Step |
Hyp |
Ref |
Expression |
1 |
|
torsubg.1 |
|- O = ( od ` G ) |
2 |
|
cnvimass |
|- ( `' O " NN ) C_ dom O |
3 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
4 |
3 1
|
odf |
|- O : ( Base ` G ) --> NN0 |
5 |
4
|
fdmi |
|- dom O = ( Base ` G ) |
6 |
2 5
|
sseqtri |
|- ( `' O " NN ) C_ ( Base ` G ) |
7 |
6
|
a1i |
|- ( G e. Abel -> ( `' O " NN ) C_ ( Base ` G ) ) |
8 |
|
ablgrp |
|- ( G e. Abel -> G e. Grp ) |
9 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
10 |
3 9
|
grpidcl |
|- ( G e. Grp -> ( 0g ` G ) e. ( Base ` G ) ) |
11 |
8 10
|
syl |
|- ( G e. Abel -> ( 0g ` G ) e. ( Base ` G ) ) |
12 |
1 9
|
od1 |
|- ( G e. Grp -> ( O ` ( 0g ` G ) ) = 1 ) |
13 |
8 12
|
syl |
|- ( G e. Abel -> ( O ` ( 0g ` G ) ) = 1 ) |
14 |
|
1nn |
|- 1 e. NN |
15 |
13 14
|
eqeltrdi |
|- ( G e. Abel -> ( O ` ( 0g ` G ) ) e. NN ) |
16 |
|
ffn |
|- ( O : ( Base ` G ) --> NN0 -> O Fn ( Base ` G ) ) |
17 |
4 16
|
ax-mp |
|- O Fn ( Base ` G ) |
18 |
|
elpreima |
|- ( O Fn ( Base ` G ) -> ( ( 0g ` G ) e. ( `' O " NN ) <-> ( ( 0g ` G ) e. ( Base ` G ) /\ ( O ` ( 0g ` G ) ) e. NN ) ) ) |
19 |
17 18
|
ax-mp |
|- ( ( 0g ` G ) e. ( `' O " NN ) <-> ( ( 0g ` G ) e. ( Base ` G ) /\ ( O ` ( 0g ` G ) ) e. NN ) ) |
20 |
11 15 19
|
sylanbrc |
|- ( G e. Abel -> ( 0g ` G ) e. ( `' O " NN ) ) |
21 |
20
|
ne0d |
|- ( G e. Abel -> ( `' O " NN ) =/= (/) ) |
22 |
8
|
ad2antrr |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> G e. Grp ) |
23 |
6
|
sseli |
|- ( x e. ( `' O " NN ) -> x e. ( Base ` G ) ) |
24 |
23
|
ad2antlr |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> x e. ( Base ` G ) ) |
25 |
6
|
sseli |
|- ( y e. ( `' O " NN ) -> y e. ( Base ` G ) ) |
26 |
25
|
adantl |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> y e. ( Base ` G ) ) |
27 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
28 |
3 27
|
grpcl |
|- ( ( G e. Grp /\ x e. ( Base ` G ) /\ y e. ( Base ` G ) ) -> ( x ( +g ` G ) y ) e. ( Base ` G ) ) |
29 |
22 24 26 28
|
syl3anc |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( x ( +g ` G ) y ) e. ( Base ` G ) ) |
30 |
|
0nnn |
|- -. 0 e. NN |
31 |
3 1
|
odcl |
|- ( x e. ( Base ` G ) -> ( O ` x ) e. NN0 ) |
32 |
24 31
|
syl |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( O ` x ) e. NN0 ) |
33 |
32
|
nn0zd |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( O ` x ) e. ZZ ) |
34 |
3 1
|
odcl |
|- ( y e. ( Base ` G ) -> ( O ` y ) e. NN0 ) |
35 |
26 34
|
syl |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( O ` y ) e. NN0 ) |
36 |
35
|
nn0zd |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( O ` y ) e. ZZ ) |
37 |
33 36
|
gcdcld |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( ( O ` x ) gcd ( O ` y ) ) e. NN0 ) |
38 |
37
|
nn0cnd |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( ( O ` x ) gcd ( O ` y ) ) e. CC ) |
39 |
38
|
mul02d |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( 0 x. ( ( O ` x ) gcd ( O ` y ) ) ) = 0 ) |
40 |
39
|
breq1d |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( ( 0 x. ( ( O ` x ) gcd ( O ` y ) ) ) || ( ( O ` x ) x. ( O ` y ) ) <-> 0 || ( ( O ` x ) x. ( O ` y ) ) ) ) |
41 |
33 36
|
zmulcld |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( ( O ` x ) x. ( O ` y ) ) e. ZZ ) |
42 |
|
0dvds |
|- ( ( ( O ` x ) x. ( O ` y ) ) e. ZZ -> ( 0 || ( ( O ` x ) x. ( O ` y ) ) <-> ( ( O ` x ) x. ( O ` y ) ) = 0 ) ) |
43 |
41 42
|
syl |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( 0 || ( ( O ` x ) x. ( O ` y ) ) <-> ( ( O ` x ) x. ( O ` y ) ) = 0 ) ) |
44 |
40 43
|
bitrd |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( ( 0 x. ( ( O ` x ) gcd ( O ` y ) ) ) || ( ( O ` x ) x. ( O ` y ) ) <-> ( ( O ` x ) x. ( O ` y ) ) = 0 ) ) |
45 |
|
elpreima |
|- ( O Fn ( Base ` G ) -> ( x e. ( `' O " NN ) <-> ( x e. ( Base ` G ) /\ ( O ` x ) e. NN ) ) ) |
46 |
17 45
|
ax-mp |
|- ( x e. ( `' O " NN ) <-> ( x e. ( Base ` G ) /\ ( O ` x ) e. NN ) ) |
47 |
46
|
simprbi |
|- ( x e. ( `' O " NN ) -> ( O ` x ) e. NN ) |
48 |
47
|
ad2antlr |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( O ` x ) e. NN ) |
49 |
|
elpreima |
|- ( O Fn ( Base ` G ) -> ( y e. ( `' O " NN ) <-> ( y e. ( Base ` G ) /\ ( O ` y ) e. NN ) ) ) |
50 |
17 49
|
ax-mp |
|- ( y e. ( `' O " NN ) <-> ( y e. ( Base ` G ) /\ ( O ` y ) e. NN ) ) |
51 |
50
|
simprbi |
|- ( y e. ( `' O " NN ) -> ( O ` y ) e. NN ) |
52 |
51
|
adantl |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( O ` y ) e. NN ) |
53 |
48 52
|
nnmulcld |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( ( O ` x ) x. ( O ` y ) ) e. NN ) |
54 |
|
eleq1 |
|- ( ( ( O ` x ) x. ( O ` y ) ) = 0 -> ( ( ( O ` x ) x. ( O ` y ) ) e. NN <-> 0 e. NN ) ) |
55 |
53 54
|
syl5ibcom |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( ( ( O ` x ) x. ( O ` y ) ) = 0 -> 0 e. NN ) ) |
56 |
44 55
|
sylbid |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( ( 0 x. ( ( O ` x ) gcd ( O ` y ) ) ) || ( ( O ` x ) x. ( O ` y ) ) -> 0 e. NN ) ) |
57 |
30 56
|
mtoi |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> -. ( 0 x. ( ( O ` x ) gcd ( O ` y ) ) ) || ( ( O ` x ) x. ( O ` y ) ) ) |
58 |
|
simpll |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> G e. Abel ) |
59 |
1 3 27
|
odadd1 |
|- ( ( G e. Abel /\ x e. ( Base ` G ) /\ y e. ( Base ` G ) ) -> ( ( O ` ( x ( +g ` G ) y ) ) x. ( ( O ` x ) gcd ( O ` y ) ) ) || ( ( O ` x ) x. ( O ` y ) ) ) |
60 |
58 24 26 59
|
syl3anc |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( ( O ` ( x ( +g ` G ) y ) ) x. ( ( O ` x ) gcd ( O ` y ) ) ) || ( ( O ` x ) x. ( O ` y ) ) ) |
61 |
|
oveq1 |
|- ( ( O ` ( x ( +g ` G ) y ) ) = 0 -> ( ( O ` ( x ( +g ` G ) y ) ) x. ( ( O ` x ) gcd ( O ` y ) ) ) = ( 0 x. ( ( O ` x ) gcd ( O ` y ) ) ) ) |
62 |
61
|
breq1d |
|- ( ( O ` ( x ( +g ` G ) y ) ) = 0 -> ( ( ( O ` ( x ( +g ` G ) y ) ) x. ( ( O ` x ) gcd ( O ` y ) ) ) || ( ( O ` x ) x. ( O ` y ) ) <-> ( 0 x. ( ( O ` x ) gcd ( O ` y ) ) ) || ( ( O ` x ) x. ( O ` y ) ) ) ) |
63 |
60 62
|
syl5ibcom |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( ( O ` ( x ( +g ` G ) y ) ) = 0 -> ( 0 x. ( ( O ` x ) gcd ( O ` y ) ) ) || ( ( O ` x ) x. ( O ` y ) ) ) ) |
64 |
57 63
|
mtod |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> -. ( O ` ( x ( +g ` G ) y ) ) = 0 ) |
65 |
3 1
|
odcl |
|- ( ( x ( +g ` G ) y ) e. ( Base ` G ) -> ( O ` ( x ( +g ` G ) y ) ) e. NN0 ) |
66 |
29 65
|
syl |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( O ` ( x ( +g ` G ) y ) ) e. NN0 ) |
67 |
|
elnn0 |
|- ( ( O ` ( x ( +g ` G ) y ) ) e. NN0 <-> ( ( O ` ( x ( +g ` G ) y ) ) e. NN \/ ( O ` ( x ( +g ` G ) y ) ) = 0 ) ) |
68 |
66 67
|
sylib |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( ( O ` ( x ( +g ` G ) y ) ) e. NN \/ ( O ` ( x ( +g ` G ) y ) ) = 0 ) ) |
69 |
68
|
ord |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( -. ( O ` ( x ( +g ` G ) y ) ) e. NN -> ( O ` ( x ( +g ` G ) y ) ) = 0 ) ) |
70 |
64 69
|
mt3d |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( O ` ( x ( +g ` G ) y ) ) e. NN ) |
71 |
|
elpreima |
|- ( O Fn ( Base ` G ) -> ( ( x ( +g ` G ) y ) e. ( `' O " NN ) <-> ( ( x ( +g ` G ) y ) e. ( Base ` G ) /\ ( O ` ( x ( +g ` G ) y ) ) e. NN ) ) ) |
72 |
17 71
|
ax-mp |
|- ( ( x ( +g ` G ) y ) e. ( `' O " NN ) <-> ( ( x ( +g ` G ) y ) e. ( Base ` G ) /\ ( O ` ( x ( +g ` G ) y ) ) e. NN ) ) |
73 |
29 70 72
|
sylanbrc |
|- ( ( ( G e. Abel /\ x e. ( `' O " NN ) ) /\ y e. ( `' O " NN ) ) -> ( x ( +g ` G ) y ) e. ( `' O " NN ) ) |
74 |
73
|
ralrimiva |
|- ( ( G e. Abel /\ x e. ( `' O " NN ) ) -> A. y e. ( `' O " NN ) ( x ( +g ` G ) y ) e. ( `' O " NN ) ) |
75 |
|
eqid |
|- ( invg ` G ) = ( invg ` G ) |
76 |
3 75
|
grpinvcl |
|- ( ( G e. Grp /\ x e. ( Base ` G ) ) -> ( ( invg ` G ) ` x ) e. ( Base ` G ) ) |
77 |
8 23 76
|
syl2an |
|- ( ( G e. Abel /\ x e. ( `' O " NN ) ) -> ( ( invg ` G ) ` x ) e. ( Base ` G ) ) |
78 |
1 75 3
|
odinv |
|- ( ( G e. Grp /\ x e. ( Base ` G ) ) -> ( O ` ( ( invg ` G ) ` x ) ) = ( O ` x ) ) |
79 |
8 23 78
|
syl2an |
|- ( ( G e. Abel /\ x e. ( `' O " NN ) ) -> ( O ` ( ( invg ` G ) ` x ) ) = ( O ` x ) ) |
80 |
47
|
adantl |
|- ( ( G e. Abel /\ x e. ( `' O " NN ) ) -> ( O ` x ) e. NN ) |
81 |
79 80
|
eqeltrd |
|- ( ( G e. Abel /\ x e. ( `' O " NN ) ) -> ( O ` ( ( invg ` G ) ` x ) ) e. NN ) |
82 |
|
elpreima |
|- ( O Fn ( Base ` G ) -> ( ( ( invg ` G ) ` x ) e. ( `' O " NN ) <-> ( ( ( invg ` G ) ` x ) e. ( Base ` G ) /\ ( O ` ( ( invg ` G ) ` x ) ) e. NN ) ) ) |
83 |
17 82
|
ax-mp |
|- ( ( ( invg ` G ) ` x ) e. ( `' O " NN ) <-> ( ( ( invg ` G ) ` x ) e. ( Base ` G ) /\ ( O ` ( ( invg ` G ) ` x ) ) e. NN ) ) |
84 |
77 81 83
|
sylanbrc |
|- ( ( G e. Abel /\ x e. ( `' O " NN ) ) -> ( ( invg ` G ) ` x ) e. ( `' O " NN ) ) |
85 |
74 84
|
jca |
|- ( ( G e. Abel /\ x e. ( `' O " NN ) ) -> ( A. y e. ( `' O " NN ) ( x ( +g ` G ) y ) e. ( `' O " NN ) /\ ( ( invg ` G ) ` x ) e. ( `' O " NN ) ) ) |
86 |
85
|
ralrimiva |
|- ( G e. Abel -> A. x e. ( `' O " NN ) ( A. y e. ( `' O " NN ) ( x ( +g ` G ) y ) e. ( `' O " NN ) /\ ( ( invg ` G ) ` x ) e. ( `' O " NN ) ) ) |
87 |
3 27 75
|
issubg2 |
|- ( G e. Grp -> ( ( `' O " NN ) e. ( SubGrp ` G ) <-> ( ( `' O " NN ) C_ ( Base ` G ) /\ ( `' O " NN ) =/= (/) /\ A. x e. ( `' O " NN ) ( A. y e. ( `' O " NN ) ( x ( +g ` G ) y ) e. ( `' O " NN ) /\ ( ( invg ` G ) ` x ) e. ( `' O " NN ) ) ) ) ) |
88 |
8 87
|
syl |
|- ( G e. Abel -> ( ( `' O " NN ) e. ( SubGrp ` G ) <-> ( ( `' O " NN ) C_ ( Base ` G ) /\ ( `' O " NN ) =/= (/) /\ A. x e. ( `' O " NN ) ( A. y e. ( `' O " NN ) ( x ( +g ` G ) y ) e. ( `' O " NN ) /\ ( ( invg ` G ) ` x ) e. ( `' O " NN ) ) ) ) ) |
89 |
7 21 86 88
|
mpbir3and |
|- ( G e. Abel -> ( `' O " NN ) e. ( SubGrp ` G ) ) |