Step |
Hyp |
Ref |
Expression |
1 |
|
pgpfi1.1 |
|- X = ( Base ` G ) |
2 |
|
simpl2 |
|- ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) -> P e. Prime ) |
3 |
|
simpl1 |
|- ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) -> G e. Grp ) |
4 |
|
simpll3 |
|- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> N e. NN0 ) |
5 |
3
|
adantr |
|- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> G e. Grp ) |
6 |
|
simplr |
|- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> ( # ` X ) = ( P ^ N ) ) |
7 |
2
|
adantr |
|- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> P e. Prime ) |
8 |
|
prmnn |
|- ( P e. Prime -> P e. NN ) |
9 |
7 8
|
syl |
|- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> P e. NN ) |
10 |
9 4
|
nnexpcld |
|- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> ( P ^ N ) e. NN ) |
11 |
10
|
nnnn0d |
|- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> ( P ^ N ) e. NN0 ) |
12 |
6 11
|
eqeltrd |
|- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> ( # ` X ) e. NN0 ) |
13 |
1
|
fvexi |
|- X e. _V |
14 |
|
hashclb |
|- ( X e. _V -> ( X e. Fin <-> ( # ` X ) e. NN0 ) ) |
15 |
13 14
|
ax-mp |
|- ( X e. Fin <-> ( # ` X ) e. NN0 ) |
16 |
12 15
|
sylibr |
|- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> X e. Fin ) |
17 |
|
simpr |
|- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> x e. X ) |
18 |
|
eqid |
|- ( od ` G ) = ( od ` G ) |
19 |
1 18
|
oddvds2 |
|- ( ( G e. Grp /\ X e. Fin /\ x e. X ) -> ( ( od ` G ) ` x ) || ( # ` X ) ) |
20 |
5 16 17 19
|
syl3anc |
|- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> ( ( od ` G ) ` x ) || ( # ` X ) ) |
21 |
20 6
|
breqtrd |
|- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> ( ( od ` G ) ` x ) || ( P ^ N ) ) |
22 |
|
oveq2 |
|- ( n = N -> ( P ^ n ) = ( P ^ N ) ) |
23 |
22
|
breq2d |
|- ( n = N -> ( ( ( od ` G ) ` x ) || ( P ^ n ) <-> ( ( od ` G ) ` x ) || ( P ^ N ) ) ) |
24 |
23
|
rspcev |
|- ( ( N e. NN0 /\ ( ( od ` G ) ` x ) || ( P ^ N ) ) -> E. n e. NN0 ( ( od ` G ) ` x ) || ( P ^ n ) ) |
25 |
4 21 24
|
syl2anc |
|- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> E. n e. NN0 ( ( od ` G ) ` x ) || ( P ^ n ) ) |
26 |
1 18
|
odcl2 |
|- ( ( G e. Grp /\ X e. Fin /\ x e. X ) -> ( ( od ` G ) ` x ) e. NN ) |
27 |
5 16 17 26
|
syl3anc |
|- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> ( ( od ` G ) ` x ) e. NN ) |
28 |
|
pcprmpw2 |
|- ( ( P e. Prime /\ ( ( od ` G ) ` x ) e. NN ) -> ( E. n e. NN0 ( ( od ` G ) ` x ) || ( P ^ n ) <-> ( ( od ` G ) ` x ) = ( P ^ ( P pCnt ( ( od ` G ) ` x ) ) ) ) ) |
29 |
|
pcprmpw |
|- ( ( P e. Prime /\ ( ( od ` G ) ` x ) e. NN ) -> ( E. n e. NN0 ( ( od ` G ) ` x ) = ( P ^ n ) <-> ( ( od ` G ) ` x ) = ( P ^ ( P pCnt ( ( od ` G ) ` x ) ) ) ) ) |
30 |
28 29
|
bitr4d |
|- ( ( P e. Prime /\ ( ( od ` G ) ` x ) e. NN ) -> ( E. n e. NN0 ( ( od ` G ) ` x ) || ( P ^ n ) <-> E. n e. NN0 ( ( od ` G ) ` x ) = ( P ^ n ) ) ) |
31 |
7 27 30
|
syl2anc |
|- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> ( E. n e. NN0 ( ( od ` G ) ` x ) || ( P ^ n ) <-> E. n e. NN0 ( ( od ` G ) ` x ) = ( P ^ n ) ) ) |
32 |
25 31
|
mpbid |
|- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> E. n e. NN0 ( ( od ` G ) ` x ) = ( P ^ n ) ) |
33 |
32
|
ralrimiva |
|- ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) -> A. x e. X E. n e. NN0 ( ( od ` G ) ` x ) = ( P ^ n ) ) |
34 |
1 18
|
ispgp |
|- ( P pGrp G <-> ( P e. Prime /\ G e. Grp /\ A. x e. X E. n e. NN0 ( ( od ` G ) ` x ) = ( P ^ n ) ) ) |
35 |
2 3 33 34
|
syl3anbrc |
|- ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) -> P pGrp G ) |
36 |
35
|
ex |
|- ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) -> ( ( # ` X ) = ( P ^ N ) -> P pGrp G ) ) |