Step |
Hyp |
Ref |
Expression |
1 |
|
ply1coe.p |
|- P = ( Poly1 ` R ) |
2 |
|
ply1coe.x |
|- X = ( var1 ` R ) |
3 |
|
ply1coe.b |
|- B = ( Base ` P ) |
4 |
|
ply1coe.n |
|- .x. = ( .s ` P ) |
5 |
|
ply1coe.m |
|- M = ( mulGrp ` P ) |
6 |
|
ply1coe.e |
|- .^ = ( .g ` M ) |
7 |
|
ply1coe.a |
|- A = ( coe1 ` K ) |
8 |
|
eqid |
|- ( 1o mPoly R ) = ( 1o mPoly R ) |
9 |
|
psr1baslem |
|- ( NN0 ^m 1o ) = { d e. ( NN0 ^m 1o ) | ( `' d " NN ) e. Fin } |
10 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
11 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
12 |
|
1onn |
|- 1o e. _om |
13 |
12
|
a1i |
|- ( ( R e. Ring /\ K e. B ) -> 1o e. _om ) |
14 |
1 3
|
ply1bas |
|- B = ( Base ` ( 1o mPoly R ) ) |
15 |
1 8 4
|
ply1vsca |
|- .x. = ( .s ` ( 1o mPoly R ) ) |
16 |
|
simpl |
|- ( ( R e. Ring /\ K e. B ) -> R e. Ring ) |
17 |
|
simpr |
|- ( ( R e. Ring /\ K e. B ) -> K e. B ) |
18 |
8 9 10 11 13 14 15 16 17
|
mplcoe1 |
|- ( ( R e. Ring /\ K e. B ) -> K = ( ( 1o mPoly R ) gsum ( a e. ( NN0 ^m 1o ) |-> ( ( K ` a ) .x. ( b e. ( NN0 ^m 1o ) |-> if ( b = a , ( 1r ` R ) , ( 0g ` R ) ) ) ) ) ) ) |
19 |
7
|
fvcoe1 |
|- ( ( K e. B /\ a e. ( NN0 ^m 1o ) ) -> ( K ` a ) = ( A ` ( a ` (/) ) ) ) |
20 |
19
|
adantll |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( K ` a ) = ( A ` ( a ` (/) ) ) ) |
21 |
12
|
a1i |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> 1o e. _om ) |
22 |
|
eqid |
|- ( mulGrp ` ( 1o mPoly R ) ) = ( mulGrp ` ( 1o mPoly R ) ) |
23 |
|
eqid |
|- ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) = ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) |
24 |
|
eqid |
|- ( 1o mVar R ) = ( 1o mVar R ) |
25 |
|
simpll |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> R e. Ring ) |
26 |
|
simpr |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> a e. ( NN0 ^m 1o ) ) |
27 |
|
eqidd |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( ( ( 1o mVar R ) ` (/) ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) ) |
28 |
|
0ex |
|- (/) e. _V |
29 |
|
fveq2 |
|- ( b = (/) -> ( ( 1o mVar R ) ` b ) = ( ( 1o mVar R ) ` (/) ) ) |
30 |
29
|
oveq1d |
|- ( b = (/) -> ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) ) |
31 |
29
|
oveq2d |
|- ( b = (/) -> ( ( ( 1o mVar R ) ` (/) ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) ) |
32 |
30 31
|
eqeq12d |
|- ( b = (/) -> ( ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) <-> ( ( ( 1o mVar R ) ` (/) ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) ) ) |
33 |
28 32
|
ralsn |
|- ( A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) <-> ( ( ( 1o mVar R ) ` (/) ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) ) |
34 |
27 33
|
sylibr |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) ) |
35 |
|
fveq2 |
|- ( x = (/) -> ( ( 1o mVar R ) ` x ) = ( ( 1o mVar R ) ` (/) ) ) |
36 |
35
|
oveq2d |
|- ( x = (/) -> ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) ) |
37 |
35
|
oveq1d |
|- ( x = (/) -> ( ( ( 1o mVar R ) ` x ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) ) |
38 |
36 37
|
eqeq12d |
|- ( x = (/) -> ( ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) <-> ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) ) ) |
39 |
38
|
ralbidv |
|- ( x = (/) -> ( A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) <-> A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) ) ) |
40 |
28 39
|
ralsn |
|- ( A. x e. { (/) } A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) <-> A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) ) |
41 |
34 40
|
sylibr |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> A. x e. { (/) } A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) ) |
42 |
|
df1o2 |
|- 1o = { (/) } |
43 |
42
|
raleqi |
|- ( A. b e. 1o ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) <-> A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) ) |
44 |
42 43
|
raleqbii |
|- ( A. x e. 1o A. b e. 1o ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) <-> A. x e. { (/) } A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) ) |
45 |
41 44
|
sylibr |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> A. x e. 1o A. b e. 1o ( ( ( 1o mVar R ) ` b ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) ) |
46 |
8 9 10 11 21 22 23 24 25 26 45
|
mplcoe5 |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( b e. ( NN0 ^m 1o ) |-> if ( b = a , ( 1r ` R ) , ( 0g ` R ) ) ) = ( ( mulGrp ` ( 1o mPoly R ) ) gsum ( c e. 1o |-> ( ( a ` c ) ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) ) ) ) |
47 |
42
|
mpteq1i |
|- ( c e. 1o |-> ( ( a ` c ) ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) ) = ( c e. { (/) } |-> ( ( a ` c ) ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) ) |
48 |
47
|
oveq2i |
|- ( ( mulGrp ` ( 1o mPoly R ) ) gsum ( c e. 1o |-> ( ( a ` c ) ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) ) ) = ( ( mulGrp ` ( 1o mPoly R ) ) gsum ( c e. { (/) } |-> ( ( a ` c ) ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) ) ) |
49 |
8
|
mplring |
|- ( ( 1o e. _om /\ R e. Ring ) -> ( 1o mPoly R ) e. Ring ) |
50 |
12 49
|
mpan |
|- ( R e. Ring -> ( 1o mPoly R ) e. Ring ) |
51 |
22
|
ringmgp |
|- ( ( 1o mPoly R ) e. Ring -> ( mulGrp ` ( 1o mPoly R ) ) e. Mnd ) |
52 |
50 51
|
syl |
|- ( R e. Ring -> ( mulGrp ` ( 1o mPoly R ) ) e. Mnd ) |
53 |
52
|
ad2antrr |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( mulGrp ` ( 1o mPoly R ) ) e. Mnd ) |
54 |
28
|
a1i |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> (/) e. _V ) |
55 |
22 14
|
mgpbas |
|- B = ( Base ` ( mulGrp ` ( 1o mPoly R ) ) ) |
56 |
55
|
a1i |
|- ( ( R e. Ring /\ K e. B ) -> B = ( Base ` ( mulGrp ` ( 1o mPoly R ) ) ) ) |
57 |
5 3
|
mgpbas |
|- B = ( Base ` M ) |
58 |
57
|
a1i |
|- ( ( R e. Ring /\ K e. B ) -> B = ( Base ` M ) ) |
59 |
|
ssv |
|- B C_ _V |
60 |
59
|
a1i |
|- ( ( R e. Ring /\ K e. B ) -> B C_ _V ) |
61 |
|
ovexd |
|- ( ( ( R e. Ring /\ K e. B ) /\ ( a e. _V /\ b e. _V ) ) -> ( a ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) b ) e. _V ) |
62 |
|
eqid |
|- ( .r ` P ) = ( .r ` P ) |
63 |
1 8 62
|
ply1mulr |
|- ( .r ` P ) = ( .r ` ( 1o mPoly R ) ) |
64 |
22 63
|
mgpplusg |
|- ( .r ` P ) = ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) |
65 |
5 62
|
mgpplusg |
|- ( .r ` P ) = ( +g ` M ) |
66 |
64 65
|
eqtr3i |
|- ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) = ( +g ` M ) |
67 |
66
|
oveqi |
|- ( a ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) b ) = ( a ( +g ` M ) b ) |
68 |
67
|
a1i |
|- ( ( ( R e. Ring /\ K e. B ) /\ ( a e. _V /\ b e. _V ) ) -> ( a ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) b ) = ( a ( +g ` M ) b ) ) |
69 |
23 6 56 58 60 61 68
|
mulgpropd |
|- ( ( R e. Ring /\ K e. B ) -> ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) = .^ ) |
70 |
69
|
oveqd |
|- ( ( R e. Ring /\ K e. B ) -> ( ( a ` (/) ) ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) X ) = ( ( a ` (/) ) .^ X ) ) |
71 |
70
|
adantr |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( ( a ` (/) ) ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) X ) = ( ( a ` (/) ) .^ X ) ) |
72 |
1
|
ply1ring |
|- ( R e. Ring -> P e. Ring ) |
73 |
5
|
ringmgp |
|- ( P e. Ring -> M e. Mnd ) |
74 |
72 73
|
syl |
|- ( R e. Ring -> M e. Mnd ) |
75 |
74
|
ad2antrr |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> M e. Mnd ) |
76 |
|
elmapi |
|- ( a e. ( NN0 ^m 1o ) -> a : 1o --> NN0 ) |
77 |
|
0lt1o |
|- (/) e. 1o |
78 |
|
ffvelcdm |
|- ( ( a : 1o --> NN0 /\ (/) e. 1o ) -> ( a ` (/) ) e. NN0 ) |
79 |
76 77 78
|
sylancl |
|- ( a e. ( NN0 ^m 1o ) -> ( a ` (/) ) e. NN0 ) |
80 |
79
|
adantl |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( a ` (/) ) e. NN0 ) |
81 |
2 1 3
|
vr1cl |
|- ( R e. Ring -> X e. B ) |
82 |
81
|
ad2antrr |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> X e. B ) |
83 |
57 6 75 80 82
|
mulgnn0cld |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( ( a ` (/) ) .^ X ) e. B ) |
84 |
71 83
|
eqeltrd |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( ( a ` (/) ) ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) X ) e. B ) |
85 |
|
fveq2 |
|- ( c = (/) -> ( a ` c ) = ( a ` (/) ) ) |
86 |
|
fveq2 |
|- ( c = (/) -> ( ( 1o mVar R ) ` c ) = ( ( 1o mVar R ) ` (/) ) ) |
87 |
2
|
vr1val |
|- X = ( ( 1o mVar R ) ` (/) ) |
88 |
86 87
|
eqtr4di |
|- ( c = (/) -> ( ( 1o mVar R ) ` c ) = X ) |
89 |
85 88
|
oveq12d |
|- ( c = (/) -> ( ( a ` c ) ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) = ( ( a ` (/) ) ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) X ) ) |
90 |
55 89
|
gsumsn |
|- ( ( ( mulGrp ` ( 1o mPoly R ) ) e. Mnd /\ (/) e. _V /\ ( ( a ` (/) ) ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) X ) e. B ) -> ( ( mulGrp ` ( 1o mPoly R ) ) gsum ( c e. { (/) } |-> ( ( a ` c ) ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) ) ) = ( ( a ` (/) ) ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) X ) ) |
91 |
53 54 84 90
|
syl3anc |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( ( mulGrp ` ( 1o mPoly R ) ) gsum ( c e. { (/) } |-> ( ( a ` c ) ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) ) ) = ( ( a ` (/) ) ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) X ) ) |
92 |
48 91
|
eqtrid |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( ( mulGrp ` ( 1o mPoly R ) ) gsum ( c e. 1o |-> ( ( a ` c ) ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) ) ) = ( ( a ` (/) ) ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) X ) ) |
93 |
46 92 71
|
3eqtrd |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( b e. ( NN0 ^m 1o ) |-> if ( b = a , ( 1r ` R ) , ( 0g ` R ) ) ) = ( ( a ` (/) ) .^ X ) ) |
94 |
20 93
|
oveq12d |
|- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( ( K ` a ) .x. ( b e. ( NN0 ^m 1o ) |-> if ( b = a , ( 1r ` R ) , ( 0g ` R ) ) ) ) = ( ( A ` ( a ` (/) ) ) .x. ( ( a ` (/) ) .^ X ) ) ) |
95 |
94
|
mpteq2dva |
|- ( ( R e. Ring /\ K e. B ) -> ( a e. ( NN0 ^m 1o ) |-> ( ( K ` a ) .x. ( b e. ( NN0 ^m 1o ) |-> if ( b = a , ( 1r ` R ) , ( 0g ` R ) ) ) ) ) = ( a e. ( NN0 ^m 1o ) |-> ( ( A ` ( a ` (/) ) ) .x. ( ( a ` (/) ) .^ X ) ) ) ) |
96 |
95
|
oveq2d |
|- ( ( R e. Ring /\ K e. B ) -> ( ( 1o mPoly R ) gsum ( a e. ( NN0 ^m 1o ) |-> ( ( K ` a ) .x. ( b e. ( NN0 ^m 1o ) |-> if ( b = a , ( 1r ` R ) , ( 0g ` R ) ) ) ) ) ) = ( ( 1o mPoly R ) gsum ( a e. ( NN0 ^m 1o ) |-> ( ( A ` ( a ` (/) ) ) .x. ( ( a ` (/) ) .^ X ) ) ) ) ) |
97 |
|
nn0ex |
|- NN0 e. _V |
98 |
97
|
mptex |
|- ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) e. _V |
99 |
98
|
a1i |
|- ( ( R e. Ring /\ K e. B ) -> ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) e. _V ) |
100 |
1
|
fvexi |
|- P e. _V |
101 |
100
|
a1i |
|- ( ( R e. Ring /\ K e. B ) -> P e. _V ) |
102 |
|
ovexd |
|- ( ( R e. Ring /\ K e. B ) -> ( 1o mPoly R ) e. _V ) |
103 |
3 14
|
eqtr3i |
|- ( Base ` P ) = ( Base ` ( 1o mPoly R ) ) |
104 |
103
|
a1i |
|- ( ( R e. Ring /\ K e. B ) -> ( Base ` P ) = ( Base ` ( 1o mPoly R ) ) ) |
105 |
|
eqid |
|- ( +g ` P ) = ( +g ` P ) |
106 |
1 8 105
|
ply1plusg |
|- ( +g ` P ) = ( +g ` ( 1o mPoly R ) ) |
107 |
106
|
a1i |
|- ( ( R e. Ring /\ K e. B ) -> ( +g ` P ) = ( +g ` ( 1o mPoly R ) ) ) |
108 |
99 101 102 104 107
|
gsumpropd |
|- ( ( R e. Ring /\ K e. B ) -> ( P gsum ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) ) = ( ( 1o mPoly R ) gsum ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) ) ) |
109 |
|
eqid |
|- ( 0g ` P ) = ( 0g ` P ) |
110 |
8 1 109
|
ply1mpl0 |
|- ( 0g ` P ) = ( 0g ` ( 1o mPoly R ) ) |
111 |
8
|
mpllmod |
|- ( ( 1o e. _om /\ R e. Ring ) -> ( 1o mPoly R ) e. LMod ) |
112 |
12 16 111
|
sylancr |
|- ( ( R e. Ring /\ K e. B ) -> ( 1o mPoly R ) e. LMod ) |
113 |
|
lmodcmn |
|- ( ( 1o mPoly R ) e. LMod -> ( 1o mPoly R ) e. CMnd ) |
114 |
112 113
|
syl |
|- ( ( R e. Ring /\ K e. B ) -> ( 1o mPoly R ) e. CMnd ) |
115 |
97
|
a1i |
|- ( ( R e. Ring /\ K e. B ) -> NN0 e. _V ) |
116 |
1
|
ply1lmod |
|- ( R e. Ring -> P e. LMod ) |
117 |
116
|
ad2antrr |
|- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> P e. LMod ) |
118 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
119 |
7 3 1 118
|
coe1f |
|- ( K e. B -> A : NN0 --> ( Base ` R ) ) |
120 |
119
|
adantl |
|- ( ( R e. Ring /\ K e. B ) -> A : NN0 --> ( Base ` R ) ) |
121 |
120
|
ffvelcdmda |
|- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> ( A ` k ) e. ( Base ` R ) ) |
122 |
1
|
ply1sca |
|- ( R e. Ring -> R = ( Scalar ` P ) ) |
123 |
122
|
eqcomd |
|- ( R e. Ring -> ( Scalar ` P ) = R ) |
124 |
123
|
ad2antrr |
|- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> ( Scalar ` P ) = R ) |
125 |
124
|
fveq2d |
|- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> ( Base ` ( Scalar ` P ) ) = ( Base ` R ) ) |
126 |
121 125
|
eleqtrrd |
|- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> ( A ` k ) e. ( Base ` ( Scalar ` P ) ) ) |
127 |
74
|
ad2antrr |
|- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> M e. Mnd ) |
128 |
|
simpr |
|- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> k e. NN0 ) |
129 |
81
|
ad2antrr |
|- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> X e. B ) |
130 |
57 6 127 128 129
|
mulgnn0cld |
|- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> ( k .^ X ) e. B ) |
131 |
|
eqid |
|- ( Scalar ` P ) = ( Scalar ` P ) |
132 |
|
eqid |
|- ( Base ` ( Scalar ` P ) ) = ( Base ` ( Scalar ` P ) ) |
133 |
3 131 4 132
|
lmodvscl |
|- ( ( P e. LMod /\ ( A ` k ) e. ( Base ` ( Scalar ` P ) ) /\ ( k .^ X ) e. B ) -> ( ( A ` k ) .x. ( k .^ X ) ) e. B ) |
134 |
117 126 130 133
|
syl3anc |
|- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> ( ( A ` k ) .x. ( k .^ X ) ) e. B ) |
135 |
134
|
fmpttd |
|- ( ( R e. Ring /\ K e. B ) -> ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) : NN0 --> B ) |
136 |
1 2 3 4 5 6 7
|
ply1coefsupp |
|- ( ( R e. Ring /\ K e. B ) -> ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) finSupp ( 0g ` P ) ) |
137 |
|
eqid |
|- ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) = ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) |
138 |
42 97 28 137
|
mapsnf1o2 |
|- ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) : ( NN0 ^m 1o ) -1-1-onto-> NN0 |
139 |
138
|
a1i |
|- ( ( R e. Ring /\ K e. B ) -> ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) : ( NN0 ^m 1o ) -1-1-onto-> NN0 ) |
140 |
14 110 114 115 135 136 139
|
gsumf1o |
|- ( ( R e. Ring /\ K e. B ) -> ( ( 1o mPoly R ) gsum ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) ) = ( ( 1o mPoly R ) gsum ( ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) o. ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ) ) ) |
141 |
|
eqidd |
|- ( ( R e. Ring /\ K e. B ) -> ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) = ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ) |
142 |
|
eqidd |
|- ( ( R e. Ring /\ K e. B ) -> ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) = ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) ) |
143 |
|
fveq2 |
|- ( k = ( a ` (/) ) -> ( A ` k ) = ( A ` ( a ` (/) ) ) ) |
144 |
|
oveq1 |
|- ( k = ( a ` (/) ) -> ( k .^ X ) = ( ( a ` (/) ) .^ X ) ) |
145 |
143 144
|
oveq12d |
|- ( k = ( a ` (/) ) -> ( ( A ` k ) .x. ( k .^ X ) ) = ( ( A ` ( a ` (/) ) ) .x. ( ( a ` (/) ) .^ X ) ) ) |
146 |
80 141 142 145
|
fmptco |
|- ( ( R e. Ring /\ K e. B ) -> ( ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) o. ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ) = ( a e. ( NN0 ^m 1o ) |-> ( ( A ` ( a ` (/) ) ) .x. ( ( a ` (/) ) .^ X ) ) ) ) |
147 |
146
|
oveq2d |
|- ( ( R e. Ring /\ K e. B ) -> ( ( 1o mPoly R ) gsum ( ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) o. ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ) ) = ( ( 1o mPoly R ) gsum ( a e. ( NN0 ^m 1o ) |-> ( ( A ` ( a ` (/) ) ) .x. ( ( a ` (/) ) .^ X ) ) ) ) ) |
148 |
108 140 147
|
3eqtrrd |
|- ( ( R e. Ring /\ K e. B ) -> ( ( 1o mPoly R ) gsum ( a e. ( NN0 ^m 1o ) |-> ( ( A ` ( a ` (/) ) ) .x. ( ( a ` (/) ) .^ X ) ) ) ) = ( P gsum ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) ) ) |
149 |
18 96 148
|
3eqtrd |
|- ( ( R e. Ring /\ K e. B ) -> K = ( P gsum ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) ) ) |