Step |
Hyp |
Ref |
Expression |
1 |
|
odadd1.1 |
|- O = ( od ` G ) |
2 |
|
odadd1.2 |
|- X = ( Base ` G ) |
3 |
|
odadd1.3 |
|- .+ = ( +g ` G ) |
4 |
|
ablgrp |
|- ( G e. Abel -> G e. Grp ) |
5 |
2 3
|
grpcl |
|- ( ( G e. Grp /\ A e. X /\ B e. X ) -> ( A .+ B ) e. X ) |
6 |
4 5
|
syl3an1 |
|- ( ( G e. Abel /\ A e. X /\ B e. X ) -> ( A .+ B ) e. X ) |
7 |
2 1
|
odcl |
|- ( ( A .+ B ) e. X -> ( O ` ( A .+ B ) ) e. NN0 ) |
8 |
6 7
|
syl |
|- ( ( G e. Abel /\ A e. X /\ B e. X ) -> ( O ` ( A .+ B ) ) e. NN0 ) |
9 |
8
|
nn0zd |
|- ( ( G e. Abel /\ A e. X /\ B e. X ) -> ( O ` ( A .+ B ) ) e. ZZ ) |
10 |
2 1
|
odcl |
|- ( A e. X -> ( O ` A ) e. NN0 ) |
11 |
10
|
3ad2ant2 |
|- ( ( G e. Abel /\ A e. X /\ B e. X ) -> ( O ` A ) e. NN0 ) |
12 |
11
|
nn0zd |
|- ( ( G e. Abel /\ A e. X /\ B e. X ) -> ( O ` A ) e. ZZ ) |
13 |
2 1
|
odcl |
|- ( B e. X -> ( O ` B ) e. NN0 ) |
14 |
13
|
3ad2ant3 |
|- ( ( G e. Abel /\ A e. X /\ B e. X ) -> ( O ` B ) e. NN0 ) |
15 |
14
|
nn0zd |
|- ( ( G e. Abel /\ A e. X /\ B e. X ) -> ( O ` B ) e. ZZ ) |
16 |
12 15
|
gcdcld |
|- ( ( G e. Abel /\ A e. X /\ B e. X ) -> ( ( O ` A ) gcd ( O ` B ) ) e. NN0 ) |
17 |
16
|
nn0zd |
|- ( ( G e. Abel /\ A e. X /\ B e. X ) -> ( ( O ` A ) gcd ( O ` B ) ) e. ZZ ) |
18 |
9 17
|
zmulcld |
|- ( ( G e. Abel /\ A e. X /\ B e. X ) -> ( ( O ` ( A .+ B ) ) x. ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ ) |
19 |
18
|
adantr |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) = 0 ) -> ( ( O ` ( A .+ B ) ) x. ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ ) |
20 |
|
dvds0 |
|- ( ( ( O ` ( A .+ B ) ) x. ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ -> ( ( O ` ( A .+ B ) ) x. ( ( O ` A ) gcd ( O ` B ) ) ) || 0 ) |
21 |
19 20
|
syl |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) = 0 ) -> ( ( O ` ( A .+ B ) ) x. ( ( O ` A ) gcd ( O ` B ) ) ) || 0 ) |
22 |
|
gcdeq0 |
|- ( ( ( O ` A ) e. ZZ /\ ( O ` B ) e. ZZ ) -> ( ( ( O ` A ) gcd ( O ` B ) ) = 0 <-> ( ( O ` A ) = 0 /\ ( O ` B ) = 0 ) ) ) |
23 |
12 15 22
|
syl2anc |
|- ( ( G e. Abel /\ A e. X /\ B e. X ) -> ( ( ( O ` A ) gcd ( O ` B ) ) = 0 <-> ( ( O ` A ) = 0 /\ ( O ` B ) = 0 ) ) ) |
24 |
23
|
biimpa |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) = 0 ) -> ( ( O ` A ) = 0 /\ ( O ` B ) = 0 ) ) |
25 |
|
oveq12 |
|- ( ( ( O ` A ) = 0 /\ ( O ` B ) = 0 ) -> ( ( O ` A ) x. ( O ` B ) ) = ( 0 x. 0 ) ) |
26 |
|
0cn |
|- 0 e. CC |
27 |
26
|
mul01i |
|- ( 0 x. 0 ) = 0 |
28 |
25 27
|
eqtrdi |
|- ( ( ( O ` A ) = 0 /\ ( O ` B ) = 0 ) -> ( ( O ` A ) x. ( O ` B ) ) = 0 ) |
29 |
24 28
|
syl |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) = 0 ) -> ( ( O ` A ) x. ( O ` B ) ) = 0 ) |
30 |
21 29
|
breqtrrd |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) = 0 ) -> ( ( O ` ( A .+ B ) ) x. ( ( O ` A ) gcd ( O ` B ) ) ) || ( ( O ` A ) x. ( O ` B ) ) ) |
31 |
|
simpl1 |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> G e. Abel ) |
32 |
17
|
adantr |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( O ` A ) gcd ( O ` B ) ) e. ZZ ) |
33 |
12
|
adantr |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( O ` A ) e. ZZ ) |
34 |
15
|
adantr |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( O ` B ) e. ZZ ) |
35 |
|
gcddvds |
|- ( ( ( O ` A ) e. ZZ /\ ( O ` B ) e. ZZ ) -> ( ( ( O ` A ) gcd ( O ` B ) ) || ( O ` A ) /\ ( ( O ` A ) gcd ( O ` B ) ) || ( O ` B ) ) ) |
36 |
33 34 35
|
syl2anc |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( ( O ` A ) gcd ( O ` B ) ) || ( O ` A ) /\ ( ( O ` A ) gcd ( O ` B ) ) || ( O ` B ) ) ) |
37 |
36
|
simpld |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( O ` A ) gcd ( O ` B ) ) || ( O ` A ) ) |
38 |
32 33 34 37
|
dvdsmultr1d |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( O ` A ) gcd ( O ` B ) ) || ( ( O ` A ) x. ( O ` B ) ) ) |
39 |
|
simpr |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) |
40 |
33 34
|
zmulcld |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( O ` A ) x. ( O ` B ) ) e. ZZ ) |
41 |
|
dvdsval2 |
|- ( ( ( ( O ` A ) gcd ( O ` B ) ) e. ZZ /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 /\ ( ( O ` A ) x. ( O ` B ) ) e. ZZ ) -> ( ( ( O ` A ) gcd ( O ` B ) ) || ( ( O ` A ) x. ( O ` B ) ) <-> ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ ) ) |
42 |
32 39 40 41
|
syl3anc |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( ( O ` A ) gcd ( O ` B ) ) || ( ( O ` A ) x. ( O ` B ) ) <-> ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ ) ) |
43 |
38 42
|
mpbid |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ ) |
44 |
|
simpl2 |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> A e. X ) |
45 |
|
simpl3 |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> B e. X ) |
46 |
|
eqid |
|- ( .g ` G ) = ( .g ` G ) |
47 |
2 46 3
|
mulgdi |
|- ( ( G e. Abel /\ ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ /\ A e. X /\ B e. X ) ) -> ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ( .g ` G ) ( A .+ B ) ) = ( ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ( .g ` G ) A ) .+ ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ( .g ` G ) B ) ) ) |
48 |
31 43 44 45 47
|
syl13anc |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ( .g ` G ) ( A .+ B ) ) = ( ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ( .g ` G ) A ) .+ ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ( .g ` G ) B ) ) ) |
49 |
36
|
simprd |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( O ` A ) gcd ( O ` B ) ) || ( O ` B ) ) |
50 |
|
dvdsval2 |
|- ( ( ( ( O ` A ) gcd ( O ` B ) ) e. ZZ /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 /\ ( O ` B ) e. ZZ ) -> ( ( ( O ` A ) gcd ( O ` B ) ) || ( O ` B ) <-> ( ( O ` B ) / ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ ) ) |
51 |
32 39 34 50
|
syl3anc |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( ( O ` A ) gcd ( O ` B ) ) || ( O ` B ) <-> ( ( O ` B ) / ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ ) ) |
52 |
49 51
|
mpbid |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( O ` B ) / ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ ) |
53 |
|
dvdsmul1 |
|- ( ( ( O ` A ) e. ZZ /\ ( ( O ` B ) / ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ ) -> ( O ` A ) || ( ( O ` A ) x. ( ( O ` B ) / ( ( O ` A ) gcd ( O ` B ) ) ) ) ) |
54 |
33 52 53
|
syl2anc |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( O ` A ) || ( ( O ` A ) x. ( ( O ` B ) / ( ( O ` A ) gcd ( O ` B ) ) ) ) ) |
55 |
33
|
zcnd |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( O ` A ) e. CC ) |
56 |
34
|
zcnd |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( O ` B ) e. CC ) |
57 |
32
|
zcnd |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( O ` A ) gcd ( O ` B ) ) e. CC ) |
58 |
55 56 57 39
|
divassd |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) = ( ( O ` A ) x. ( ( O ` B ) / ( ( O ` A ) gcd ( O ` B ) ) ) ) ) |
59 |
54 58
|
breqtrrd |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( O ` A ) || ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ) |
60 |
31 4
|
syl |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> G e. Grp ) |
61 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
62 |
2 1 46 61
|
oddvds |
|- ( ( G e. Grp /\ A e. X /\ ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ ) -> ( ( O ` A ) || ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) <-> ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ( .g ` G ) A ) = ( 0g ` G ) ) ) |
63 |
60 44 43 62
|
syl3anc |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( O ` A ) || ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) <-> ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ( .g ` G ) A ) = ( 0g ` G ) ) ) |
64 |
59 63
|
mpbid |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ( .g ` G ) A ) = ( 0g ` G ) ) |
65 |
|
dvdsval2 |
|- ( ( ( ( O ` A ) gcd ( O ` B ) ) e. ZZ /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 /\ ( O ` A ) e. ZZ ) -> ( ( ( O ` A ) gcd ( O ` B ) ) || ( O ` A ) <-> ( ( O ` A ) / ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ ) ) |
66 |
32 39 33 65
|
syl3anc |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( ( O ` A ) gcd ( O ` B ) ) || ( O ` A ) <-> ( ( O ` A ) / ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ ) ) |
67 |
37 66
|
mpbid |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( O ` A ) / ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ ) |
68 |
|
dvdsmul1 |
|- ( ( ( O ` B ) e. ZZ /\ ( ( O ` A ) / ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ ) -> ( O ` B ) || ( ( O ` B ) x. ( ( O ` A ) / ( ( O ` A ) gcd ( O ` B ) ) ) ) ) |
69 |
34 67 68
|
syl2anc |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( O ` B ) || ( ( O ` B ) x. ( ( O ` A ) / ( ( O ` A ) gcd ( O ` B ) ) ) ) ) |
70 |
55 56
|
mulcomd |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( O ` A ) x. ( O ` B ) ) = ( ( O ` B ) x. ( O ` A ) ) ) |
71 |
70
|
oveq1d |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) = ( ( ( O ` B ) x. ( O ` A ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ) |
72 |
56 55 57 39
|
divassd |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( ( O ` B ) x. ( O ` A ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) = ( ( O ` B ) x. ( ( O ` A ) / ( ( O ` A ) gcd ( O ` B ) ) ) ) ) |
73 |
71 72
|
eqtrd |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) = ( ( O ` B ) x. ( ( O ` A ) / ( ( O ` A ) gcd ( O ` B ) ) ) ) ) |
74 |
69 73
|
breqtrrd |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( O ` B ) || ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ) |
75 |
2 1 46 61
|
oddvds |
|- ( ( G e. Grp /\ B e. X /\ ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ ) -> ( ( O ` B ) || ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) <-> ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ( .g ` G ) B ) = ( 0g ` G ) ) ) |
76 |
60 45 43 75
|
syl3anc |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( O ` B ) || ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) <-> ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ( .g ` G ) B ) = ( 0g ` G ) ) ) |
77 |
74 76
|
mpbid |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ( .g ` G ) B ) = ( 0g ` G ) ) |
78 |
64 77
|
oveq12d |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ( .g ` G ) A ) .+ ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ( .g ` G ) B ) ) = ( ( 0g ` G ) .+ ( 0g ` G ) ) ) |
79 |
2 61
|
grpidcl |
|- ( G e. Grp -> ( 0g ` G ) e. X ) |
80 |
2 3 61
|
grplid |
|- ( ( G e. Grp /\ ( 0g ` G ) e. X ) -> ( ( 0g ` G ) .+ ( 0g ` G ) ) = ( 0g ` G ) ) |
81 |
60 79 80
|
syl2anc2 |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( 0g ` G ) .+ ( 0g ` G ) ) = ( 0g ` G ) ) |
82 |
48 78 81
|
3eqtrd |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ( .g ` G ) ( A .+ B ) ) = ( 0g ` G ) ) |
83 |
6
|
adantr |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( A .+ B ) e. X ) |
84 |
2 1 46 61
|
oddvds |
|- ( ( G e. Grp /\ ( A .+ B ) e. X /\ ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ ) -> ( ( O ` ( A .+ B ) ) || ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) <-> ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ( .g ` G ) ( A .+ B ) ) = ( 0g ` G ) ) ) |
85 |
60 83 43 84
|
syl3anc |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( O ` ( A .+ B ) ) || ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) <-> ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ( .g ` G ) ( A .+ B ) ) = ( 0g ` G ) ) ) |
86 |
82 85
|
mpbird |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( O ` ( A .+ B ) ) || ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ) |
87 |
9
|
adantr |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( O ` ( A .+ B ) ) e. ZZ ) |
88 |
|
dvdsmulcr |
|- ( ( ( O ` ( A .+ B ) ) e. ZZ /\ ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) e. ZZ /\ ( ( ( O ` A ) gcd ( O ` B ) ) e. ZZ /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) ) -> ( ( ( O ` ( A .+ B ) ) x. ( ( O ` A ) gcd ( O ` B ) ) ) || ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) x. ( ( O ` A ) gcd ( O ` B ) ) ) <-> ( O ` ( A .+ B ) ) || ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ) ) |
89 |
87 43 32 39 88
|
syl112anc |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( ( O ` ( A .+ B ) ) x. ( ( O ` A ) gcd ( O ` B ) ) ) || ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) x. ( ( O ` A ) gcd ( O ` B ) ) ) <-> ( O ` ( A .+ B ) ) || ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) ) ) |
90 |
86 89
|
mpbird |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( O ` ( A .+ B ) ) x. ( ( O ` A ) gcd ( O ` B ) ) ) || ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) x. ( ( O ` A ) gcd ( O ` B ) ) ) ) |
91 |
40
|
zcnd |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( O ` A ) x. ( O ` B ) ) e. CC ) |
92 |
91 57 39
|
divcan1d |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( ( ( O ` A ) x. ( O ` B ) ) / ( ( O ` A ) gcd ( O ` B ) ) ) x. ( ( O ` A ) gcd ( O ` B ) ) ) = ( ( O ` A ) x. ( O ` B ) ) ) |
93 |
90 92
|
breqtrd |
|- ( ( ( G e. Abel /\ A e. X /\ B e. X ) /\ ( ( O ` A ) gcd ( O ` B ) ) =/= 0 ) -> ( ( O ` ( A .+ B ) ) x. ( ( O ` A ) gcd ( O ` B ) ) ) || ( ( O ` A ) x. ( O ` B ) ) ) |
94 |
30 93
|
pm2.61dane |
|- ( ( G e. Abel /\ A e. X /\ B e. X ) -> ( ( O ` ( A .+ B ) ) x. ( ( O ` A ) gcd ( O ` B ) ) ) || ( ( O ` A ) x. ( O ` B ) ) ) |