Step |
Hyp |
Ref |
Expression |
1 |
|
elqaa.1 |
|- ( ph -> A e. CC ) |
2 |
|
elqaa.2 |
|- ( ph -> F e. ( ( Poly ` QQ ) \ { 0p } ) ) |
3 |
|
elqaa.3 |
|- ( ph -> ( F ` A ) = 0 ) |
4 |
|
elqaa.4 |
|- B = ( coeff ` F ) |
5 |
|
elqaa.5 |
|- N = ( k e. NN0 |-> inf ( { n e. NN | ( ( B ` k ) x. n ) e. ZZ } , RR , < ) ) |
6 |
|
elqaa.6 |
|- R = ( seq 0 ( x. , N ) ` ( deg ` F ) ) |
7 |
|
elqaa.7 |
|- P = ( x e. _V , y e. _V |-> ( ( x x. y ) mod ( N ` K ) ) ) |
8 |
|
elfznn0 |
|- ( K e. ( 0 ... ( deg ` F ) ) -> K e. NN0 ) |
9 |
6
|
fveq2i |
|- ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` R ) = ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` ( seq 0 ( x. , N ) ` ( deg ` F ) ) ) |
10 |
|
nnmulcl |
|- ( ( i e. NN /\ j e. NN ) -> ( i x. j ) e. NN ) |
11 |
10
|
adantl |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> ( i x. j ) e. NN ) |
12 |
|
elfznn0 |
|- ( i e. ( 0 ... ( deg ` F ) ) -> i e. NN0 ) |
13 |
1 2 3 4 5 6
|
elqaalem1 |
|- ( ( ph /\ i e. NN0 ) -> ( ( N ` i ) e. NN /\ ( ( B ` i ) x. ( N ` i ) ) e. ZZ ) ) |
14 |
13
|
simpld |
|- ( ( ph /\ i e. NN0 ) -> ( N ` i ) e. NN ) |
15 |
14
|
adantlr |
|- ( ( ( ph /\ K e. NN0 ) /\ i e. NN0 ) -> ( N ` i ) e. NN ) |
16 |
12 15
|
sylan2 |
|- ( ( ( ph /\ K e. NN0 ) /\ i e. ( 0 ... ( deg ` F ) ) ) -> ( N ` i ) e. NN ) |
17 |
|
eldifi |
|- ( F e. ( ( Poly ` QQ ) \ { 0p } ) -> F e. ( Poly ` QQ ) ) |
18 |
|
dgrcl |
|- ( F e. ( Poly ` QQ ) -> ( deg ` F ) e. NN0 ) |
19 |
2 17 18
|
3syl |
|- ( ph -> ( deg ` F ) e. NN0 ) |
20 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
21 |
19 20
|
eleqtrdi |
|- ( ph -> ( deg ` F ) e. ( ZZ>= ` 0 ) ) |
22 |
21
|
adantr |
|- ( ( ph /\ K e. NN0 ) -> ( deg ` F ) e. ( ZZ>= ` 0 ) ) |
23 |
|
nnz |
|- ( i e. NN -> i e. ZZ ) |
24 |
23
|
ad2antrl |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> i e. ZZ ) |
25 |
1 2 3 4 5 6
|
elqaalem1 |
|- ( ( ph /\ K e. NN0 ) -> ( ( N ` K ) e. NN /\ ( ( B ` K ) x. ( N ` K ) ) e. ZZ ) ) |
26 |
25
|
simpld |
|- ( ( ph /\ K e. NN0 ) -> ( N ` K ) e. NN ) |
27 |
26
|
adantr |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> ( N ` K ) e. NN ) |
28 |
24 27
|
zmodcld |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> ( i mod ( N ` K ) ) e. NN0 ) |
29 |
28
|
nn0zd |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> ( i mod ( N ` K ) ) e. ZZ ) |
30 |
|
nnz |
|- ( j e. NN -> j e. ZZ ) |
31 |
30
|
ad2antll |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> j e. ZZ ) |
32 |
31 27
|
zmodcld |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> ( j mod ( N ` K ) ) e. NN0 ) |
33 |
32
|
nn0zd |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> ( j mod ( N ` K ) ) e. ZZ ) |
34 |
27
|
nnrpd |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> ( N ` K ) e. RR+ ) |
35 |
|
nnre |
|- ( i e. NN -> i e. RR ) |
36 |
35
|
ad2antrl |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> i e. RR ) |
37 |
|
modabs2 |
|- ( ( i e. RR /\ ( N ` K ) e. RR+ ) -> ( ( i mod ( N ` K ) ) mod ( N ` K ) ) = ( i mod ( N ` K ) ) ) |
38 |
36 34 37
|
syl2anc |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> ( ( i mod ( N ` K ) ) mod ( N ` K ) ) = ( i mod ( N ` K ) ) ) |
39 |
|
nnre |
|- ( j e. NN -> j e. RR ) |
40 |
39
|
ad2antll |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> j e. RR ) |
41 |
|
modabs2 |
|- ( ( j e. RR /\ ( N ` K ) e. RR+ ) -> ( ( j mod ( N ` K ) ) mod ( N ` K ) ) = ( j mod ( N ` K ) ) ) |
42 |
40 34 41
|
syl2anc |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> ( ( j mod ( N ` K ) ) mod ( N ` K ) ) = ( j mod ( N ` K ) ) ) |
43 |
29 24 33 31 34 38 42
|
modmul12d |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> ( ( ( i mod ( N ` K ) ) x. ( j mod ( N ` K ) ) ) mod ( N ` K ) ) = ( ( i x. j ) mod ( N ` K ) ) ) |
44 |
|
oveq1 |
|- ( k = i -> ( k mod ( N ` K ) ) = ( i mod ( N ` K ) ) ) |
45 |
|
eqid |
|- ( k e. NN |-> ( k mod ( N ` K ) ) ) = ( k e. NN |-> ( k mod ( N ` K ) ) ) |
46 |
|
ovex |
|- ( i mod ( N ` K ) ) e. _V |
47 |
44 45 46
|
fvmpt |
|- ( i e. NN -> ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` i ) = ( i mod ( N ` K ) ) ) |
48 |
47
|
ad2antrl |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` i ) = ( i mod ( N ` K ) ) ) |
49 |
|
oveq1 |
|- ( k = j -> ( k mod ( N ` K ) ) = ( j mod ( N ` K ) ) ) |
50 |
|
ovex |
|- ( j mod ( N ` K ) ) e. _V |
51 |
49 45 50
|
fvmpt |
|- ( j e. NN -> ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` j ) = ( j mod ( N ` K ) ) ) |
52 |
51
|
ad2antll |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` j ) = ( j mod ( N ` K ) ) ) |
53 |
48 52
|
oveq12d |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> ( ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` i ) P ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` j ) ) = ( ( i mod ( N ` K ) ) P ( j mod ( N ` K ) ) ) ) |
54 |
|
oveq12 |
|- ( ( x = ( i mod ( N ` K ) ) /\ y = ( j mod ( N ` K ) ) ) -> ( x x. y ) = ( ( i mod ( N ` K ) ) x. ( j mod ( N ` K ) ) ) ) |
55 |
54
|
oveq1d |
|- ( ( x = ( i mod ( N ` K ) ) /\ y = ( j mod ( N ` K ) ) ) -> ( ( x x. y ) mod ( N ` K ) ) = ( ( ( i mod ( N ` K ) ) x. ( j mod ( N ` K ) ) ) mod ( N ` K ) ) ) |
56 |
|
ovex |
|- ( ( ( i mod ( N ` K ) ) x. ( j mod ( N ` K ) ) ) mod ( N ` K ) ) e. _V |
57 |
55 7 56
|
ovmpoa |
|- ( ( ( i mod ( N ` K ) ) e. _V /\ ( j mod ( N ` K ) ) e. _V ) -> ( ( i mod ( N ` K ) ) P ( j mod ( N ` K ) ) ) = ( ( ( i mod ( N ` K ) ) x. ( j mod ( N ` K ) ) ) mod ( N ` K ) ) ) |
58 |
46 50 57
|
mp2an |
|- ( ( i mod ( N ` K ) ) P ( j mod ( N ` K ) ) ) = ( ( ( i mod ( N ` K ) ) x. ( j mod ( N ` K ) ) ) mod ( N ` K ) ) |
59 |
53 58
|
eqtrdi |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> ( ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` i ) P ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` j ) ) = ( ( ( i mod ( N ` K ) ) x. ( j mod ( N ` K ) ) ) mod ( N ` K ) ) ) |
60 |
|
oveq1 |
|- ( k = ( i x. j ) -> ( k mod ( N ` K ) ) = ( ( i x. j ) mod ( N ` K ) ) ) |
61 |
|
ovex |
|- ( ( i x. j ) mod ( N ` K ) ) e. _V |
62 |
60 45 61
|
fvmpt |
|- ( ( i x. j ) e. NN -> ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` ( i x. j ) ) = ( ( i x. j ) mod ( N ` K ) ) ) |
63 |
11 62
|
syl |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` ( i x. j ) ) = ( ( i x. j ) mod ( N ` K ) ) ) |
64 |
43 59 63
|
3eqtr4rd |
|- ( ( ( ph /\ K e. NN0 ) /\ ( i e. NN /\ j e. NN ) ) -> ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` ( i x. j ) ) = ( ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` i ) P ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` j ) ) ) |
65 |
|
oveq1 |
|- ( k = ( N ` i ) -> ( k mod ( N ` K ) ) = ( ( N ` i ) mod ( N ` K ) ) ) |
66 |
|
ovex |
|- ( ( N ` i ) mod ( N ` K ) ) e. _V |
67 |
65 45 66
|
fvmpt |
|- ( ( N ` i ) e. NN -> ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` ( N ` i ) ) = ( ( N ` i ) mod ( N ` K ) ) ) |
68 |
15 67
|
syl |
|- ( ( ( ph /\ K e. NN0 ) /\ i e. NN0 ) -> ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` ( N ` i ) ) = ( ( N ` i ) mod ( N ` K ) ) ) |
69 |
|
fveq2 |
|- ( k = i -> ( N ` k ) = ( N ` i ) ) |
70 |
69
|
oveq1d |
|- ( k = i -> ( ( N ` k ) mod ( N ` K ) ) = ( ( N ` i ) mod ( N ` K ) ) ) |
71 |
|
eqid |
|- ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) = ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) |
72 |
70 71 66
|
fvmpt |
|- ( i e. NN0 -> ( ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) ` i ) = ( ( N ` i ) mod ( N ` K ) ) ) |
73 |
72
|
adantl |
|- ( ( ( ph /\ K e. NN0 ) /\ i e. NN0 ) -> ( ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) ` i ) = ( ( N ` i ) mod ( N ` K ) ) ) |
74 |
68 73
|
eqtr4d |
|- ( ( ( ph /\ K e. NN0 ) /\ i e. NN0 ) -> ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` ( N ` i ) ) = ( ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) ` i ) ) |
75 |
12 74
|
sylan2 |
|- ( ( ( ph /\ K e. NN0 ) /\ i e. ( 0 ... ( deg ` F ) ) ) -> ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` ( N ` i ) ) = ( ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) ` i ) ) |
76 |
11 16 22 64 75
|
seqhomo |
|- ( ( ph /\ K e. NN0 ) -> ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` ( seq 0 ( x. , N ) ` ( deg ` F ) ) ) = ( seq 0 ( P , ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) ) ` ( deg ` F ) ) ) |
77 |
9 76
|
syl5eq |
|- ( ( ph /\ K e. NN0 ) -> ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` R ) = ( seq 0 ( P , ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) ) ` ( deg ` F ) ) ) |
78 |
8 77
|
sylan2 |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` R ) = ( seq 0 ( P , ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) ) ` ( deg ` F ) ) ) |
79 |
|
0zd |
|- ( ph -> 0 e. ZZ ) |
80 |
10
|
adantl |
|- ( ( ph /\ ( i e. NN /\ j e. NN ) ) -> ( i x. j ) e. NN ) |
81 |
20 79 14 80
|
seqf |
|- ( ph -> seq 0 ( x. , N ) : NN0 --> NN ) |
82 |
81 19
|
ffvelrnd |
|- ( ph -> ( seq 0 ( x. , N ) ` ( deg ` F ) ) e. NN ) |
83 |
6 82
|
eqeltrid |
|- ( ph -> R e. NN ) |
84 |
83
|
adantr |
|- ( ( ph /\ K e. NN0 ) -> R e. NN ) |
85 |
|
oveq1 |
|- ( k = R -> ( k mod ( N ` K ) ) = ( R mod ( N ` K ) ) ) |
86 |
|
ovex |
|- ( R mod ( N ` K ) ) e. _V |
87 |
85 45 86
|
fvmpt |
|- ( R e. NN -> ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` R ) = ( R mod ( N ` K ) ) ) |
88 |
84 87
|
syl |
|- ( ( ph /\ K e. NN0 ) -> ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` R ) = ( R mod ( N ` K ) ) ) |
89 |
8 88
|
sylan2 |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( ( k e. NN |-> ( k mod ( N ` K ) ) ) ` R ) = ( R mod ( N ` K ) ) ) |
90 |
|
oveq12 |
|- ( ( x = i /\ y = j ) -> ( x x. y ) = ( i x. j ) ) |
91 |
90
|
oveq1d |
|- ( ( x = i /\ y = j ) -> ( ( x x. y ) mod ( N ` K ) ) = ( ( i x. j ) mod ( N ` K ) ) ) |
92 |
91 7 61
|
ovmpoa |
|- ( ( i e. _V /\ j e. _V ) -> ( i P j ) = ( ( i x. j ) mod ( N ` K ) ) ) |
93 |
92
|
el2v |
|- ( i P j ) = ( ( i x. j ) mod ( N ` K ) ) |
94 |
|
nn0mulcl |
|- ( ( i e. NN0 /\ j e. NN0 ) -> ( i x. j ) e. NN0 ) |
95 |
94
|
nn0zd |
|- ( ( i e. NN0 /\ j e. NN0 ) -> ( i x. j ) e. ZZ ) |
96 |
8 26
|
sylan2 |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( N ` K ) e. NN ) |
97 |
|
zmodcl |
|- ( ( ( i x. j ) e. ZZ /\ ( N ` K ) e. NN ) -> ( ( i x. j ) mod ( N ` K ) ) e. NN0 ) |
98 |
95 96 97
|
syl2anr |
|- ( ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) /\ ( i e. NN0 /\ j e. NN0 ) ) -> ( ( i x. j ) mod ( N ` K ) ) e. NN0 ) |
99 |
93 98
|
eqeltrid |
|- ( ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) /\ ( i e. NN0 /\ j e. NN0 ) ) -> ( i P j ) e. NN0 ) |
100 |
|
fveq2 |
|- ( k = m -> ( B ` k ) = ( B ` m ) ) |
101 |
100
|
oveq1d |
|- ( k = m -> ( ( B ` k ) x. n ) = ( ( B ` m ) x. n ) ) |
102 |
101
|
eleq1d |
|- ( k = m -> ( ( ( B ` k ) x. n ) e. ZZ <-> ( ( B ` m ) x. n ) e. ZZ ) ) |
103 |
102
|
rabbidv |
|- ( k = m -> { n e. NN | ( ( B ` k ) x. n ) e. ZZ } = { n e. NN | ( ( B ` m ) x. n ) e. ZZ } ) |
104 |
103
|
infeq1d |
|- ( k = m -> inf ( { n e. NN | ( ( B ` k ) x. n ) e. ZZ } , RR , < ) = inf ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } , RR , < ) ) |
105 |
104
|
cbvmptv |
|- ( k e. NN0 |-> inf ( { n e. NN | ( ( B ` k ) x. n ) e. ZZ } , RR , < ) ) = ( m e. NN0 |-> inf ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } , RR , < ) ) |
106 |
5 105
|
eqtri |
|- N = ( m e. NN0 |-> inf ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } , RR , < ) ) |
107 |
1 2 3 4 106 6
|
elqaalem1 |
|- ( ( ph /\ k e. NN0 ) -> ( ( N ` k ) e. NN /\ ( ( B ` k ) x. ( N ` k ) ) e. ZZ ) ) |
108 |
107
|
simpld |
|- ( ( ph /\ k e. NN0 ) -> ( N ` k ) e. NN ) |
109 |
108
|
adantlr |
|- ( ( ( ph /\ K e. NN0 ) /\ k e. NN0 ) -> ( N ` k ) e. NN ) |
110 |
109
|
nnzd |
|- ( ( ( ph /\ K e. NN0 ) /\ k e. NN0 ) -> ( N ` k ) e. ZZ ) |
111 |
26
|
adantr |
|- ( ( ( ph /\ K e. NN0 ) /\ k e. NN0 ) -> ( N ` K ) e. NN ) |
112 |
110 111
|
zmodcld |
|- ( ( ( ph /\ K e. NN0 ) /\ k e. NN0 ) -> ( ( N ` k ) mod ( N ` K ) ) e. NN0 ) |
113 |
112
|
fmpttd |
|- ( ( ph /\ K e. NN0 ) -> ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) : NN0 --> NN0 ) |
114 |
8 113
|
sylan2 |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) : NN0 --> NN0 ) |
115 |
|
ffvelrn |
|- ( ( ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) : NN0 --> NN0 /\ i e. NN0 ) -> ( ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) ` i ) e. NN0 ) |
116 |
114 12 115
|
syl2an |
|- ( ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) /\ i e. ( 0 ... ( deg ` F ) ) ) -> ( ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) ` i ) e. NN0 ) |
117 |
|
c0ex |
|- 0 e. _V |
118 |
|
vex |
|- i e. _V |
119 |
|
oveq12 |
|- ( ( x = 0 /\ y = i ) -> ( x x. y ) = ( 0 x. i ) ) |
120 |
119
|
oveq1d |
|- ( ( x = 0 /\ y = i ) -> ( ( x x. y ) mod ( N ` K ) ) = ( ( 0 x. i ) mod ( N ` K ) ) ) |
121 |
|
ovex |
|- ( ( 0 x. i ) mod ( N ` K ) ) e. _V |
122 |
120 7 121
|
ovmpoa |
|- ( ( 0 e. _V /\ i e. _V ) -> ( 0 P i ) = ( ( 0 x. i ) mod ( N ` K ) ) ) |
123 |
117 118 122
|
mp2an |
|- ( 0 P i ) = ( ( 0 x. i ) mod ( N ` K ) ) |
124 |
|
nn0cn |
|- ( i e. NN0 -> i e. CC ) |
125 |
124
|
mul02d |
|- ( i e. NN0 -> ( 0 x. i ) = 0 ) |
126 |
125
|
oveq1d |
|- ( i e. NN0 -> ( ( 0 x. i ) mod ( N ` K ) ) = ( 0 mod ( N ` K ) ) ) |
127 |
96
|
nnrpd |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( N ` K ) e. RR+ ) |
128 |
|
0mod |
|- ( ( N ` K ) e. RR+ -> ( 0 mod ( N ` K ) ) = 0 ) |
129 |
127 128
|
syl |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( 0 mod ( N ` K ) ) = 0 ) |
130 |
126 129
|
sylan9eqr |
|- ( ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) /\ i e. NN0 ) -> ( ( 0 x. i ) mod ( N ` K ) ) = 0 ) |
131 |
123 130
|
syl5eq |
|- ( ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) /\ i e. NN0 ) -> ( 0 P i ) = 0 ) |
132 |
|
oveq12 |
|- ( ( x = i /\ y = 0 ) -> ( x x. y ) = ( i x. 0 ) ) |
133 |
132
|
oveq1d |
|- ( ( x = i /\ y = 0 ) -> ( ( x x. y ) mod ( N ` K ) ) = ( ( i x. 0 ) mod ( N ` K ) ) ) |
134 |
|
ovex |
|- ( ( i x. 0 ) mod ( N ` K ) ) e. _V |
135 |
133 7 134
|
ovmpoa |
|- ( ( i e. _V /\ 0 e. _V ) -> ( i P 0 ) = ( ( i x. 0 ) mod ( N ` K ) ) ) |
136 |
118 117 135
|
mp2an |
|- ( i P 0 ) = ( ( i x. 0 ) mod ( N ` K ) ) |
137 |
124
|
mul01d |
|- ( i e. NN0 -> ( i x. 0 ) = 0 ) |
138 |
137
|
oveq1d |
|- ( i e. NN0 -> ( ( i x. 0 ) mod ( N ` K ) ) = ( 0 mod ( N ` K ) ) ) |
139 |
138 129
|
sylan9eqr |
|- ( ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) /\ i e. NN0 ) -> ( ( i x. 0 ) mod ( N ` K ) ) = 0 ) |
140 |
136 139
|
syl5eq |
|- ( ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) /\ i e. NN0 ) -> ( i P 0 ) = 0 ) |
141 |
|
simpr |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> K e. ( 0 ... ( deg ` F ) ) ) |
142 |
19
|
adantr |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( deg ` F ) e. NN0 ) |
143 |
8
|
adantl |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> K e. NN0 ) |
144 |
|
fveq2 |
|- ( k = K -> ( N ` k ) = ( N ` K ) ) |
145 |
144
|
oveq1d |
|- ( k = K -> ( ( N ` k ) mod ( N ` K ) ) = ( ( N ` K ) mod ( N ` K ) ) ) |
146 |
|
ovex |
|- ( ( N ` K ) mod ( N ` K ) ) e. _V |
147 |
145 71 146
|
fvmpt |
|- ( K e. NN0 -> ( ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) ` K ) = ( ( N ` K ) mod ( N ` K ) ) ) |
148 |
143 147
|
syl |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) ` K ) = ( ( N ` K ) mod ( N ` K ) ) ) |
149 |
96
|
nncnd |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( N ` K ) e. CC ) |
150 |
96
|
nnne0d |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( N ` K ) =/= 0 ) |
151 |
149 150
|
dividd |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( ( N ` K ) / ( N ` K ) ) = 1 ) |
152 |
|
1z |
|- 1 e. ZZ |
153 |
151 152
|
eqeltrdi |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( ( N ` K ) / ( N ` K ) ) e. ZZ ) |
154 |
96
|
nnred |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( N ` K ) e. RR ) |
155 |
|
mod0 |
|- ( ( ( N ` K ) e. RR /\ ( N ` K ) e. RR+ ) -> ( ( ( N ` K ) mod ( N ` K ) ) = 0 <-> ( ( N ` K ) / ( N ` K ) ) e. ZZ ) ) |
156 |
154 127 155
|
syl2anc |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( ( ( N ` K ) mod ( N ` K ) ) = 0 <-> ( ( N ` K ) / ( N ` K ) ) e. ZZ ) ) |
157 |
153 156
|
mpbird |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( ( N ` K ) mod ( N ` K ) ) = 0 ) |
158 |
148 157
|
eqtrd |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) ` K ) = 0 ) |
159 |
99 116 131 140 141 142 158
|
seqz |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( seq 0 ( P , ( k e. NN0 |-> ( ( N ` k ) mod ( N ` K ) ) ) ) ` ( deg ` F ) ) = 0 ) |
160 |
78 89 159
|
3eqtr3d |
|- ( ( ph /\ K e. ( 0 ... ( deg ` F ) ) ) -> ( R mod ( N ` K ) ) = 0 ) |