Step |
Hyp |
Ref |
Expression |
1 |
|
mplasclf.p |
|- P = ( I mPoly R ) |
2 |
|
mplasclf.b |
|- B = ( Base ` P ) |
3 |
|
mplasclf.k |
|- K = ( Base ` R ) |
4 |
|
mplasclf.a |
|- A = ( algSc ` P ) |
5 |
|
mplasclf.i |
|- ( ph -> I e. W ) |
6 |
|
mplasclf.r |
|- ( ph -> R e. Ring ) |
7 |
|
eqid |
|- ( Scalar ` P ) = ( Scalar ` P ) |
8 |
1
|
mplring |
|- ( ( I e. W /\ R e. Ring ) -> P e. Ring ) |
9 |
1
|
mpllmod |
|- ( ( I e. W /\ R e. Ring ) -> P e. LMod ) |
10 |
|
eqid |
|- ( Base ` ( Scalar ` P ) ) = ( Base ` ( Scalar ` P ) ) |
11 |
4 7 8 9 10 2
|
asclf |
|- ( ( I e. W /\ R e. Ring ) -> A : ( Base ` ( Scalar ` P ) ) --> B ) |
12 |
5 6 11
|
syl2anc |
|- ( ph -> A : ( Base ` ( Scalar ` P ) ) --> B ) |
13 |
1 5 6
|
mplsca |
|- ( ph -> R = ( Scalar ` P ) ) |
14 |
13
|
fveq2d |
|- ( ph -> ( Base ` R ) = ( Base ` ( Scalar ` P ) ) ) |
15 |
3 14
|
syl5eq |
|- ( ph -> K = ( Base ` ( Scalar ` P ) ) ) |
16 |
15
|
feq2d |
|- ( ph -> ( A : K --> B <-> A : ( Base ` ( Scalar ` P ) ) --> B ) ) |
17 |
12 16
|
mpbird |
|- ( ph -> A : K --> B ) |