| Step |
Hyp |
Ref |
Expression |
| 1 |
|
mplmon2cl.p |
|- P = ( I mPoly R ) |
| 2 |
|
mplmon2cl.d |
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } |
| 3 |
|
mplmon2cl.z |
|- .0. = ( 0g ` R ) |
| 4 |
|
mplmon2cl.c |
|- C = ( Base ` R ) |
| 5 |
|
mplmon2cl.i |
|- ( ph -> I e. W ) |
| 6 |
|
mplmon2cl.r |
|- ( ph -> R e. Ring ) |
| 7 |
|
mplmon2cl.b |
|- B = ( Base ` P ) |
| 8 |
|
mplmon2cl.x |
|- ( ph -> X e. C ) |
| 9 |
|
mplmon2cl.k |
|- ( ph -> K e. D ) |
| 10 |
|
eqid |
|- ( .s ` P ) = ( .s ` P ) |
| 11 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
| 12 |
1 10 2 11 3 4 5 6 9 8
|
mplmon2 |
|- ( ph -> ( X ( .s ` P ) ( y e. D |-> if ( y = K , ( 1r ` R ) , .0. ) ) ) = ( y e. D |-> if ( y = K , X , .0. ) ) ) |
| 13 |
1 5 6
|
mpllmodd |
|- ( ph -> P e. LMod ) |
| 14 |
1 5 6
|
mplsca |
|- ( ph -> R = ( Scalar ` P ) ) |
| 15 |
14
|
fveq2d |
|- ( ph -> ( Base ` R ) = ( Base ` ( Scalar ` P ) ) ) |
| 16 |
4 15
|
eqtrid |
|- ( ph -> C = ( Base ` ( Scalar ` P ) ) ) |
| 17 |
8 16
|
eleqtrd |
|- ( ph -> X e. ( Base ` ( Scalar ` P ) ) ) |
| 18 |
1 7 3 11 2 5 6 9
|
mplmon |
|- ( ph -> ( y e. D |-> if ( y = K , ( 1r ` R ) , .0. ) ) e. B ) |
| 19 |
|
eqid |
|- ( Scalar ` P ) = ( Scalar ` P ) |
| 20 |
|
eqid |
|- ( Base ` ( Scalar ` P ) ) = ( Base ` ( Scalar ` P ) ) |
| 21 |
7 19 10 20
|
lmodvscl |
|- ( ( P e. LMod /\ X e. ( Base ` ( Scalar ` P ) ) /\ ( y e. D |-> if ( y = K , ( 1r ` R ) , .0. ) ) e. B ) -> ( X ( .s ` P ) ( y e. D |-> if ( y = K , ( 1r ` R ) , .0. ) ) ) e. B ) |
| 22 |
13 17 18 21
|
syl3anc |
|- ( ph -> ( X ( .s ` P ) ( y e. D |-> if ( y = K , ( 1r ` R ) , .0. ) ) ) e. B ) |
| 23 |
12 22
|
eqeltrrd |
|- ( ph -> ( y e. D |-> if ( y = K , X , .0. ) ) e. B ) |