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 |
|
ablfac1.m |
|- M = ( P ^ ( P pCnt ( # ` B ) ) ) |
8 |
|
ablfac1.n |
|- N = ( ( # ` B ) / M ) |
9 |
6
|
sselda |
|- ( ( ph /\ P e. A ) -> P e. Prime ) |
10 |
|
prmnn |
|- ( P e. Prime -> P e. NN ) |
11 |
9 10
|
syl |
|- ( ( ph /\ P e. A ) -> P e. NN ) |
12 |
|
ablgrp |
|- ( G e. Abel -> G e. Grp ) |
13 |
1
|
grpbn0 |
|- ( G e. Grp -> B =/= (/) ) |
14 |
4 12 13
|
3syl |
|- ( ph -> B =/= (/) ) |
15 |
|
hashnncl |
|- ( B e. Fin -> ( ( # ` B ) e. NN <-> B =/= (/) ) ) |
16 |
5 15
|
syl |
|- ( ph -> ( ( # ` B ) e. NN <-> B =/= (/) ) ) |
17 |
14 16
|
mpbird |
|- ( ph -> ( # ` B ) e. NN ) |
18 |
17
|
adantr |
|- ( ( ph /\ P e. A ) -> ( # ` B ) e. NN ) |
19 |
9 18
|
pccld |
|- ( ( ph /\ P e. A ) -> ( P pCnt ( # ` B ) ) e. NN0 ) |
20 |
11 19
|
nnexpcld |
|- ( ( ph /\ P e. A ) -> ( P ^ ( P pCnt ( # ` B ) ) ) e. NN ) |
21 |
7 20
|
eqeltrid |
|- ( ( ph /\ P e. A ) -> M e. NN ) |
22 |
|
pcdvds |
|- ( ( P e. Prime /\ ( # ` B ) e. NN ) -> ( P ^ ( P pCnt ( # ` B ) ) ) || ( # ` B ) ) |
23 |
9 18 22
|
syl2anc |
|- ( ( ph /\ P e. A ) -> ( P ^ ( P pCnt ( # ` B ) ) ) || ( # ` B ) ) |
24 |
7 23
|
eqbrtrid |
|- ( ( ph /\ P e. A ) -> M || ( # ` B ) ) |
25 |
|
nndivdvds |
|- ( ( ( # ` B ) e. NN /\ M e. NN ) -> ( M || ( # ` B ) <-> ( ( # ` B ) / M ) e. NN ) ) |
26 |
18 21 25
|
syl2anc |
|- ( ( ph /\ P e. A ) -> ( M || ( # ` B ) <-> ( ( # ` B ) / M ) e. NN ) ) |
27 |
24 26
|
mpbid |
|- ( ( ph /\ P e. A ) -> ( ( # ` B ) / M ) e. NN ) |
28 |
8 27
|
eqeltrid |
|- ( ( ph /\ P e. A ) -> N e. NN ) |
29 |
21 28
|
jca |
|- ( ( ph /\ P e. A ) -> ( M e. NN /\ N e. NN ) ) |
30 |
7
|
oveq1i |
|- ( M gcd N ) = ( ( P ^ ( P pCnt ( # ` B ) ) ) gcd N ) |
31 |
|
pcndvds2 |
|- ( ( P e. Prime /\ ( # ` B ) e. NN ) -> -. P || ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) ) |
32 |
9 18 31
|
syl2anc |
|- ( ( ph /\ P e. A ) -> -. P || ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) ) |
33 |
7
|
oveq2i |
|- ( ( # ` B ) / M ) = ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) |
34 |
8 33
|
eqtri |
|- N = ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) |
35 |
34
|
breq2i |
|- ( P || N <-> P || ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) ) |
36 |
32 35
|
sylnibr |
|- ( ( ph /\ P e. A ) -> -. P || N ) |
37 |
28
|
nnzd |
|- ( ( ph /\ P e. A ) -> N e. ZZ ) |
38 |
|
coprm |
|- ( ( P e. Prime /\ N e. ZZ ) -> ( -. P || N <-> ( P gcd N ) = 1 ) ) |
39 |
9 37 38
|
syl2anc |
|- ( ( ph /\ P e. A ) -> ( -. P || N <-> ( P gcd N ) = 1 ) ) |
40 |
36 39
|
mpbid |
|- ( ( ph /\ P e. A ) -> ( P gcd N ) = 1 ) |
41 |
|
prmz |
|- ( P e. Prime -> P e. ZZ ) |
42 |
9 41
|
syl |
|- ( ( ph /\ P e. A ) -> P e. ZZ ) |
43 |
|
rpexp1i |
|- ( ( P e. ZZ /\ N e. ZZ /\ ( P pCnt ( # ` B ) ) e. NN0 ) -> ( ( P gcd N ) = 1 -> ( ( P ^ ( P pCnt ( # ` B ) ) ) gcd N ) = 1 ) ) |
44 |
42 37 19 43
|
syl3anc |
|- ( ( ph /\ P e. A ) -> ( ( P gcd N ) = 1 -> ( ( P ^ ( P pCnt ( # ` B ) ) ) gcd N ) = 1 ) ) |
45 |
40 44
|
mpd |
|- ( ( ph /\ P e. A ) -> ( ( P ^ ( P pCnt ( # ` B ) ) ) gcd N ) = 1 ) |
46 |
30 45
|
syl5eq |
|- ( ( ph /\ P e. A ) -> ( M gcd N ) = 1 ) |
47 |
8
|
oveq2i |
|- ( M x. N ) = ( M x. ( ( # ` B ) / M ) ) |
48 |
18
|
nncnd |
|- ( ( ph /\ P e. A ) -> ( # ` B ) e. CC ) |
49 |
21
|
nncnd |
|- ( ( ph /\ P e. A ) -> M e. CC ) |
50 |
21
|
nnne0d |
|- ( ( ph /\ P e. A ) -> M =/= 0 ) |
51 |
48 49 50
|
divcan2d |
|- ( ( ph /\ P e. A ) -> ( M x. ( ( # ` B ) / M ) ) = ( # ` B ) ) |
52 |
47 51
|
eqtr2id |
|- ( ( ph /\ P e. A ) -> ( # ` B ) = ( M x. N ) ) |
53 |
29 46 52
|
3jca |
|- ( ( ph /\ P e. A ) -> ( ( M e. NN /\ N e. NN ) /\ ( M gcd N ) = 1 /\ ( # ` B ) = ( M x. N ) ) ) |