Step |
Hyp |
Ref |
Expression |
1 |
|
ablfac1.b |
|- B = ( Base ` G ) |
2 |
|
ablfac1.o |
|- O = ( od ` G ) |
3 |
|
ablfac1.s |
|- S = ( p e. A |-> { x e. B | ( O ` x ) || ( p ^ ( p pCnt ( # ` B ) ) ) } ) |
4 |
|
ablfac1.g |
|- ( ph -> G e. Abel ) |
5 |
|
ablfac1.f |
|- ( ph -> B e. Fin ) |
6 |
|
ablfac1.1 |
|- ( ph -> A C_ Prime ) |
7 |
|
id |
|- ( p = P -> p = P ) |
8 |
|
oveq1 |
|- ( p = P -> ( p pCnt ( # ` B ) ) = ( P pCnt ( # ` B ) ) ) |
9 |
7 8
|
oveq12d |
|- ( p = P -> ( p ^ ( p pCnt ( # ` B ) ) ) = ( P ^ ( P pCnt ( # ` B ) ) ) ) |
10 |
9
|
breq2d |
|- ( p = P -> ( ( O ` x ) || ( p ^ ( p pCnt ( # ` B ) ) ) <-> ( O ` x ) || ( P ^ ( P pCnt ( # ` B ) ) ) ) ) |
11 |
10
|
rabbidv |
|- ( p = P -> { x e. B | ( O ` x ) || ( p ^ ( p pCnt ( # ` B ) ) ) } = { x e. B | ( O ` x ) || ( P ^ ( P pCnt ( # ` B ) ) ) } ) |
12 |
1
|
fvexi |
|- B e. _V |
13 |
12
|
rabex |
|- { x e. B | ( O ` x ) || ( p ^ ( p pCnt ( # ` B ) ) ) } e. _V |
14 |
11 3 13
|
fvmpt3i |
|- ( P e. A -> ( S ` P ) = { x e. B | ( O ` x ) || ( P ^ ( P pCnt ( # ` B ) ) ) } ) |
15 |
14
|
adantl |
|- ( ( ph /\ P e. A ) -> ( S ` P ) = { x e. B | ( O ` x ) || ( P ^ ( P pCnt ( # ` B ) ) ) } ) |
16 |
15
|
fveq2d |
|- ( ( ph /\ P e. A ) -> ( # ` ( S ` P ) ) = ( # ` { x e. B | ( O ` x ) || ( P ^ ( P pCnt ( # ` B ) ) ) } ) ) |
17 |
|
eqid |
|- { x e. B | ( O ` x ) || ( P ^ ( P pCnt ( # ` B ) ) ) } = { x e. B | ( O ` x ) || ( P ^ ( P pCnt ( # ` B ) ) ) } |
18 |
|
eqid |
|- { x e. B | ( O ` x ) || ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) } = { x e. B | ( O ` x ) || ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) } |
19 |
4
|
adantr |
|- ( ( ph /\ P e. A ) -> G e. Abel ) |
20 |
|
eqid |
|- ( P ^ ( P pCnt ( # ` B ) ) ) = ( P ^ ( P pCnt ( # ` B ) ) ) |
21 |
|
eqid |
|- ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) = ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) |
22 |
1 2 3 4 5 6 20 21
|
ablfac1lem |
|- ( ( ph /\ P e. A ) -> ( ( ( P ^ ( P pCnt ( # ` B ) ) ) e. NN /\ ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) e. NN ) /\ ( ( P ^ ( P pCnt ( # ` B ) ) ) gcd ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) ) = 1 /\ ( # ` B ) = ( ( P ^ ( P pCnt ( # ` B ) ) ) x. ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) ) ) ) |
23 |
22
|
simp1d |
|- ( ( ph /\ P e. A ) -> ( ( P ^ ( P pCnt ( # ` B ) ) ) e. NN /\ ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) e. NN ) ) |
24 |
23
|
simpld |
|- ( ( ph /\ P e. A ) -> ( P ^ ( P pCnt ( # ` B ) ) ) e. NN ) |
25 |
23
|
simprd |
|- ( ( ph /\ P e. A ) -> ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) e. NN ) |
26 |
22
|
simp2d |
|- ( ( ph /\ P e. A ) -> ( ( P ^ ( P pCnt ( # ` B ) ) ) gcd ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) ) = 1 ) |
27 |
22
|
simp3d |
|- ( ( ph /\ P e. A ) -> ( # ` B ) = ( ( P ^ ( P pCnt ( # ` B ) ) ) x. ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) ) ) |
28 |
1 2 17 18 19 24 25 26 27
|
ablfacrp2 |
|- ( ( ph /\ P e. A ) -> ( ( # ` { x e. B | ( O ` x ) || ( P ^ ( P pCnt ( # ` B ) ) ) } ) = ( P ^ ( P pCnt ( # ` B ) ) ) /\ ( # ` { x e. B | ( O ` x ) || ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) } ) = ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) ) ) |
29 |
28
|
simpld |
|- ( ( ph /\ P e. A ) -> ( # ` { x e. B | ( O ` x ) || ( P ^ ( P pCnt ( # ` B ) ) ) } ) = ( P ^ ( P pCnt ( # ` B ) ) ) ) |
30 |
16 29
|
eqtrd |
|- ( ( ph /\ P e. A ) -> ( # ` ( S ` P ) ) = ( P ^ ( P pCnt ( # ` B ) ) ) ) |