Step |
Hyp |
Ref |
Expression |
1 |
|
coe1pwmul.z |
|- .0. = ( 0g ` R ) |
2 |
|
coe1pwmul.p |
|- P = ( Poly1 ` R ) |
3 |
|
coe1pwmul.x |
|- X = ( var1 ` R ) |
4 |
|
coe1pwmul.n |
|- N = ( mulGrp ` P ) |
5 |
|
coe1pwmul.e |
|- .^ = ( .g ` N ) |
6 |
|
coe1pwmul.b |
|- B = ( Base ` P ) |
7 |
|
coe1pwmul.t |
|- .x. = ( .r ` P ) |
8 |
|
coe1pwmul.r |
|- ( ph -> R e. Ring ) |
9 |
|
coe1pwmul.a |
|- ( ph -> A e. B ) |
10 |
|
coe1pwmul.d |
|- ( ph -> D e. NN0 ) |
11 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
12 |
|
eqid |
|- ( .s ` P ) = ( .s ` P ) |
13 |
|
eqid |
|- ( .r ` R ) = ( .r ` R ) |
14 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
15 |
11 14
|
ringidcl |
|- ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) ) |
16 |
8 15
|
syl |
|- ( ph -> ( 1r ` R ) e. ( Base ` R ) ) |
17 |
1 11 2 3 12 4 5 6 7 13 9 8 16 10
|
coe1tmmul |
|- ( ph -> ( coe1 ` ( ( ( 1r ` R ) ( .s ` P ) ( D .^ X ) ) .x. A ) ) = ( x e. NN0 |-> if ( D <_ x , ( ( 1r ` R ) ( .r ` R ) ( ( coe1 ` A ) ` ( x - D ) ) ) , .0. ) ) ) |
18 |
2
|
ply1sca |
|- ( R e. Ring -> R = ( Scalar ` P ) ) |
19 |
8 18
|
syl |
|- ( ph -> R = ( Scalar ` P ) ) |
20 |
19
|
fveq2d |
|- ( ph -> ( 1r ` R ) = ( 1r ` ( Scalar ` P ) ) ) |
21 |
20
|
oveq1d |
|- ( ph -> ( ( 1r ` R ) ( .s ` P ) ( D .^ X ) ) = ( ( 1r ` ( Scalar ` P ) ) ( .s ` P ) ( D .^ X ) ) ) |
22 |
2
|
ply1lmod |
|- ( R e. Ring -> P e. LMod ) |
23 |
8 22
|
syl |
|- ( ph -> P e. LMod ) |
24 |
2
|
ply1ring |
|- ( R e. Ring -> P e. Ring ) |
25 |
4
|
ringmgp |
|- ( P e. Ring -> N e. Mnd ) |
26 |
8 24 25
|
3syl |
|- ( ph -> N e. Mnd ) |
27 |
3 2 6
|
vr1cl |
|- ( R e. Ring -> X e. B ) |
28 |
8 27
|
syl |
|- ( ph -> X e. B ) |
29 |
4 6
|
mgpbas |
|- B = ( Base ` N ) |
30 |
29 5
|
mulgnn0cl |
|- ( ( N e. Mnd /\ D e. NN0 /\ X e. B ) -> ( D .^ X ) e. B ) |
31 |
26 10 28 30
|
syl3anc |
|- ( ph -> ( D .^ X ) e. B ) |
32 |
|
eqid |
|- ( Scalar ` P ) = ( Scalar ` P ) |
33 |
|
eqid |
|- ( 1r ` ( Scalar ` P ) ) = ( 1r ` ( Scalar ` P ) ) |
34 |
6 32 12 33
|
lmodvs1 |
|- ( ( P e. LMod /\ ( D .^ X ) e. B ) -> ( ( 1r ` ( Scalar ` P ) ) ( .s ` P ) ( D .^ X ) ) = ( D .^ X ) ) |
35 |
23 31 34
|
syl2anc |
|- ( ph -> ( ( 1r ` ( Scalar ` P ) ) ( .s ` P ) ( D .^ X ) ) = ( D .^ X ) ) |
36 |
21 35
|
eqtrd |
|- ( ph -> ( ( 1r ` R ) ( .s ` P ) ( D .^ X ) ) = ( D .^ X ) ) |
37 |
36
|
fvoveq1d |
|- ( ph -> ( coe1 ` ( ( ( 1r ` R ) ( .s ` P ) ( D .^ X ) ) .x. A ) ) = ( coe1 ` ( ( D .^ X ) .x. A ) ) ) |
38 |
8
|
ad2antrr |
|- ( ( ( ph /\ x e. NN0 ) /\ D <_ x ) -> R e. Ring ) |
39 |
|
eqid |
|- ( coe1 ` A ) = ( coe1 ` A ) |
40 |
39 6 2 11
|
coe1f |
|- ( A e. B -> ( coe1 ` A ) : NN0 --> ( Base ` R ) ) |
41 |
9 40
|
syl |
|- ( ph -> ( coe1 ` A ) : NN0 --> ( Base ` R ) ) |
42 |
41
|
ad2antrr |
|- ( ( ( ph /\ x e. NN0 ) /\ D <_ x ) -> ( coe1 ` A ) : NN0 --> ( Base ` R ) ) |
43 |
10
|
ad2antrr |
|- ( ( ( ph /\ x e. NN0 ) /\ D <_ x ) -> D e. NN0 ) |
44 |
|
simplr |
|- ( ( ( ph /\ x e. NN0 ) /\ D <_ x ) -> x e. NN0 ) |
45 |
|
simpr |
|- ( ( ( ph /\ x e. NN0 ) /\ D <_ x ) -> D <_ x ) |
46 |
|
nn0sub2 |
|- ( ( D e. NN0 /\ x e. NN0 /\ D <_ x ) -> ( x - D ) e. NN0 ) |
47 |
43 44 45 46
|
syl3anc |
|- ( ( ( ph /\ x e. NN0 ) /\ D <_ x ) -> ( x - D ) e. NN0 ) |
48 |
42 47
|
ffvelrnd |
|- ( ( ( ph /\ x e. NN0 ) /\ D <_ x ) -> ( ( coe1 ` A ) ` ( x - D ) ) e. ( Base ` R ) ) |
49 |
11 13 14
|
ringlidm |
|- ( ( R e. Ring /\ ( ( coe1 ` A ) ` ( x - D ) ) e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) ( ( coe1 ` A ) ` ( x - D ) ) ) = ( ( coe1 ` A ) ` ( x - D ) ) ) |
50 |
38 48 49
|
syl2anc |
|- ( ( ( ph /\ x e. NN0 ) /\ D <_ x ) -> ( ( 1r ` R ) ( .r ` R ) ( ( coe1 ` A ) ` ( x - D ) ) ) = ( ( coe1 ` A ) ` ( x - D ) ) ) |
51 |
50
|
ifeq1da |
|- ( ( ph /\ x e. NN0 ) -> if ( D <_ x , ( ( 1r ` R ) ( .r ` R ) ( ( coe1 ` A ) ` ( x - D ) ) ) , .0. ) = if ( D <_ x , ( ( coe1 ` A ) ` ( x - D ) ) , .0. ) ) |
52 |
51
|
mpteq2dva |
|- ( ph -> ( x e. NN0 |-> if ( D <_ x , ( ( 1r ` R ) ( .r ` R ) ( ( coe1 ` A ) ` ( x - D ) ) ) , .0. ) ) = ( x e. NN0 |-> if ( D <_ x , ( ( coe1 ` A ) ` ( x - D ) ) , .0. ) ) ) |
53 |
17 37 52
|
3eqtr3d |
|- ( ph -> ( coe1 ` ( ( D .^ X ) .x. A ) ) = ( x e. NN0 |-> if ( D <_ x , ( ( coe1 ` A ) ` ( x - D ) ) , .0. ) ) ) |