 133, 55, 134 | ovmpoa 7515 |
. . . . 5
β’ ((π β V β§ 0 β V)
β (ππ0) = ((π Β· 0) mod (πβπΎ))) |
136 | 118, 117,
135 | mp2an 691 |
. . . 4
β’ (ππ0) = ((π Β· 0) mod (πβπΎ)) |
137 | 124 | mul01d 11361 |
. . . . . 6
β’ (π β β0
β (π Β· 0) =
0) |
138 | 137 | oveq1d 7377 |
. . . . 5
β’ (π β β0
β ((π Β· 0) mod
(πβπΎ)) = (0 mod (πβπΎ))) |
139 | 138, 129 | sylan9eqr 2799 |
. . . 4
β’ (((π β§ πΎ β (0...(degβπΉ))) β§ π β β0) β ((π Β· 0) mod (πβπΎ)) = 0) |
140 | 136, 139 | eqtrid 2789 |
. . 3
β’ (((π β§ πΎ β (0...(degβπΉ))) β§ π β β0) β (ππ0) = 0) |
141 | | simpr 486 |
. . 3
β’ ((π β§ πΎ β (0...(degβπΉ))) β πΎ β (0...(degβπΉ))) |
142 | 18 | adantr 482 |
. . 3
β’ ((π β§ πΎ β (0...(degβπΉ))) β (degβπΉ) β
β0) |
143 | 1 | adantl 483 |
. . . . 5
β’ ((π β§ πΎ β (0...(degβπΉ))) β πΎ β
β0) |
144 | | fveq2 6847 |
. . . . . . 7
β’ (π = πΎ β (πβπ) = (πβπΎ)) |
145 | 144 | oveq1d 7377 |
. . . . . 6
β’ (π = πΎ β ((πβπ) mod (πβπΎ)) = ((πβπΎ) mod (πβπΎ))) |
146 | | ovex 7395 |
. . . . . 6
β’ ((πβπΎ) mod (πβπΎ)) β V |
147 | 145, 71, 146 | fvmpt 6953 |
. . . . 5
β’ (πΎ β β0
β ((π β
β0 β¦ ((πβπ) mod (πβπΎ)))βπΎ) = ((πβπΎ) mod (πβπΎ))) |
148 | 143, 147 | syl 17 |
. . . 4
β’ ((π β§ πΎ β (0...(degβπΉ))) β ((π β β0 β¦ ((πβπ) mod (πβπΎ)))βπΎ) = ((πβπΎ) mod (πβπΎ))) |
149 | 96 | nncnd 12176 |
. . . . . . 7
β’ ((π β§ πΎ β (0...(degβπΉ))) β (πβπΎ) β β) |
150 | 96 | nnne0d 12210 |
. . . . . . 7
β’ ((π β§ πΎ β (0...(degβπΉ))) β (πβπΎ) β 0) |
151 | 149, 150 | dividd 11936 |
. . . . . 6
β’ ((π β§ πΎ β (0...(degβπΉ))) β ((πβπΎ) / (πβπΎ)) = 1) |
152 | | 1z 12540 |
. . . . . 6
β’ 1 β
β€ |
153 | 151, 152 | eqeltrdi 2846 |
. . . . 5
β’ ((π β§ πΎ β (0...(degβπΉ))) β ((πβπΎ) / (πβπΎ)) β β€) |
154 | 96 | nnred 12175 |
. . . . . 6
β’ ((π β§ πΎ β (0...(degβπΉ))) β (πβπΎ) β β) |
155 | | mod0 13788 |
. . . . . 6
β’ (((πβπΎ) β β β§ (πβπΎ) β β+) β
(((πβπΎ) mod (πβπΎ)) = 0 β ((πβπΎ) / (πβπΎ)) β β€)) |
156 | 154, 127,
155 | syl2anc 585 |
. . . . 5
β’ ((π β§ πΎ β (0...(degβπΉ))) β (((πβπΎ) mod (πβπΎ)) = 0 β ((πβπΎ) / (πβπΎ)) β β€)) |
157 | 153, 156 | mpbird 257 |
. . . 4
β’ ((π β§ πΎ β (0...(degβπΉ))) β ((πβπΎ) mod (πβπΎ)) = 0) |
158 | 148, 157 | eqtrd 2777 |
. . 3
β’ ((π β§ πΎ β (0...(degβπΉ))) β ((π β β0 β¦ ((πβπ) mod (πβπΎ)))βπΎ) = 0) |
159 | 99, 116, 131, 140, 141, 142, 158 | seqz 13963 |
. 2
β’ ((π β§ πΎ β (0...(degβπΉ))) β (seq0(π, (π β β0 β¦ ((πβπ) mod (πβπΎ))))β(degβπΉ)) = 0) |
160 | 78, 89, 159 | 3eqtr3d 2785 |
1
β’ ((π β§ πΎ β (0...(degβπΉ))) β (π
mod (πβπΎ)) = 0) |