Step |
Hyp |
Ref |
Expression |
1 |
|
mhpsubg.h |
|- H = ( I mHomP R ) |
2 |
|
mhpsubg.p |
|- P = ( I mPoly R ) |
3 |
|
mhpsubg.i |
|- ( ph -> I e. V ) |
4 |
|
mhpsubg.r |
|- ( ph -> R e. Grp ) |
5 |
|
mhpsubg.n |
|- ( ph -> N e. NN0 ) |
6 |
|
eqid |
|- ( Base ` P ) = ( Base ` P ) |
7 |
|
simpr |
|- ( ( ph /\ x e. ( H ` N ) ) -> x e. ( H ` N ) ) |
8 |
1 2 6 7
|
mhpmpl |
|- ( ( ph /\ x e. ( H ` N ) ) -> x e. ( Base ` P ) ) |
9 |
8
|
ex |
|- ( ph -> ( x e. ( H ` N ) -> x e. ( Base ` P ) ) ) |
10 |
9
|
ssrdv |
|- ( ph -> ( H ` N ) C_ ( Base ` P ) ) |
11 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
12 |
|
eqid |
|- { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } |
13 |
1 11 12 3 4 5
|
mhp0cl |
|- ( ph -> ( { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } X. { ( 0g ` R ) } ) e. ( H ` N ) ) |
14 |
13
|
ne0d |
|- ( ph -> ( H ` N ) =/= (/) ) |
15 |
|
eqid |
|- ( +g ` P ) = ( +g ` P ) |
16 |
4
|
adantr |
|- ( ( ph /\ x e. ( H ` N ) ) -> R e. Grp ) |
17 |
16
|
adantr |
|- ( ( ( ph /\ x e. ( H ` N ) ) /\ y e. ( H ` N ) ) -> R e. Grp ) |
18 |
|
simplr |
|- ( ( ( ph /\ x e. ( H ` N ) ) /\ y e. ( H ` N ) ) -> x e. ( H ` N ) ) |
19 |
|
simpr |
|- ( ( ( ph /\ x e. ( H ` N ) ) /\ y e. ( H ` N ) ) -> y e. ( H ` N ) ) |
20 |
1 2 15 17 18 19
|
mhpaddcl |
|- ( ( ( ph /\ x e. ( H ` N ) ) /\ y e. ( H ` N ) ) -> ( x ( +g ` P ) y ) e. ( H ` N ) ) |
21 |
20
|
ralrimiva |
|- ( ( ph /\ x e. ( H ` N ) ) -> A. y e. ( H ` N ) ( x ( +g ` P ) y ) e. ( H ` N ) ) |
22 |
|
eqid |
|- ( invg ` P ) = ( invg ` P ) |
23 |
1 2 22 16 7
|
mhpinvcl |
|- ( ( ph /\ x e. ( H ` N ) ) -> ( ( invg ` P ) ` x ) e. ( H ` N ) ) |
24 |
21 23
|
jca |
|- ( ( ph /\ x e. ( H ` N ) ) -> ( A. y e. ( H ` N ) ( x ( +g ` P ) y ) e. ( H ` N ) /\ ( ( invg ` P ) ` x ) e. ( H ` N ) ) ) |
25 |
24
|
ralrimiva |
|- ( ph -> A. x e. ( H ` N ) ( A. y e. ( H ` N ) ( x ( +g ` P ) y ) e. ( H ` N ) /\ ( ( invg ` P ) ` x ) e. ( H ` N ) ) ) |
26 |
2
|
mplgrp |
|- ( ( I e. V /\ R e. Grp ) -> P e. Grp ) |
27 |
3 4 26
|
syl2anc |
|- ( ph -> P e. Grp ) |
28 |
6 15 22
|
issubg2 |
|- ( P e. Grp -> ( ( H ` N ) e. ( SubGrp ` P ) <-> ( ( H ` N ) C_ ( Base ` P ) /\ ( H ` N ) =/= (/) /\ A. x e. ( H ` N ) ( A. y e. ( H ` N ) ( x ( +g ` P ) y ) e. ( H ` N ) /\ ( ( invg ` P ) ` x ) e. ( H ` N ) ) ) ) ) |
29 |
27 28
|
syl |
|- ( ph -> ( ( H ` N ) e. ( SubGrp ` P ) <-> ( ( H ` N ) C_ ( Base ` P ) /\ ( H ` N ) =/= (/) /\ A. x e. ( H ` N ) ( A. y e. ( H ` N ) ( x ( +g ` P ) y ) e. ( H ` N ) /\ ( ( invg ` P ) ` x ) e. ( H ` N ) ) ) ) ) |
30 |
10 14 25 29
|
mpbir3and |
|- ( ph -> ( H ` N ) e. ( SubGrp ` P ) ) |