Step |
Hyp |
Ref |
Expression |
0 |
|
cslw |
|- pSyl |
1 |
|
vp |
|- p |
2 |
|
cprime |
|- Prime |
3 |
|
vg |
|- g |
4 |
|
cgrp |
|- Grp |
5 |
|
vh |
|- h |
6 |
|
csubg |
|- SubGrp |
7 |
3
|
cv |
|- g |
8 |
7 6
|
cfv |
|- ( SubGrp ` g ) |
9 |
|
vk |
|- k |
10 |
5
|
cv |
|- h |
11 |
9
|
cv |
|- k |
12 |
10 11
|
wss |
|- h C_ k |
13 |
1
|
cv |
|- p |
14 |
|
cpgp |
|- pGrp |
15 |
|
cress |
|- |`s |
16 |
7 11 15
|
co |
|- ( g |`s k ) |
17 |
13 16 14
|
wbr |
|- p pGrp ( g |`s k ) |
18 |
12 17
|
wa |
|- ( h C_ k /\ p pGrp ( g |`s k ) ) |
19 |
10 11
|
wceq |
|- h = k |
20 |
18 19
|
wb |
|- ( ( h C_ k /\ p pGrp ( g |`s k ) ) <-> h = k ) |
21 |
20 9 8
|
wral |
|- A. k e. ( SubGrp ` g ) ( ( h C_ k /\ p pGrp ( g |`s k ) ) <-> h = k ) |
22 |
21 5 8
|
crab |
|- { h e. ( SubGrp ` g ) | A. k e. ( SubGrp ` g ) ( ( h C_ k /\ p pGrp ( g |`s k ) ) <-> h = k ) } |
23 |
1 3 2 4 22
|
cmpo |
|- ( p e. Prime , g e. Grp |-> { h e. ( SubGrp ` g ) | A. k e. ( SubGrp ` g ) ( ( h C_ k /\ p pGrp ( g |`s k ) ) <-> h = k ) } ) |
24 |
0 23
|
wceq |
|- pSyl = ( p e. Prime , g e. Grp |-> { h e. ( SubGrp ` g ) | A. k e. ( SubGrp ` g ) ( ( h C_ k /\ p pGrp ( g |`s k ) ) <-> h = k ) } ) |