| Step |
Hyp |
Ref |
Expression |
| 1 |
|
mplascl.p |
|- P = ( I mPoly R ) |
| 2 |
|
mplascl.d |
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } |
| 3 |
|
mplascl.z |
|- .0. = ( 0g ` R ) |
| 4 |
|
mplascl.b |
|- B = ( Base ` R ) |
| 5 |
|
mplascl.a |
|- A = ( algSc ` P ) |
| 6 |
|
mplascl.i |
|- ( ph -> I e. W ) |
| 7 |
|
mplascl.r |
|- ( ph -> R e. Ring ) |
| 8 |
|
mplascl.x |
|- ( ph -> X e. B ) |
| 9 |
1 6 7
|
mplsca |
|- ( ph -> R = ( Scalar ` P ) ) |
| 10 |
9
|
fveq2d |
|- ( ph -> ( Base ` R ) = ( Base ` ( Scalar ` P ) ) ) |
| 11 |
4 10
|
eqtrid |
|- ( ph -> B = ( Base ` ( Scalar ` P ) ) ) |
| 12 |
8 11
|
eleqtrd |
|- ( ph -> X e. ( Base ` ( Scalar ` P ) ) ) |
| 13 |
|
eqid |
|- ( Scalar ` P ) = ( Scalar ` P ) |
| 14 |
|
eqid |
|- ( Base ` ( Scalar ` P ) ) = ( Base ` ( Scalar ` P ) ) |
| 15 |
|
eqid |
|- ( .s ` P ) = ( .s ` P ) |
| 16 |
|
eqid |
|- ( 1r ` P ) = ( 1r ` P ) |
| 17 |
5 13 14 15 16
|
asclval |
|- ( X e. ( Base ` ( Scalar ` P ) ) -> ( A ` X ) = ( X ( .s ` P ) ( 1r ` P ) ) ) |
| 18 |
12 17
|
syl |
|- ( ph -> ( A ` X ) = ( X ( .s ` P ) ( 1r ` P ) ) ) |
| 19 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
| 20 |
1 2 3 19 16 6 7
|
mpl1 |
|- ( ph -> ( 1r ` P ) = ( y e. D |-> if ( y = ( I X. { 0 } ) , ( 1r ` R ) , .0. ) ) ) |
| 21 |
20
|
oveq2d |
|- ( ph -> ( X ( .s ` P ) ( 1r ` P ) ) = ( X ( .s ` P ) ( y e. D |-> if ( y = ( I X. { 0 } ) , ( 1r ` R ) , .0. ) ) ) ) |
| 22 |
2
|
psrbag0 |
|- ( I e. W -> ( I X. { 0 } ) e. D ) |
| 23 |
6 22
|
syl |
|- ( ph -> ( I X. { 0 } ) e. D ) |
| 24 |
1 15 2 19 3 4 6 7 23 8
|
mplmon2 |
|- ( ph -> ( X ( .s ` P ) ( y e. D |-> if ( y = ( I X. { 0 } ) , ( 1r ` R ) , .0. ) ) ) = ( y e. D |-> if ( y = ( I X. { 0 } ) , X , .0. ) ) ) |
| 25 |
18 21 24
|
3eqtrd |
|- ( ph -> ( A ` X ) = ( y e. D |-> if ( y = ( I X. { 0 } ) , X , .0. ) ) ) |