| 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 |
|
coe1tmfv2.r |
|- ( ph -> R e. Ring ) |
| 9 |
|
coe1tmfv2.c |
|- ( ph -> C e. K ) |
| 10 |
|
coe1tmfv2.d |
|- ( ph -> D e. NN0 ) |
| 11 |
|
coe1tmfv2.f |
|- ( ph -> F e. NN0 ) |
| 12 |
|
coe1tmfv2.q |
|- ( ph -> D =/= F ) |
| 13 |
1 2 3 4 5 6 7
|
coe1tm |
|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( coe1 ` ( C .x. ( D .^ X ) ) ) = ( x e. NN0 |-> if ( x = D , C , .0. ) ) ) |
| 14 |
8 9 10 13
|
syl3anc |
|- ( ph -> ( coe1 ` ( C .x. ( D .^ X ) ) ) = ( x e. NN0 |-> if ( x = D , C , .0. ) ) ) |
| 15 |
14
|
fveq1d |
|- ( ph -> ( ( coe1 ` ( C .x. ( D .^ X ) ) ) ` F ) = ( ( x e. NN0 |-> if ( x = D , C , .0. ) ) ` F ) ) |
| 16 |
|
eqid |
|- ( x e. NN0 |-> if ( x = D , C , .0. ) ) = ( x e. NN0 |-> if ( x = D , C , .0. ) ) |
| 17 |
|
eqeq1 |
|- ( x = F -> ( x = D <-> F = D ) ) |
| 18 |
17
|
ifbid |
|- ( x = F -> if ( x = D , C , .0. ) = if ( F = D , C , .0. ) ) |
| 19 |
2 1
|
ring0cl |
|- ( R e. Ring -> .0. e. K ) |
| 20 |
8 19
|
syl |
|- ( ph -> .0. e. K ) |
| 21 |
9 20
|
ifcld |
|- ( ph -> if ( F = D , C , .0. ) e. K ) |
| 22 |
16 18 11 21
|
fvmptd3 |
|- ( ph -> ( ( x e. NN0 |-> if ( x = D , C , .0. ) ) ` F ) = if ( F = D , C , .0. ) ) |
| 23 |
12
|
necomd |
|- ( ph -> F =/= D ) |
| 24 |
23
|
neneqd |
|- ( ph -> -. F = D ) |
| 25 |
24
|
iffalsed |
|- ( ph -> if ( F = D , C , .0. ) = .0. ) |
| 26 |
15 22 25
|
3eqtrd |
|- ( ph -> ( ( coe1 ` ( C .x. ( D .^ X ) ) ) ` F ) = .0. ) |