Step |
Hyp |
Ref |
Expression |
1 |
|
proot1hash.g |
|- G = ( ( mulGrp ` R ) |`s ( Unit ` R ) ) |
2 |
|
proot1hash.o |
|- O = ( od ` G ) |
3 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
4 |
3 2
|
odf |
|- O : ( Base ` G ) --> NN0 |
5 |
|
ffn |
|- ( O : ( Base ` G ) --> NN0 -> O Fn ( Base ` G ) ) |
6 |
|
fniniseg2 |
|- ( O Fn ( Base ` G ) -> ( `' O " { N } ) = { x e. ( Base ` G ) | ( O ` x ) = N } ) |
7 |
4 5 6
|
mp2b |
|- ( `' O " { N } ) = { x e. ( Base ` G ) | ( O ` x ) = N } |
8 |
|
simp3 |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> X e. ( `' O " { N } ) ) |
9 |
|
fniniseg |
|- ( O Fn ( Base ` G ) -> ( X e. ( `' O " { N } ) <-> ( X e. ( Base ` G ) /\ ( O ` X ) = N ) ) ) |
10 |
4 5 9
|
mp2b |
|- ( X e. ( `' O " { N } ) <-> ( X e. ( Base ` G ) /\ ( O ` X ) = N ) ) |
11 |
8 10
|
sylib |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> ( X e. ( Base ` G ) /\ ( O ` X ) = N ) ) |
12 |
11
|
simprd |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> ( O ` X ) = N ) |
13 |
12
|
eqeq2d |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> ( ( O ` x ) = ( O ` X ) <-> ( O ` x ) = N ) ) |
14 |
13
|
rabbidv |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> { x e. ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) | ( O ` x ) = ( O ` X ) } = { x e. ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) | ( O ` x ) = N } ) |
15 |
|
isidom |
|- ( R e. IDomn <-> ( R e. CRing /\ R e. Domn ) ) |
16 |
15
|
simprbi |
|- ( R e. IDomn -> R e. Domn ) |
17 |
16
|
3ad2ant1 |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> R e. Domn ) |
18 |
|
domnring |
|- ( R e. Domn -> R e. Ring ) |
19 |
|
eqid |
|- ( Unit ` R ) = ( Unit ` R ) |
20 |
19 1
|
unitgrp |
|- ( R e. Ring -> G e. Grp ) |
21 |
17 18 20
|
3syl |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> G e. Grp ) |
22 |
3
|
subgacs |
|- ( G e. Grp -> ( SubGrp ` G ) e. ( ACS ` ( Base ` G ) ) ) |
23 |
|
acsmre |
|- ( ( SubGrp ` G ) e. ( ACS ` ( Base ` G ) ) -> ( SubGrp ` G ) e. ( Moore ` ( Base ` G ) ) ) |
24 |
21 22 23
|
3syl |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> ( SubGrp ` G ) e. ( Moore ` ( Base ` G ) ) ) |
25 |
|
eqid |
|- ( mrCls ` ( SubGrp ` G ) ) = ( mrCls ` ( SubGrp ` G ) ) |
26 |
25
|
mrcssv |
|- ( ( SubGrp ` G ) e. ( Moore ` ( Base ` G ) ) -> ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) C_ ( Base ` G ) ) |
27 |
|
dfrab3ss |
|- ( ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) C_ ( Base ` G ) -> { x e. ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) | ( O ` x ) = N } = ( ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) i^i { x e. ( Base ` G ) | ( O ` x ) = N } ) ) |
28 |
24 26 27
|
3syl |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> { x e. ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) | ( O ` x ) = N } = ( ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) i^i { x e. ( Base ` G ) | ( O ` x ) = N } ) ) |
29 |
|
incom |
|- ( ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) i^i { x e. ( Base ` G ) | ( O ` x ) = N } ) = ( { x e. ( Base ` G ) | ( O ` x ) = N } i^i ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) ) |
30 |
|
simpl1 |
|- ( ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) /\ x e. ( `' O " { N } ) ) -> R e. IDomn ) |
31 |
|
simpl2 |
|- ( ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) /\ x e. ( `' O " { N } ) ) -> N e. NN ) |
32 |
|
simpr |
|- ( ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) /\ x e. ( `' O " { N } ) ) -> x e. ( `' O " { N } ) ) |
33 |
|
simpl3 |
|- ( ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) /\ x e. ( `' O " { N } ) ) -> X e. ( `' O " { N } ) ) |
34 |
1 2 25
|
proot1mul |
|- ( ( ( R e. IDomn /\ N e. NN ) /\ ( x e. ( `' O " { N } ) /\ X e. ( `' O " { N } ) ) ) -> x e. ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) ) |
35 |
30 31 32 33 34
|
syl22anc |
|- ( ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) /\ x e. ( `' O " { N } ) ) -> x e. ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) ) |
36 |
35
|
ex |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> ( x e. ( `' O " { N } ) -> x e. ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) ) ) |
37 |
36
|
ssrdv |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> ( `' O " { N } ) C_ ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) ) |
38 |
7 37
|
eqsstrrid |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> { x e. ( Base ` G ) | ( O ` x ) = N } C_ ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) ) |
39 |
|
df-ss |
|- ( { x e. ( Base ` G ) | ( O ` x ) = N } C_ ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) <-> ( { x e. ( Base ` G ) | ( O ` x ) = N } i^i ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) ) = { x e. ( Base ` G ) | ( O ` x ) = N } ) |
40 |
38 39
|
sylib |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> ( { x e. ( Base ` G ) | ( O ` x ) = N } i^i ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) ) = { x e. ( Base ` G ) | ( O ` x ) = N } ) |
41 |
29 40
|
syl5eq |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> ( ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) i^i { x e. ( Base ` G ) | ( O ` x ) = N } ) = { x e. ( Base ` G ) | ( O ` x ) = N } ) |
42 |
14 28 41
|
3eqtrrd |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> { x e. ( Base ` G ) | ( O ` x ) = N } = { x e. ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) | ( O ` x ) = ( O ` X ) } ) |
43 |
7 42
|
syl5eq |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> ( `' O " { N } ) = { x e. ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) | ( O ` x ) = ( O ` X ) } ) |
44 |
43
|
fveq2d |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> ( # ` ( `' O " { N } ) ) = ( # ` { x e. ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) | ( O ` x ) = ( O ` X ) } ) ) |
45 |
11
|
simpld |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> X e. ( Base ` G ) ) |
46 |
|
simp2 |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> N e. NN ) |
47 |
12 46
|
eqeltrd |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> ( O ` X ) e. NN ) |
48 |
3 2 25
|
odngen |
|- ( ( G e. Grp /\ X e. ( Base ` G ) /\ ( O ` X ) e. NN ) -> ( # ` { x e. ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) | ( O ` x ) = ( O ` X ) } ) = ( phi ` ( O ` X ) ) ) |
49 |
21 45 47 48
|
syl3anc |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> ( # ` { x e. ( ( mrCls ` ( SubGrp ` G ) ) ` { X } ) | ( O ` x ) = ( O ` X ) } ) = ( phi ` ( O ` X ) ) ) |
50 |
12
|
fveq2d |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> ( phi ` ( O ` X ) ) = ( phi ` N ) ) |
51 |
44 49 50
|
3eqtrd |
|- ( ( R e. IDomn /\ N e. NN /\ X e. ( `' O " { N } ) ) -> ( # ` ( `' O " { N } ) ) = ( phi ` N ) ) |