| Step |
Hyp |
Ref |
Expression |
| 1 |
|
cycsubg2.x |
|- X = ( Base ` G ) |
| 2 |
|
cycsubg2.t |
|- .x. = ( .g ` G ) |
| 3 |
|
cycsubg2.f |
|- F = ( x e. ZZ |-> ( x .x. A ) ) |
| 4 |
|
cycsubg2.k |
|- K = ( mrCls ` ( SubGrp ` G ) ) |
| 5 |
|
snssg |
|- ( A e. X -> ( A e. y <-> { A } C_ y ) ) |
| 6 |
5
|
bicomd |
|- ( A e. X -> ( { A } C_ y <-> A e. y ) ) |
| 7 |
6
|
adantl |
|- ( ( G e. Grp /\ A e. X ) -> ( { A } C_ y <-> A e. y ) ) |
| 8 |
7
|
rabbidv |
|- ( ( G e. Grp /\ A e. X ) -> { y e. ( SubGrp ` G ) | { A } C_ y } = { y e. ( SubGrp ` G ) | A e. y } ) |
| 9 |
8
|
inteqd |
|- ( ( G e. Grp /\ A e. X ) -> |^| { y e. ( SubGrp ` G ) | { A } C_ y } = |^| { y e. ( SubGrp ` G ) | A e. y } ) |
| 10 |
1
|
subgacs |
|- ( G e. Grp -> ( SubGrp ` G ) e. ( ACS ` X ) ) |
| 11 |
10
|
acsmred |
|- ( G e. Grp -> ( SubGrp ` G ) e. ( Moore ` X ) ) |
| 12 |
|
snssi |
|- ( A e. X -> { A } C_ X ) |
| 13 |
4
|
mrcval |
|- ( ( ( SubGrp ` G ) e. ( Moore ` X ) /\ { A } C_ X ) -> ( K ` { A } ) = |^| { y e. ( SubGrp ` G ) | { A } C_ y } ) |
| 14 |
11 12 13
|
syl2an |
|- ( ( G e. Grp /\ A e. X ) -> ( K ` { A } ) = |^| { y e. ( SubGrp ` G ) | { A } C_ y } ) |
| 15 |
1 2 3
|
cycsubg |
|- ( ( G e. Grp /\ A e. X ) -> ran F = |^| { y e. ( SubGrp ` G ) | A e. y } ) |
| 16 |
9 14 15
|
3eqtr4d |
|- ( ( G e. Grp /\ A e. X ) -> ( K ` { A } ) = ran F ) |