Step |
Hyp |
Ref |
Expression |
1 |
|
mplmul.p |
|- P = ( I mPoly R ) |
2 |
|
mplmul.b |
|- B = ( Base ` P ) |
3 |
|
mplmul.m |
|- .x. = ( .r ` R ) |
4 |
|
mplmul.t |
|- .xb = ( .r ` P ) |
5 |
|
mplmul.d |
|- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } |
6 |
|
mplmul.f |
|- ( ph -> F e. B ) |
7 |
|
mplmul.g |
|- ( ph -> G e. B ) |
8 |
|
eqid |
|- ( I mPwSer R ) = ( I mPwSer R ) |
9 |
|
eqid |
|- ( Base ` ( I mPwSer R ) ) = ( Base ` ( I mPwSer R ) ) |
10 |
1 8 4
|
mplmulr |
|- .xb = ( .r ` ( I mPwSer R ) ) |
11 |
1 8 2 9
|
mplbasss |
|- B C_ ( Base ` ( I mPwSer R ) ) |
12 |
11 6
|
sselid |
|- ( ph -> F e. ( Base ` ( I mPwSer R ) ) ) |
13 |
11 7
|
sselid |
|- ( ph -> G e. ( Base ` ( I mPwSer R ) ) ) |
14 |
8 9 3 10 5 12 13
|
psrmulfval |
|- ( ph -> ( F .xb G ) = ( k e. D |-> ( R gsum ( x e. { y e. D | y oR <_ k } |-> ( ( F ` x ) .x. ( G ` ( k oF - x ) ) ) ) ) ) ) |