Step |
Hyp |
Ref |
Expression |
1 |
|
odmulgid.1 |
|- X = ( Base ` G ) |
2 |
|
odmulgid.2 |
|- O = ( od ` G ) |
3 |
|
odmulgid.3 |
|- .x. = ( .g ` G ) |
4 |
1 3
|
mulgcl |
|- ( ( G e. Grp /\ N e. ZZ /\ A e. X ) -> ( N .x. A ) e. X ) |
5 |
4
|
3com23 |
|- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> ( N .x. A ) e. X ) |
6 |
1 2
|
odcl |
|- ( ( N .x. A ) e. X -> ( O ` ( N .x. A ) ) e. NN0 ) |
7 |
5 6
|
syl |
|- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> ( O ` ( N .x. A ) ) e. NN0 ) |
8 |
7
|
nn0cnd |
|- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> ( O ` ( N .x. A ) ) e. CC ) |
9 |
8
|
adantr |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) = 0 ) -> ( O ` ( N .x. A ) ) e. CC ) |
10 |
9
|
mul02d |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) = 0 ) -> ( 0 x. ( O ` ( N .x. A ) ) ) = 0 ) |
11 |
|
simpr |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) = 0 ) -> ( N gcd ( O ` A ) ) = 0 ) |
12 |
11
|
oveq1d |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) = 0 ) -> ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) = ( 0 x. ( O ` ( N .x. A ) ) ) ) |
13 |
|
simp3 |
|- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> N e. ZZ ) |
14 |
1 2
|
odcl |
|- ( A e. X -> ( O ` A ) e. NN0 ) |
15 |
14
|
3ad2ant2 |
|- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> ( O ` A ) e. NN0 ) |
16 |
15
|
nn0zd |
|- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> ( O ` A ) e. ZZ ) |
17 |
|
gcdeq0 |
|- ( ( N e. ZZ /\ ( O ` A ) e. ZZ ) -> ( ( N gcd ( O ` A ) ) = 0 <-> ( N = 0 /\ ( O ` A ) = 0 ) ) ) |
18 |
13 16 17
|
syl2anc |
|- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> ( ( N gcd ( O ` A ) ) = 0 <-> ( N = 0 /\ ( O ` A ) = 0 ) ) ) |
19 |
18
|
simplbda |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) = 0 ) -> ( O ` A ) = 0 ) |
20 |
10 12 19
|
3eqtr4rd |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) = 0 ) -> ( O ` A ) = ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) ) |
21 |
|
simpll3 |
|- ( ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) /\ x e. NN0 ) -> N e. ZZ ) |
22 |
16
|
ad2antrr |
|- ( ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) /\ x e. NN0 ) -> ( O ` A ) e. ZZ ) |
23 |
|
gcddvds |
|- ( ( N e. ZZ /\ ( O ` A ) e. ZZ ) -> ( ( N gcd ( O ` A ) ) || N /\ ( N gcd ( O ` A ) ) || ( O ` A ) ) ) |
24 |
21 22 23
|
syl2anc |
|- ( ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) /\ x e. NN0 ) -> ( ( N gcd ( O ` A ) ) || N /\ ( N gcd ( O ` A ) ) || ( O ` A ) ) ) |
25 |
24
|
simprd |
|- ( ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) /\ x e. NN0 ) -> ( N gcd ( O ` A ) ) || ( O ` A ) ) |
26 |
13 16
|
gcdcld |
|- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> ( N gcd ( O ` A ) ) e. NN0 ) |
27 |
26
|
adantr |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) -> ( N gcd ( O ` A ) ) e. NN0 ) |
28 |
27
|
nn0zd |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) -> ( N gcd ( O ` A ) ) e. ZZ ) |
29 |
28
|
adantr |
|- ( ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) /\ x e. NN0 ) -> ( N gcd ( O ` A ) ) e. ZZ ) |
30 |
|
nn0z |
|- ( x e. NN0 -> x e. ZZ ) |
31 |
30
|
adantl |
|- ( ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) /\ x e. NN0 ) -> x e. ZZ ) |
32 |
|
dvdstr |
|- ( ( ( N gcd ( O ` A ) ) e. ZZ /\ ( O ` A ) e. ZZ /\ x e. ZZ ) -> ( ( ( N gcd ( O ` A ) ) || ( O ` A ) /\ ( O ` A ) || x ) -> ( N gcd ( O ` A ) ) || x ) ) |
33 |
29 22 31 32
|
syl3anc |
|- ( ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) /\ x e. NN0 ) -> ( ( ( N gcd ( O ` A ) ) || ( O ` A ) /\ ( O ` A ) || x ) -> ( N gcd ( O ` A ) ) || x ) ) |
34 |
25 33
|
mpand |
|- ( ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) /\ x e. NN0 ) -> ( ( O ` A ) || x -> ( N gcd ( O ` A ) ) || x ) ) |
35 |
7
|
nn0zd |
|- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> ( O ` ( N .x. A ) ) e. ZZ ) |
36 |
35
|
ad2antrr |
|- ( ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) /\ x e. NN0 ) -> ( O ` ( N .x. A ) ) e. ZZ ) |
37 |
|
muldvds1 |
|- ( ( ( N gcd ( O ` A ) ) e. ZZ /\ ( O ` ( N .x. A ) ) e. ZZ /\ x e. ZZ ) -> ( ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || x -> ( N gcd ( O ` A ) ) || x ) ) |
38 |
29 36 31 37
|
syl3anc |
|- ( ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) /\ x e. NN0 ) -> ( ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || x -> ( N gcd ( O ` A ) ) || x ) ) |
39 |
|
dvdszrcl |
|- ( ( N gcd ( O ` A ) ) || x -> ( ( N gcd ( O ` A ) ) e. ZZ /\ x e. ZZ ) ) |
40 |
|
divides |
|- ( ( ( N gcd ( O ` A ) ) e. ZZ /\ x e. ZZ ) -> ( ( N gcd ( O ` A ) ) || x <-> E. y e. ZZ ( y x. ( N gcd ( O ` A ) ) ) = x ) ) |
41 |
39 40
|
syl |
|- ( ( N gcd ( O ` A ) ) || x -> ( ( N gcd ( O ` A ) ) || x <-> E. y e. ZZ ( y x. ( N gcd ( O ` A ) ) ) = x ) ) |
42 |
41
|
ibi |
|- ( ( N gcd ( O ` A ) ) || x -> E. y e. ZZ ( y x. ( N gcd ( O ` A ) ) ) = x ) |
43 |
35
|
adantr |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( ( N gcd ( O ` A ) ) =/= 0 /\ y e. ZZ ) ) -> ( O ` ( N .x. A ) ) e. ZZ ) |
44 |
|
simprr |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( ( N gcd ( O ` A ) ) =/= 0 /\ y e. ZZ ) ) -> y e. ZZ ) |
45 |
28
|
adantrr |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( ( N gcd ( O ` A ) ) =/= 0 /\ y e. ZZ ) ) -> ( N gcd ( O ` A ) ) e. ZZ ) |
46 |
|
simprl |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( ( N gcd ( O ` A ) ) =/= 0 /\ y e. ZZ ) ) -> ( N gcd ( O ` A ) ) =/= 0 ) |
47 |
|
dvdscmulr |
|- ( ( ( O ` ( N .x. A ) ) e. ZZ /\ y e. ZZ /\ ( ( N gcd ( O ` A ) ) e. ZZ /\ ( N gcd ( O ` A ) ) =/= 0 ) ) -> ( ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || ( ( N gcd ( O ` A ) ) x. y ) <-> ( O ` ( N .x. A ) ) || y ) ) |
48 |
43 44 45 46 47
|
syl112anc |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( ( N gcd ( O ` A ) ) =/= 0 /\ y e. ZZ ) ) -> ( ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || ( ( N gcd ( O ` A ) ) x. y ) <-> ( O ` ( N .x. A ) ) || y ) ) |
49 |
1 2 3
|
odmulgid |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ y e. ZZ ) -> ( ( O ` ( N .x. A ) ) || y <-> ( O ` A ) || ( y x. N ) ) ) |
50 |
49
|
adantrl |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( ( N gcd ( O ` A ) ) =/= 0 /\ y e. ZZ ) ) -> ( ( O ` ( N .x. A ) ) || y <-> ( O ` A ) || ( y x. N ) ) ) |
51 |
|
simpl3 |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( ( N gcd ( O ` A ) ) =/= 0 /\ y e. ZZ ) ) -> N e. ZZ ) |
52 |
|
dvdsmulgcd |
|- ( ( y e. ZZ /\ N e. ZZ ) -> ( ( O ` A ) || ( y x. N ) <-> ( O ` A ) || ( y x. ( N gcd ( O ` A ) ) ) ) ) |
53 |
44 51 52
|
syl2anc |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( ( N gcd ( O ` A ) ) =/= 0 /\ y e. ZZ ) ) -> ( ( O ` A ) || ( y x. N ) <-> ( O ` A ) || ( y x. ( N gcd ( O ` A ) ) ) ) ) |
54 |
48 50 53
|
3bitrrd |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( ( N gcd ( O ` A ) ) =/= 0 /\ y e. ZZ ) ) -> ( ( O ` A ) || ( y x. ( N gcd ( O ` A ) ) ) <-> ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || ( ( N gcd ( O ` A ) ) x. y ) ) ) |
55 |
45
|
zcnd |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( ( N gcd ( O ` A ) ) =/= 0 /\ y e. ZZ ) ) -> ( N gcd ( O ` A ) ) e. CC ) |
56 |
44
|
zcnd |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( ( N gcd ( O ` A ) ) =/= 0 /\ y e. ZZ ) ) -> y e. CC ) |
57 |
55 56
|
mulcomd |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( ( N gcd ( O ` A ) ) =/= 0 /\ y e. ZZ ) ) -> ( ( N gcd ( O ` A ) ) x. y ) = ( y x. ( N gcd ( O ` A ) ) ) ) |
58 |
57
|
breq2d |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( ( N gcd ( O ` A ) ) =/= 0 /\ y e. ZZ ) ) -> ( ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || ( ( N gcd ( O ` A ) ) x. y ) <-> ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || ( y x. ( N gcd ( O ` A ) ) ) ) ) |
59 |
54 58
|
bitrd |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( ( N gcd ( O ` A ) ) =/= 0 /\ y e. ZZ ) ) -> ( ( O ` A ) || ( y x. ( N gcd ( O ` A ) ) ) <-> ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || ( y x. ( N gcd ( O ` A ) ) ) ) ) |
60 |
59
|
anassrs |
|- ( ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) /\ y e. ZZ ) -> ( ( O ` A ) || ( y x. ( N gcd ( O ` A ) ) ) <-> ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || ( y x. ( N gcd ( O ` A ) ) ) ) ) |
61 |
|
breq2 |
|- ( ( y x. ( N gcd ( O ` A ) ) ) = x -> ( ( O ` A ) || ( y x. ( N gcd ( O ` A ) ) ) <-> ( O ` A ) || x ) ) |
62 |
|
breq2 |
|- ( ( y x. ( N gcd ( O ` A ) ) ) = x -> ( ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || ( y x. ( N gcd ( O ` A ) ) ) <-> ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || x ) ) |
63 |
61 62
|
bibi12d |
|- ( ( y x. ( N gcd ( O ` A ) ) ) = x -> ( ( ( O ` A ) || ( y x. ( N gcd ( O ` A ) ) ) <-> ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || ( y x. ( N gcd ( O ` A ) ) ) ) <-> ( ( O ` A ) || x <-> ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || x ) ) ) |
64 |
60 63
|
syl5ibcom |
|- ( ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) /\ y e. ZZ ) -> ( ( y x. ( N gcd ( O ` A ) ) ) = x -> ( ( O ` A ) || x <-> ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || x ) ) ) |
65 |
64
|
rexlimdva |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) -> ( E. y e. ZZ ( y x. ( N gcd ( O ` A ) ) ) = x -> ( ( O ` A ) || x <-> ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || x ) ) ) |
66 |
42 65
|
syl5 |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) -> ( ( N gcd ( O ` A ) ) || x -> ( ( O ` A ) || x <-> ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || x ) ) ) |
67 |
66
|
adantr |
|- ( ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) /\ x e. NN0 ) -> ( ( N gcd ( O ` A ) ) || x -> ( ( O ` A ) || x <-> ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || x ) ) ) |
68 |
34 38 67
|
pm5.21ndd |
|- ( ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) /\ x e. NN0 ) -> ( ( O ` A ) || x <-> ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || x ) ) |
69 |
68
|
ralrimiva |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) -> A. x e. NN0 ( ( O ` A ) || x <-> ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || x ) ) |
70 |
15
|
adantr |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) -> ( O ` A ) e. NN0 ) |
71 |
7
|
adantr |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) -> ( O ` ( N .x. A ) ) e. NN0 ) |
72 |
27 71
|
nn0mulcld |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) -> ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) e. NN0 ) |
73 |
|
dvdsext |
|- ( ( ( O ` A ) e. NN0 /\ ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) e. NN0 ) -> ( ( O ` A ) = ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) <-> A. x e. NN0 ( ( O ` A ) || x <-> ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || x ) ) ) |
74 |
70 72 73
|
syl2anc |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) -> ( ( O ` A ) = ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) <-> A. x e. NN0 ( ( O ` A ) || x <-> ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) || x ) ) ) |
75 |
69 74
|
mpbird |
|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N gcd ( O ` A ) ) =/= 0 ) -> ( O ` A ) = ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) ) |
76 |
20 75
|
pm2.61dane |
|- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> ( O ` A ) = ( ( N gcd ( O ` A ) ) x. ( O ` ( N .x. A ) ) ) ) |