Step |
Hyp |
Ref |
Expression |
1 |
|
torsubg.1 |
⊢ 𝑂 = ( od ‘ 𝐺 ) |
2 |
|
cnvimass |
⊢ ( ◡ 𝑂 “ ℕ ) ⊆ dom 𝑂 |
3 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
4 |
3 1
|
odf |
⊢ 𝑂 : ( Base ‘ 𝐺 ) ⟶ ℕ0 |
5 |
4
|
fdmi |
⊢ dom 𝑂 = ( Base ‘ 𝐺 ) |
6 |
2 5
|
sseqtri |
⊢ ( ◡ 𝑂 “ ℕ ) ⊆ ( Base ‘ 𝐺 ) |
7 |
6
|
a1i |
⊢ ( 𝐺 ∈ Abel → ( ◡ 𝑂 “ ℕ ) ⊆ ( Base ‘ 𝐺 ) ) |
8 |
|
ablgrp |
⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp ) |
9 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
10 |
3 9
|
grpidcl |
⊢ ( 𝐺 ∈ Grp → ( 0g ‘ 𝐺 ) ∈ ( Base ‘ 𝐺 ) ) |
11 |
8 10
|
syl |
⊢ ( 𝐺 ∈ Abel → ( 0g ‘ 𝐺 ) ∈ ( Base ‘ 𝐺 ) ) |
12 |
1 9
|
od1 |
⊢ ( 𝐺 ∈ Grp → ( 𝑂 ‘ ( 0g ‘ 𝐺 ) ) = 1 ) |
13 |
8 12
|
syl |
⊢ ( 𝐺 ∈ Abel → ( 𝑂 ‘ ( 0g ‘ 𝐺 ) ) = 1 ) |
14 |
|
1nn |
⊢ 1 ∈ ℕ |
15 |
13 14
|
eqeltrdi |
⊢ ( 𝐺 ∈ Abel → ( 𝑂 ‘ ( 0g ‘ 𝐺 ) ) ∈ ℕ ) |
16 |
|
ffn |
⊢ ( 𝑂 : ( Base ‘ 𝐺 ) ⟶ ℕ0 → 𝑂 Fn ( Base ‘ 𝐺 ) ) |
17 |
4 16
|
ax-mp |
⊢ 𝑂 Fn ( Base ‘ 𝐺 ) |
18 |
|
elpreima |
⊢ ( 𝑂 Fn ( Base ‘ 𝐺 ) → ( ( 0g ‘ 𝐺 ) ∈ ( ◡ 𝑂 “ ℕ ) ↔ ( ( 0g ‘ 𝐺 ) ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ ( 0g ‘ 𝐺 ) ) ∈ ℕ ) ) ) |
19 |
17 18
|
ax-mp |
⊢ ( ( 0g ‘ 𝐺 ) ∈ ( ◡ 𝑂 “ ℕ ) ↔ ( ( 0g ‘ 𝐺 ) ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ ( 0g ‘ 𝐺 ) ) ∈ ℕ ) ) |
20 |
11 15 19
|
sylanbrc |
⊢ ( 𝐺 ∈ Abel → ( 0g ‘ 𝐺 ) ∈ ( ◡ 𝑂 “ ℕ ) ) |
21 |
20
|
ne0d |
⊢ ( 𝐺 ∈ Abel → ( ◡ 𝑂 “ ℕ ) ≠ ∅ ) |
22 |
8
|
ad2antrr |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → 𝐺 ∈ Grp ) |
23 |
6
|
sseli |
⊢ ( 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) → 𝑥 ∈ ( Base ‘ 𝐺 ) ) |
24 |
23
|
ad2antlr |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → 𝑥 ∈ ( Base ‘ 𝐺 ) ) |
25 |
6
|
sseli |
⊢ ( 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) → 𝑦 ∈ ( Base ‘ 𝐺 ) ) |
26 |
25
|
adantl |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → 𝑦 ∈ ( Base ‘ 𝐺 ) ) |
27 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
28 |
3 27
|
grpcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( Base ‘ 𝐺 ) ) |
29 |
22 24 26 28
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( Base ‘ 𝐺 ) ) |
30 |
|
0nnn |
⊢ ¬ 0 ∈ ℕ |
31 |
3 1
|
odcl |
⊢ ( 𝑥 ∈ ( Base ‘ 𝐺 ) → ( 𝑂 ‘ 𝑥 ) ∈ ℕ0 ) |
32 |
24 31
|
syl |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( 𝑂 ‘ 𝑥 ) ∈ ℕ0 ) |
33 |
32
|
nn0zd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( 𝑂 ‘ 𝑥 ) ∈ ℤ ) |
34 |
3 1
|
odcl |
⊢ ( 𝑦 ∈ ( Base ‘ 𝐺 ) → ( 𝑂 ‘ 𝑦 ) ∈ ℕ0 ) |
35 |
26 34
|
syl |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( 𝑂 ‘ 𝑦 ) ∈ ℕ0 ) |
36 |
35
|
nn0zd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( 𝑂 ‘ 𝑦 ) ∈ ℤ ) |
37 |
33 36
|
gcdcld |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( ( 𝑂 ‘ 𝑥 ) gcd ( 𝑂 ‘ 𝑦 ) ) ∈ ℕ0 ) |
38 |
37
|
nn0cnd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( ( 𝑂 ‘ 𝑥 ) gcd ( 𝑂 ‘ 𝑦 ) ) ∈ ℂ ) |
39 |
38
|
mul02d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( 0 · ( ( 𝑂 ‘ 𝑥 ) gcd ( 𝑂 ‘ 𝑦 ) ) ) = 0 ) |
40 |
39
|
breq1d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( ( 0 · ( ( 𝑂 ‘ 𝑥 ) gcd ( 𝑂 ‘ 𝑦 ) ) ) ∥ ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) ↔ 0 ∥ ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) ) ) |
41 |
33 36
|
zmulcld |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) ∈ ℤ ) |
42 |
|
0dvds |
⊢ ( ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) ∈ ℤ → ( 0 ∥ ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) ↔ ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) = 0 ) ) |
43 |
41 42
|
syl |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( 0 ∥ ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) ↔ ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) = 0 ) ) |
44 |
40 43
|
bitrd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( ( 0 · ( ( 𝑂 ‘ 𝑥 ) gcd ( 𝑂 ‘ 𝑦 ) ) ) ∥ ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) ↔ ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) = 0 ) ) |
45 |
|
elpreima |
⊢ ( 𝑂 Fn ( Base ‘ 𝐺 ) → ( 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ↔ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ ℕ ) ) ) |
46 |
17 45
|
ax-mp |
⊢ ( 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ↔ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ 𝑥 ) ∈ ℕ ) ) |
47 |
46
|
simprbi |
⊢ ( 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) → ( 𝑂 ‘ 𝑥 ) ∈ ℕ ) |
48 |
47
|
ad2antlr |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( 𝑂 ‘ 𝑥 ) ∈ ℕ ) |
49 |
|
elpreima |
⊢ ( 𝑂 Fn ( Base ‘ 𝐺 ) → ( 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ↔ ( 𝑦 ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ 𝑦 ) ∈ ℕ ) ) ) |
50 |
17 49
|
ax-mp |
⊢ ( 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ↔ ( 𝑦 ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ 𝑦 ) ∈ ℕ ) ) |
51 |
50
|
simprbi |
⊢ ( 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) → ( 𝑂 ‘ 𝑦 ) ∈ ℕ ) |
52 |
51
|
adantl |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( 𝑂 ‘ 𝑦 ) ∈ ℕ ) |
53 |
48 52
|
nnmulcld |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) ∈ ℕ ) |
54 |
|
eleq1 |
⊢ ( ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) = 0 → ( ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) ∈ ℕ ↔ 0 ∈ ℕ ) ) |
55 |
53 54
|
syl5ibcom |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) = 0 → 0 ∈ ℕ ) ) |
56 |
44 55
|
sylbid |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( ( 0 · ( ( 𝑂 ‘ 𝑥 ) gcd ( 𝑂 ‘ 𝑦 ) ) ) ∥ ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) → 0 ∈ ℕ ) ) |
57 |
30 56
|
mtoi |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ¬ ( 0 · ( ( 𝑂 ‘ 𝑥 ) gcd ( 𝑂 ‘ 𝑦 ) ) ) ∥ ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) ) |
58 |
|
simpll |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → 𝐺 ∈ Abel ) |
59 |
1 3 27
|
odadd1 |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) → ( ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) · ( ( 𝑂 ‘ 𝑥 ) gcd ( 𝑂 ‘ 𝑦 ) ) ) ∥ ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) ) |
60 |
58 24 26 59
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) · ( ( 𝑂 ‘ 𝑥 ) gcd ( 𝑂 ‘ 𝑦 ) ) ) ∥ ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) ) |
61 |
|
oveq1 |
⊢ ( ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) = 0 → ( ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) · ( ( 𝑂 ‘ 𝑥 ) gcd ( 𝑂 ‘ 𝑦 ) ) ) = ( 0 · ( ( 𝑂 ‘ 𝑥 ) gcd ( 𝑂 ‘ 𝑦 ) ) ) ) |
62 |
61
|
breq1d |
⊢ ( ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) = 0 → ( ( ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) · ( ( 𝑂 ‘ 𝑥 ) gcd ( 𝑂 ‘ 𝑦 ) ) ) ∥ ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) ↔ ( 0 · ( ( 𝑂 ‘ 𝑥 ) gcd ( 𝑂 ‘ 𝑦 ) ) ) ∥ ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) ) ) |
63 |
60 62
|
syl5ibcom |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) = 0 → ( 0 · ( ( 𝑂 ‘ 𝑥 ) gcd ( 𝑂 ‘ 𝑦 ) ) ) ∥ ( ( 𝑂 ‘ 𝑥 ) · ( 𝑂 ‘ 𝑦 ) ) ) ) |
64 |
57 63
|
mtod |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ¬ ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) = 0 ) |
65 |
3 1
|
odcl |
⊢ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( Base ‘ 𝐺 ) → ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) ∈ ℕ0 ) |
66 |
29 65
|
syl |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) ∈ ℕ0 ) |
67 |
|
elnn0 |
⊢ ( ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) ∈ ℕ0 ↔ ( ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) ∈ ℕ ∨ ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) = 0 ) ) |
68 |
66 67
|
sylib |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) ∈ ℕ ∨ ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) = 0 ) ) |
69 |
68
|
ord |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( ¬ ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) ∈ ℕ → ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) = 0 ) ) |
70 |
64 69
|
mt3d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) ∈ ℕ ) |
71 |
|
elpreima |
⊢ ( 𝑂 Fn ( Base ‘ 𝐺 ) → ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( ◡ 𝑂 “ ℕ ) ↔ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) ∈ ℕ ) ) ) |
72 |
17 71
|
ax-mp |
⊢ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( ◡ 𝑂 “ ℕ ) ↔ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) ∈ ℕ ) ) |
73 |
29 70 72
|
sylanbrc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) ∧ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( ◡ 𝑂 “ ℕ ) ) |
74 |
73
|
ralrimiva |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) → ∀ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( ◡ 𝑂 “ ℕ ) ) |
75 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
76 |
3 75
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ( Base ‘ 𝐺 ) ) |
77 |
8 23 76
|
syl2an |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ( Base ‘ 𝐺 ) ) |
78 |
1 75 3
|
odinv |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑂 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) = ( 𝑂 ‘ 𝑥 ) ) |
79 |
8 23 78
|
syl2an |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( 𝑂 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) = ( 𝑂 ‘ 𝑥 ) ) |
80 |
47
|
adantl |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( 𝑂 ‘ 𝑥 ) ∈ ℕ ) |
81 |
79 80
|
eqeltrd |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( 𝑂 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) ∈ ℕ ) |
82 |
|
elpreima |
⊢ ( 𝑂 Fn ( Base ‘ 𝐺 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ( ◡ 𝑂 “ ℕ ) ↔ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) ∈ ℕ ) ) ) |
83 |
17 82
|
ax-mp |
⊢ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ( ◡ 𝑂 “ ℕ ) ↔ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) ∈ ℕ ) ) |
84 |
77 81 83
|
sylanbrc |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ( ◡ 𝑂 “ ℕ ) ) |
85 |
74 84
|
jca |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ) → ( ∀ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( ◡ 𝑂 “ ℕ ) ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ( ◡ 𝑂 “ ℕ ) ) ) |
86 |
85
|
ralrimiva |
⊢ ( 𝐺 ∈ Abel → ∀ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ( ∀ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( ◡ 𝑂 “ ℕ ) ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ( ◡ 𝑂 “ ℕ ) ) ) |
87 |
3 27 75
|
issubg2 |
⊢ ( 𝐺 ∈ Grp → ( ( ◡ 𝑂 “ ℕ ) ∈ ( SubGrp ‘ 𝐺 ) ↔ ( ( ◡ 𝑂 “ ℕ ) ⊆ ( Base ‘ 𝐺 ) ∧ ( ◡ 𝑂 “ ℕ ) ≠ ∅ ∧ ∀ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ( ∀ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( ◡ 𝑂 “ ℕ ) ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ( ◡ 𝑂 “ ℕ ) ) ) ) ) |
88 |
8 87
|
syl |
⊢ ( 𝐺 ∈ Abel → ( ( ◡ 𝑂 “ ℕ ) ∈ ( SubGrp ‘ 𝐺 ) ↔ ( ( ◡ 𝑂 “ ℕ ) ⊆ ( Base ‘ 𝐺 ) ∧ ( ◡ 𝑂 “ ℕ ) ≠ ∅ ∧ ∀ 𝑥 ∈ ( ◡ 𝑂 “ ℕ ) ( ∀ 𝑦 ∈ ( ◡ 𝑂 “ ℕ ) ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( ◡ 𝑂 “ ℕ ) ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ( ◡ 𝑂 “ ℕ ) ) ) ) ) |
89 |
7 21 86 88
|
mpbir3and |
⊢ ( 𝐺 ∈ Abel → ( ◡ 𝑂 “ ℕ ) ∈ ( SubGrp ‘ 𝐺 ) ) |