Step |
Hyp |
Ref |
Expression |
1 |
|
mhplss.h |
|- H = ( I mHomP R ) |
2 |
|
mhplss.p |
|- P = ( I mPoly R ) |
3 |
|
mhplss.i |
|- ( ph -> I e. V ) |
4 |
|
mhplss.r |
|- ( ph -> R e. Ring ) |
5 |
|
mhplss.n |
|- ( ph -> N e. NN0 ) |
6 |
|
ringgrp |
|- ( R e. Ring -> R e. Grp ) |
7 |
4 6
|
syl |
|- ( ph -> R e. Grp ) |
8 |
1 2 3 7 5
|
mhpsubg |
|- ( ph -> ( H ` N ) e. ( SubGrp ` P ) ) |
9 |
|
eqid |
|- ( .s ` P ) = ( .s ` P ) |
10 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
11 |
3
|
adantr |
|- ( ( ph /\ ( a e. ( Base ` ( Scalar ` P ) ) /\ b e. ( H ` N ) ) ) -> I e. V ) |
12 |
4
|
adantr |
|- ( ( ph /\ ( a e. ( Base ` ( Scalar ` P ) ) /\ b e. ( H ` N ) ) ) -> R e. Ring ) |
13 |
5
|
adantr |
|- ( ( ph /\ ( a e. ( Base ` ( Scalar ` P ) ) /\ b e. ( H ` N ) ) ) -> N e. NN0 ) |
14 |
2 3 4
|
mplsca |
|- ( ph -> R = ( Scalar ` P ) ) |
15 |
14
|
fveq2d |
|- ( ph -> ( Base ` R ) = ( Base ` ( Scalar ` P ) ) ) |
16 |
15
|
eleq2d |
|- ( ph -> ( a e. ( Base ` R ) <-> a e. ( Base ` ( Scalar ` P ) ) ) ) |
17 |
16
|
biimpar |
|- ( ( ph /\ a e. ( Base ` ( Scalar ` P ) ) ) -> a e. ( Base ` R ) ) |
18 |
17
|
adantrr |
|- ( ( ph /\ ( a e. ( Base ` ( Scalar ` P ) ) /\ b e. ( H ` N ) ) ) -> a e. ( Base ` R ) ) |
19 |
|
simprr |
|- ( ( ph /\ ( a e. ( Base ` ( Scalar ` P ) ) /\ b e. ( H ` N ) ) ) -> b e. ( H ` N ) ) |
20 |
1 2 9 10 11 12 13 18 19
|
mhpvscacl |
|- ( ( ph /\ ( a e. ( Base ` ( Scalar ` P ) ) /\ b e. ( H ` N ) ) ) -> ( a ( .s ` P ) b ) e. ( H ` N ) ) |
21 |
20
|
ralrimivva |
|- ( ph -> A. a e. ( Base ` ( Scalar ` P ) ) A. b e. ( H ` N ) ( a ( .s ` P ) b ) e. ( H ` N ) ) |
22 |
2
|
mpllmod |
|- ( ( I e. V /\ R e. Ring ) -> P e. LMod ) |
23 |
3 4 22
|
syl2anc |
|- ( ph -> P e. LMod ) |
24 |
|
eqid |
|- ( Scalar ` P ) = ( Scalar ` P ) |
25 |
|
eqid |
|- ( Base ` ( Scalar ` P ) ) = ( Base ` ( Scalar ` P ) ) |
26 |
|
eqid |
|- ( Base ` P ) = ( Base ` P ) |
27 |
|
eqid |
|- ( LSubSp ` P ) = ( LSubSp ` P ) |
28 |
24 25 26 9 27
|
islss4 |
|- ( P e. LMod -> ( ( H ` N ) e. ( LSubSp ` P ) <-> ( ( H ` N ) e. ( SubGrp ` P ) /\ A. a e. ( Base ` ( Scalar ` P ) ) A. b e. ( H ` N ) ( a ( .s ` P ) b ) e. ( H ` N ) ) ) ) |
29 |
23 28
|
syl |
|- ( ph -> ( ( H ` N ) e. ( LSubSp ` P ) <-> ( ( H ` N ) e. ( SubGrp ` P ) /\ A. a e. ( Base ` ( Scalar ` P ) ) A. b e. ( H ` N ) ( a ( .s ` P ) b ) e. ( H ` N ) ) ) ) |
30 |
8 21 29
|
mpbir2and |
|- ( ph -> ( H ` N ) e. ( LSubSp ` P ) ) |