Step |
Hyp |
Ref |
Expression |
1 |
|
coe1tm.z |
|- .0. = ( 0g ` R ) |
2 |
|
coe1tm.k |
|- K = ( Base ` R ) |
3 |
|
coe1tm.p |
|- P = ( Poly1 ` R ) |
4 |
|
coe1tm.x |
|- X = ( var1 ` R ) |
5 |
|
coe1tm.m |
|- .x. = ( .s ` P ) |
6 |
|
coe1tm.n |
|- N = ( mulGrp ` P ) |
7 |
|
coe1tm.e |
|- .^ = ( .g ` N ) |
8 |
|
eqid |
|- ( Base ` P ) = ( Base ` P ) |
9 |
2 3 4 5 6 7 8
|
ply1tmcl |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( C .x. ( D .^ X ) ) e. ( Base ` P ) ) |
10 |
|
eqid |
|- ( coe1 ` ( C .x. ( D .^ X ) ) ) = ( coe1 ` ( C .x. ( D .^ X ) ) ) |
11 |
|
eqid |
|- ( x e. NN0 |-> ( 1o X. { x } ) ) = ( x e. NN0 |-> ( 1o X. { x } ) ) |
12 |
10 8 3 11
|
coe1fval2 |
|- ( ( C .x. ( D .^ X ) ) e. ( Base ` P ) -> ( coe1 ` ( C .x. ( D .^ X ) ) ) = ( ( C .x. ( D .^ X ) ) o. ( x e. NN0 |-> ( 1o X. { x } ) ) ) ) |
13 |
9 12
|
syl |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( coe1 ` ( C .x. ( D .^ X ) ) ) = ( ( C .x. ( D .^ X ) ) o. ( x e. NN0 |-> ( 1o X. { x } ) ) ) ) |
14 |
|
fconst6g |
|- ( x e. NN0 -> ( 1o X. { x } ) : 1o --> NN0 ) |
15 |
|
nn0ex |
|- NN0 e. _V |
16 |
|
1oex |
|- 1o e. _V |
17 |
15 16
|
elmap |
|- ( ( 1o X. { x } ) e. ( NN0 ^m 1o ) <-> ( 1o X. { x } ) : 1o --> NN0 ) |
18 |
14 17
|
sylibr |
|- ( x e. NN0 -> ( 1o X. { x } ) e. ( NN0 ^m 1o ) ) |
19 |
18
|
adantl |
|- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( 1o X. { x } ) e. ( NN0 ^m 1o ) ) |
20 |
|
eqidd |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( x e. NN0 |-> ( 1o X. { x } ) ) = ( x e. NN0 |-> ( 1o X. { x } ) ) ) |
21 |
|
eqid |
|- ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) = ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) |
22 |
6 8
|
mgpbas |
|- ( Base ` P ) = ( Base ` N ) |
23 |
22
|
a1i |
|- ( R e. Ring -> ( Base ` P ) = ( Base ` N ) ) |
24 |
|
eqid |
|- ( mulGrp ` ( 1o mPoly R ) ) = ( mulGrp ` ( 1o mPoly R ) ) |
25 |
3 8
|
ply1bas |
|- ( Base ` P ) = ( Base ` ( 1o mPoly R ) ) |
26 |
24 25
|
mgpbas |
|- ( Base ` P ) = ( Base ` ( mulGrp ` ( 1o mPoly R ) ) ) |
27 |
26
|
a1i |
|- ( R e. Ring -> ( Base ` P ) = ( Base ` ( mulGrp ` ( 1o mPoly R ) ) ) ) |
28 |
|
ssv |
|- ( Base ` P ) C_ _V |
29 |
28
|
a1i |
|- ( R e. Ring -> ( Base ` P ) C_ _V ) |
30 |
|
ovexd |
|- ( ( R e. Ring /\ ( x e. _V /\ y e. _V ) ) -> ( x ( +g ` N ) y ) e. _V ) |
31 |
|
eqid |
|- ( .r ` P ) = ( .r ` P ) |
32 |
6 31
|
mgpplusg |
|- ( .r ` P ) = ( +g ` N ) |
33 |
|
eqid |
|- ( 1o mPoly R ) = ( 1o mPoly R ) |
34 |
3 33 31
|
ply1mulr |
|- ( .r ` P ) = ( .r ` ( 1o mPoly R ) ) |
35 |
24 34
|
mgpplusg |
|- ( .r ` P ) = ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) |
36 |
32 35
|
eqtr3i |
|- ( +g ` N ) = ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) |
37 |
36
|
a1i |
|- ( R e. Ring -> ( +g ` N ) = ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) ) |
38 |
37
|
oveqdr |
|- ( ( R e. Ring /\ ( x e. _V /\ y e. _V ) ) -> ( x ( +g ` N ) y ) = ( x ( +g ` ( mulGrp ` ( 1o mPoly R ) ) ) y ) ) |
39 |
7 21 23 27 29 30 38
|
mulgpropd |
|- ( R e. Ring -> .^ = ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) ) |
40 |
39
|
3ad2ant1 |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> .^ = ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) ) |
41 |
|
eqidd |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> D = D ) |
42 |
4
|
vr1val |
|- X = ( ( 1o mVar R ) ` (/) ) |
43 |
42
|
a1i |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> X = ( ( 1o mVar R ) ` (/) ) ) |
44 |
40 41 43
|
oveq123d |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( D .^ X ) = ( D ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) ) |
45 |
44
|
oveq2d |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( C .x. ( D .^ X ) ) = ( C .x. ( D ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) ) ) |
46 |
|
psr1baslem |
|- ( NN0 ^m 1o ) = { a e. ( NN0 ^m 1o ) | ( `' a " NN ) e. Fin } |
47 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
48 |
|
1on |
|- 1o e. On |
49 |
48
|
a1i |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> 1o e. On ) |
50 |
|
eqid |
|- ( 1o mVar R ) = ( 1o mVar R ) |
51 |
|
simp1 |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> R e. Ring ) |
52 |
|
0lt1o |
|- (/) e. 1o |
53 |
52
|
a1i |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> (/) e. 1o ) |
54 |
|
simp3 |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> D e. NN0 ) |
55 |
33 46 1 47 49 24 21 50 51 53 54
|
mplcoe3 |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( y e. ( NN0 ^m 1o ) |-> if ( y = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , ( 1r ` R ) , .0. ) ) = ( D ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) ) |
56 |
55
|
oveq2d |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( C .x. ( y e. ( NN0 ^m 1o ) |-> if ( y = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , ( 1r ` R ) , .0. ) ) ) = ( C .x. ( D ( .g ` ( mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) ) ) |
57 |
3 33 5
|
ply1vsca |
|- .x. = ( .s ` ( 1o mPoly R ) ) |
58 |
|
elsni |
|- ( b e. { (/) } -> b = (/) ) |
59 |
|
df1o2 |
|- 1o = { (/) } |
60 |
58 59
|
eleq2s |
|- ( b e. 1o -> b = (/) ) |
61 |
60
|
iftrued |
|- ( b e. 1o -> if ( b = (/) , D , 0 ) = D ) |
62 |
61
|
adantl |
|- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ b e. 1o ) -> if ( b = (/) , D , 0 ) = D ) |
63 |
62
|
mpteq2dva |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( b e. 1o |-> if ( b = (/) , D , 0 ) ) = ( b e. 1o |-> D ) ) |
64 |
|
fconstmpt |
|- ( 1o X. { D } ) = ( b e. 1o |-> D ) |
65 |
63 64
|
eqtr4di |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( b e. 1o |-> if ( b = (/) , D , 0 ) ) = ( 1o X. { D } ) ) |
66 |
|
fconst6g |
|- ( D e. NN0 -> ( 1o X. { D } ) : 1o --> NN0 ) |
67 |
15 16
|
elmap |
|- ( ( 1o X. { D } ) e. ( NN0 ^m 1o ) <-> ( 1o X. { D } ) : 1o --> NN0 ) |
68 |
66 67
|
sylibr |
|- ( D e. NN0 -> ( 1o X. { D } ) e. ( NN0 ^m 1o ) ) |
69 |
68
|
3ad2ant3 |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( 1o X. { D } ) e. ( NN0 ^m 1o ) ) |
70 |
65 69
|
eqeltrd |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( b e. 1o |-> if ( b = (/) , D , 0 ) ) e. ( NN0 ^m 1o ) ) |
71 |
|
simp2 |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> C e. K ) |
72 |
33 57 46 47 1 2 49 51 70 71
|
mplmon2 |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( C .x. ( y e. ( NN0 ^m 1o ) |-> if ( y = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , ( 1r ` R ) , .0. ) ) ) = ( y e. ( NN0 ^m 1o ) |-> if ( y = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , C , .0. ) ) ) |
73 |
45 56 72
|
3eqtr2d |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( C .x. ( D .^ X ) ) = ( y e. ( NN0 ^m 1o ) |-> if ( y = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , C , .0. ) ) ) |
74 |
|
eqeq1 |
|- ( y = ( 1o X. { x } ) -> ( y = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) <-> ( 1o X. { x } ) = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) ) ) |
75 |
74
|
ifbid |
|- ( y = ( 1o X. { x } ) -> if ( y = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , C , .0. ) = if ( ( 1o X. { x } ) = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , C , .0. ) ) |
76 |
19 20 73 75
|
fmptco |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( ( C .x. ( D .^ X ) ) o. ( x e. NN0 |-> ( 1o X. { x } ) ) ) = ( x e. NN0 |-> if ( ( 1o X. { x } ) = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , C , .0. ) ) ) |
77 |
65
|
adantr |
|- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( b e. 1o |-> if ( b = (/) , D , 0 ) ) = ( 1o X. { D } ) ) |
78 |
77
|
eqeq2d |
|- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( ( 1o X. { x } ) = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) <-> ( 1o X. { x } ) = ( 1o X. { D } ) ) ) |
79 |
|
fveq1 |
|- ( ( 1o X. { x } ) = ( 1o X. { D } ) -> ( ( 1o X. { x } ) ` (/) ) = ( ( 1o X. { D } ) ` (/) ) ) |
80 |
|
vex |
|- x e. _V |
81 |
80
|
fvconst2 |
|- ( (/) e. 1o -> ( ( 1o X. { x } ) ` (/) ) = x ) |
82 |
52 81
|
mp1i |
|- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( ( 1o X. { x } ) ` (/) ) = x ) |
83 |
|
simpl3 |
|- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> D e. NN0 ) |
84 |
|
fvconst2g |
|- ( ( D e. NN0 /\ (/) e. 1o ) -> ( ( 1o X. { D } ) ` (/) ) = D ) |
85 |
83 52 84
|
sylancl |
|- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( ( 1o X. { D } ) ` (/) ) = D ) |
86 |
82 85
|
eqeq12d |
|- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( ( ( 1o X. { x } ) ` (/) ) = ( ( 1o X. { D } ) ` (/) ) <-> x = D ) ) |
87 |
79 86
|
imbitrid |
|- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( ( 1o X. { x } ) = ( 1o X. { D } ) -> x = D ) ) |
88 |
|
sneq |
|- ( x = D -> { x } = { D } ) |
89 |
88
|
xpeq2d |
|- ( x = D -> ( 1o X. { x } ) = ( 1o X. { D } ) ) |
90 |
87 89
|
impbid1 |
|- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( ( 1o X. { x } ) = ( 1o X. { D } ) <-> x = D ) ) |
91 |
78 90
|
bitrd |
|- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( ( 1o X. { x } ) = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) <-> x = D ) ) |
92 |
91
|
ifbid |
|- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> if ( ( 1o X. { x } ) = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , C , .0. ) = if ( x = D , C , .0. ) ) |
93 |
92
|
mpteq2dva |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( x e. NN0 |-> if ( ( 1o X. { x } ) = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , C , .0. ) ) = ( x e. NN0 |-> if ( x = D , C , .0. ) ) ) |
94 |
13 76 93
|
3eqtrd |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( coe1 ` ( C .x. ( D .^ X ) ) ) = ( x e. NN0 |-> if ( x = D , C , .0. ) ) ) |