| Step |
Hyp |
Ref |
Expression |
| 1 |
|
coe1vr1.1 |
|- P = ( Poly1 ` R ) |
| 2 |
|
coe1vr1.2 |
|- X = ( var1 ` R ) |
| 3 |
|
coe1vr1.3 |
|- ( ph -> R e. Ring ) |
| 4 |
|
coe1vr1.4 |
|- .0. = ( 0g ` R ) |
| 5 |
|
coe1vr1.5 |
|- .1. = ( 1r ` R ) |
| 6 |
|
eqid |
|- ( Base ` P ) = ( Base ` P ) |
| 7 |
2 1 6
|
vr1cl |
|- ( R e. Ring -> X e. ( Base ` P ) ) |
| 8 |
|
eqid |
|- ( mulGrp ` P ) = ( mulGrp ` P ) |
| 9 |
8 6
|
mgpbas |
|- ( Base ` P ) = ( Base ` ( mulGrp ` P ) ) |
| 10 |
|
eqid |
|- ( .g ` ( mulGrp ` P ) ) = ( .g ` ( mulGrp ` P ) ) |
| 11 |
9 10
|
mulg1 |
|- ( X e. ( Base ` P ) -> ( 1 ( .g ` ( mulGrp ` P ) ) X ) = X ) |
| 12 |
3 7 11
|
3syl |
|- ( ph -> ( 1 ( .g ` ( mulGrp ` P ) ) X ) = X ) |
| 13 |
12
|
fveq2d |
|- ( ph -> ( coe1 ` ( 1 ( .g ` ( mulGrp ` P ) ) X ) ) = ( coe1 ` X ) ) |
| 14 |
|
1nn0 |
|- 1 e. NN0 |
| 15 |
14
|
a1i |
|- ( ph -> 1 e. NN0 ) |
| 16 |
1 2 10 3 15 4 5
|
coe1mon |
|- ( ph -> ( coe1 ` ( 1 ( .g ` ( mulGrp ` P ) ) X ) ) = ( k e. NN0 |-> if ( k = 1 , .1. , .0. ) ) ) |
| 17 |
13 16
|
eqtr3d |
|- ( ph -> ( coe1 ` X ) = ( k e. NN0 |-> if ( k = 1 , .1. , .0. ) ) ) |