| Step |
Hyp |
Ref |
Expression |
| 1 |
|
mplvsca.p |
|- P = ( I mPoly R ) |
| 2 |
|
mplvsca.n |
|- .xb = ( .s ` P ) |
| 3 |
|
mplvsca.k |
|- K = ( Base ` R ) |
| 4 |
|
mplvsca.b |
|- B = ( Base ` P ) |
| 5 |
|
mplvsca.m |
|- .x. = ( .r ` R ) |
| 6 |
|
mplvsca.d |
|- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } |
| 7 |
|
mplvsca.x |
|- ( ph -> X e. K ) |
| 8 |
|
mplvsca.f |
|- ( ph -> F e. B ) |
| 9 |
|
mplvscaval.y |
|- ( ph -> Y e. D ) |
| 10 |
1 2 3 4 5 6 7 8
|
mplvsca |
|- ( ph -> ( X .xb F ) = ( ( D X. { X } ) oF .x. F ) ) |
| 11 |
10
|
fveq1d |
|- ( ph -> ( ( X .xb F ) ` Y ) = ( ( ( D X. { X } ) oF .x. F ) ` Y ) ) |
| 12 |
|
ovex |
|- ( NN0 ^m I ) e. _V |
| 13 |
6 12
|
rabex2 |
|- D e. _V |
| 14 |
13
|
a1i |
|- ( ph -> D e. _V ) |
| 15 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
| 16 |
1 15 4 6 8
|
mplelf |
|- ( ph -> F : D --> ( Base ` R ) ) |
| 17 |
16
|
ffnd |
|- ( ph -> F Fn D ) |
| 18 |
|
eqidd |
|- ( ( ph /\ Y e. D ) -> ( F ` Y ) = ( F ` Y ) ) |
| 19 |
14 7 17 18
|
ofc1 |
|- ( ( ph /\ Y e. D ) -> ( ( ( D X. { X } ) oF .x. F ) ` Y ) = ( X .x. ( F ` Y ) ) ) |
| 20 |
9 19
|
mpdan |
|- ( ph -> ( ( ( D X. { X } ) oF .x. F ) ` Y ) = ( X .x. ( F ` Y ) ) ) |
| 21 |
11 20
|
eqtrd |
|- ( ph -> ( ( X .xb F ) ` Y ) = ( X .x. ( F ` Y ) ) ) |