Step |
Hyp |
Ref |
Expression |
1 |
|
odadd1.1 |
⊢ 𝑂 = ( od ‘ 𝐺 ) |
2 |
|
odadd1.2 |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
3 |
|
odadd1.3 |
⊢ + = ( +g ‘ 𝐺 ) |
4 |
2 1
|
odcl |
⊢ ( 𝐴 ∈ 𝑋 → ( 𝑂 ‘ 𝐴 ) ∈ ℕ0 ) |
5 |
4
|
3ad2ant2 |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑂 ‘ 𝐴 ) ∈ ℕ0 ) |
6 |
5
|
nn0zd |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑂 ‘ 𝐴 ) ∈ ℤ ) |
7 |
2 1
|
odcl |
⊢ ( 𝐵 ∈ 𝑋 → ( 𝑂 ‘ 𝐵 ) ∈ ℕ0 ) |
8 |
7
|
3ad2ant3 |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑂 ‘ 𝐵 ) ∈ ℕ0 ) |
9 |
8
|
nn0zd |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑂 ‘ 𝐵 ) ∈ ℤ ) |
10 |
6 9
|
zmulcld |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ∈ ℤ ) |
11 |
10
|
adantr |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 0 ) → ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ∈ ℤ ) |
12 |
|
dvds0 |
⊢ ( ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ∈ ℤ → ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ∥ 0 ) |
13 |
11 12
|
syl |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 0 ) → ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ∥ 0 ) |
14 |
|
simpr |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 0 ) → ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 0 ) |
15 |
14
|
sq0id |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 0 ) → ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) = 0 ) |
16 |
15
|
oveq2d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 0 ) → ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ) = ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · 0 ) ) |
17 |
|
ablgrp |
⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp ) |
18 |
2 3
|
grpcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 + 𝐵 ) ∈ 𝑋 ) |
19 |
17 18
|
syl3an1 |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 + 𝐵 ) ∈ 𝑋 ) |
20 |
2 1
|
odcl |
⊢ ( ( 𝐴 + 𝐵 ) ∈ 𝑋 → ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℕ0 ) |
21 |
19 20
|
syl |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℕ0 ) |
22 |
21
|
nn0zd |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℤ ) |
23 |
22
|
adantr |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 0 ) → ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℤ ) |
24 |
23
|
zcnd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 0 ) → ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℂ ) |
25 |
24
|
mul01d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 0 ) → ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · 0 ) = 0 ) |
26 |
16 25
|
eqtrd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 0 ) → ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ) = 0 ) |
27 |
13 26
|
breqtrrd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 0 ) → ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ) ) |
28 |
6
|
adantr |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 𝑂 ‘ 𝐴 ) ∈ ℤ ) |
29 |
9
|
adantr |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 𝑂 ‘ 𝐵 ) ∈ ℤ ) |
30 |
28 29
|
gcdcld |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∈ ℕ0 ) |
31 |
30
|
nn0cnd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∈ ℂ ) |
32 |
31
|
sqvald |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) = ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) |
33 |
32
|
oveq2d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ) = ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) ) |
34 |
|
gcddvds |
⊢ ( ( ( 𝑂 ‘ 𝐴 ) ∈ ℤ ∧ ( 𝑂 ‘ 𝐵 ) ∈ ℤ ) → ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∥ ( 𝑂 ‘ 𝐴 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∥ ( 𝑂 ‘ 𝐵 ) ) ) |
35 |
28 29 34
|
syl2anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∥ ( 𝑂 ‘ 𝐴 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∥ ( 𝑂 ‘ 𝐵 ) ) ) |
36 |
35
|
simpld |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∥ ( 𝑂 ‘ 𝐴 ) ) |
37 |
30
|
nn0zd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∈ ℤ ) |
38 |
|
simpr |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) |
39 |
|
dvdsval2 |
⊢ ( ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∈ ℤ ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ∧ ( 𝑂 ‘ 𝐴 ) ∈ ℤ ) → ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∥ ( 𝑂 ‘ 𝐴 ) ↔ ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∈ ℤ ) ) |
40 |
37 38 28 39
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∥ ( 𝑂 ‘ 𝐴 ) ↔ ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∈ ℤ ) ) |
41 |
36 40
|
mpbid |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∈ ℤ ) |
42 |
41
|
zcnd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∈ ℂ ) |
43 |
35
|
simprd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∥ ( 𝑂 ‘ 𝐵 ) ) |
44 |
|
dvdsval2 |
⊢ ( ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∈ ℤ ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ∧ ( 𝑂 ‘ 𝐵 ) ∈ ℤ ) → ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∥ ( 𝑂 ‘ 𝐵 ) ↔ ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∈ ℤ ) ) |
45 |
37 38 29 44
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∥ ( 𝑂 ‘ 𝐵 ) ↔ ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∈ ℤ ) ) |
46 |
43 45
|
mpbid |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∈ ℤ ) |
47 |
46
|
zcnd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∈ ℂ ) |
48 |
42 31 47 31
|
mul4d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) = ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) ) |
49 |
28
|
zcnd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 𝑂 ‘ 𝐴 ) ∈ ℂ ) |
50 |
49 31 38
|
divcan1d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) = ( 𝑂 ‘ 𝐴 ) ) |
51 |
29
|
zcnd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 𝑂 ‘ 𝐵 ) ∈ ℂ ) |
52 |
51 31 38
|
divcan1d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) = ( 𝑂 ‘ 𝐵 ) ) |
53 |
50 52
|
oveq12d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) = ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ) |
54 |
33 48 53
|
3eqtr2d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ) = ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ) |
55 |
22
|
adantr |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℤ ) |
56 |
|
dvdsmul2 |
⊢ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℤ ∧ ( 𝑂 ‘ 𝐴 ) ∈ ℤ ) → ( 𝑂 ‘ 𝐴 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ) |
57 |
55 28 56
|
syl2anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 𝑂 ‘ 𝐴 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ) |
58 |
|
simpl1 |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → 𝐺 ∈ Abel ) |
59 |
55 29
|
zmulcld |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ∈ ℤ ) |
60 |
|
simpl2 |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → 𝐴 ∈ 𝑋 ) |
61 |
|
simpl3 |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → 𝐵 ∈ 𝑋 ) |
62 |
|
eqid |
⊢ ( .g ‘ 𝐺 ) = ( .g ‘ 𝐺 ) |
63 |
2 62 3
|
mulgdi |
⊢ ( ( 𝐺 ∈ Abel ∧ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ∈ ℤ ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) ( 𝐴 + 𝐵 ) ) = ( ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐴 ) + ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐵 ) ) ) |
64 |
58 59 60 61 63
|
syl13anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) ( 𝐴 + 𝐵 ) ) = ( ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐴 ) + ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐵 ) ) ) |
65 |
|
dvdsmul2 |
⊢ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℤ ∧ ( 𝑂 ‘ 𝐵 ) ∈ ℤ ) → ( 𝑂 ‘ 𝐵 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ) |
66 |
55 29 65
|
syl2anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 𝑂 ‘ 𝐵 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ) |
67 |
58 17
|
syl |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → 𝐺 ∈ Grp ) |
68 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
69 |
2 1 62 68
|
oddvds |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ∈ ℤ ) → ( ( 𝑂 ‘ 𝐵 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ↔ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐵 ) = ( 0g ‘ 𝐺 ) ) ) |
70 |
67 61 59 69
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ 𝐵 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ↔ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐵 ) = ( 0g ‘ 𝐺 ) ) ) |
71 |
66 70
|
mpbid |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐵 ) = ( 0g ‘ 𝐺 ) ) |
72 |
71
|
oveq2d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐴 ) + ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐵 ) ) = ( ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐴 ) + ( 0g ‘ 𝐺 ) ) ) |
73 |
64 72
|
eqtrd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) ( 𝐴 + 𝐵 ) ) = ( ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐴 ) + ( 0g ‘ 𝐺 ) ) ) |
74 |
|
dvdsmul1 |
⊢ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℤ ∧ ( 𝑂 ‘ 𝐵 ) ∈ ℤ ) → ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ) |
75 |
55 29 74
|
syl2anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ) |
76 |
19
|
adantr |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 𝐴 + 𝐵 ) ∈ 𝑋 ) |
77 |
2 1 62 68
|
oddvds |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝐴 + 𝐵 ) ∈ 𝑋 ∧ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ∈ ℤ ) → ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ↔ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) ( 𝐴 + 𝐵 ) ) = ( 0g ‘ 𝐺 ) ) ) |
78 |
67 76 59 77
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ↔ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) ( 𝐴 + 𝐵 ) ) = ( 0g ‘ 𝐺 ) ) ) |
79 |
75 78
|
mpbid |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) ( 𝐴 + 𝐵 ) ) = ( 0g ‘ 𝐺 ) ) |
80 |
2 62
|
mulgcl |
⊢ ( ( 𝐺 ∈ Grp ∧ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ∈ ℤ ∧ 𝐴 ∈ 𝑋 ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐴 ) ∈ 𝑋 ) |
81 |
67 59 60 80
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐴 ) ∈ 𝑋 ) |
82 |
2 3 68
|
grprid |
⊢ ( ( 𝐺 ∈ Grp ∧ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐴 ) ∈ 𝑋 ) → ( ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐴 ) + ( 0g ‘ 𝐺 ) ) = ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐴 ) ) |
83 |
67 81 82
|
syl2anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐴 ) + ( 0g ‘ 𝐺 ) ) = ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐴 ) ) |
84 |
73 79 83
|
3eqtr3rd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) ) |
85 |
2 1 62 68
|
oddvds |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ∈ ℤ ) → ( ( 𝑂 ‘ 𝐴 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ↔ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) ) ) |
86 |
67 60 59 85
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ 𝐴 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ↔ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) ) ) |
87 |
84 86
|
mpbird |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 𝑂 ‘ 𝐴 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ) |
88 |
55 28
|
zmulcld |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ∈ ℤ ) |
89 |
|
dvdsgcd |
⊢ ( ( ( 𝑂 ‘ 𝐴 ) ∈ ℤ ∧ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ∈ ℤ ∧ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ∈ ℤ ) → ( ( ( 𝑂 ‘ 𝐴 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ∧ ( 𝑂 ‘ 𝐴 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ) → ( 𝑂 ‘ 𝐴 ) ∥ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) gcd ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ) ) ) |
90 |
28 88 59 89
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ 𝐴 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ∧ ( 𝑂 ‘ 𝐴 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ) → ( 𝑂 ‘ 𝐴 ) ∥ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) gcd ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ) ) ) |
91 |
57 87 90
|
mp2and |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 𝑂 ‘ 𝐴 ) ∥ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) gcd ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ) ) |
92 |
21
|
adantr |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℕ0 ) |
93 |
|
mulgcd |
⊢ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℕ0 ∧ ( 𝑂 ‘ 𝐴 ) ∈ ℤ ∧ ( 𝑂 ‘ 𝐵 ) ∈ ℤ ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) gcd ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ) = ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) |
94 |
92 28 29 93
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) gcd ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ) = ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) |
95 |
91 94
|
breqtrd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 𝑂 ‘ 𝐴 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) |
96 |
50 95
|
eqbrtrd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) |
97 |
|
dvdsmulcr |
⊢ ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∈ ℤ ∧ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℤ ∧ ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∈ ℤ ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) ) → ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ↔ ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ) ) |
98 |
41 55 37 38 97
|
syl112anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ↔ ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ) ) |
99 |
96 98
|
mpbid |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ) |
100 |
2 62 3
|
mulgdi |
⊢ ( ( 𝐺 ∈ Abel ∧ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ∈ ℤ ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) ( 𝐴 + 𝐵 ) ) = ( ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐴 ) + ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐵 ) ) ) |
101 |
58 88 60 61 100
|
syl13anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) ( 𝐴 + 𝐵 ) ) = ( ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐴 ) + ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐵 ) ) ) |
102 |
2 1 62 68
|
oddvds |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ∈ ℤ ) → ( ( 𝑂 ‘ 𝐴 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ↔ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) ) ) |
103 |
67 60 88 102
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ 𝐴 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ↔ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) ) ) |
104 |
57 103
|
mpbid |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) ) |
105 |
104
|
oveq1d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐴 ) + ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐵 ) ) = ( ( 0g ‘ 𝐺 ) + ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐵 ) ) ) |
106 |
101 105
|
eqtrd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) ( 𝐴 + 𝐵 ) ) = ( ( 0g ‘ 𝐺 ) + ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐵 ) ) ) |
107 |
|
dvdsmul1 |
⊢ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℤ ∧ ( 𝑂 ‘ 𝐴 ) ∈ ℤ ) → ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ) |
108 |
55 28 107
|
syl2anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ) |
109 |
2 1 62 68
|
oddvds |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝐴 + 𝐵 ) ∈ 𝑋 ∧ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ∈ ℤ ) → ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ↔ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) ( 𝐴 + 𝐵 ) ) = ( 0g ‘ 𝐺 ) ) ) |
110 |
67 76 88 109
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ↔ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) ( 𝐴 + 𝐵 ) ) = ( 0g ‘ 𝐺 ) ) ) |
111 |
108 110
|
mpbid |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) ( 𝐴 + 𝐵 ) ) = ( 0g ‘ 𝐺 ) ) |
112 |
2 62
|
mulgcl |
⊢ ( ( 𝐺 ∈ Grp ∧ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ∈ ℤ ∧ 𝐵 ∈ 𝑋 ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐵 ) ∈ 𝑋 ) |
113 |
67 88 61 112
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐵 ) ∈ 𝑋 ) |
114 |
2 3 68
|
grplid |
⊢ ( ( 𝐺 ∈ Grp ∧ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐵 ) ∈ 𝑋 ) → ( ( 0g ‘ 𝐺 ) + ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐵 ) ) = ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐵 ) ) |
115 |
67 113 114
|
syl2anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 0g ‘ 𝐺 ) + ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐵 ) ) = ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐵 ) ) |
116 |
106 111 115
|
3eqtr3rd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐵 ) = ( 0g ‘ 𝐺 ) ) |
117 |
2 1 62 68
|
oddvds |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ∈ ℤ ) → ( ( 𝑂 ‘ 𝐵 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ↔ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐵 ) = ( 0g ‘ 𝐺 ) ) ) |
118 |
67 61 88 117
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ 𝐵 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ↔ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ( .g ‘ 𝐺 ) 𝐵 ) = ( 0g ‘ 𝐺 ) ) ) |
119 |
116 118
|
mpbird |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 𝑂 ‘ 𝐵 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ) |
120 |
|
dvdsgcd |
⊢ ( ( ( 𝑂 ‘ 𝐵 ) ∈ ℤ ∧ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ∈ ℤ ∧ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ∈ ℤ ) → ( ( ( 𝑂 ‘ 𝐵 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ∧ ( 𝑂 ‘ 𝐵 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ) → ( 𝑂 ‘ 𝐵 ) ∥ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) gcd ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ) ) ) |
121 |
29 88 59 120
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ 𝐵 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) ∧ ( 𝑂 ‘ 𝐵 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ) → ( 𝑂 ‘ 𝐵 ) ∥ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) gcd ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ) ) ) |
122 |
119 66 121
|
mp2and |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 𝑂 ‘ 𝐵 ) ∥ ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐴 ) ) gcd ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( 𝑂 ‘ 𝐵 ) ) ) ) |
123 |
122 94
|
breqtrd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 𝑂 ‘ 𝐵 ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) |
124 |
52 123
|
eqbrtrd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) |
125 |
|
dvdsmulcr |
⊢ ( ( ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∈ ℤ ∧ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℤ ∧ ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∈ ℤ ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) ) → ( ( ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ↔ ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ) ) |
126 |
46 55 37 38 125
|
syl112anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ↔ ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ) ) |
127 |
124 126
|
mpbid |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ) |
128 |
41 46
|
gcdcld |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) gcd ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) ∈ ℕ0 ) |
129 |
128
|
nn0cnd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) gcd ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) ∈ ℂ ) |
130 |
|
1cnd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → 1 ∈ ℂ ) |
131 |
31
|
mulid2d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( 1 · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) = ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) |
132 |
50 52
|
oveq12d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) gcd ( ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) = ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) |
133 |
|
mulgcdr |
⊢ ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∈ ℤ ∧ ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∈ ℤ ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∈ ℕ0 ) → ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) gcd ( ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) = ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) gcd ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) |
134 |
41 46 30 133
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) gcd ( ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) = ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) gcd ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) |
135 |
131 132 134
|
3eqtr2rd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) gcd ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) = ( 1 · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) |
136 |
129 130 31 38 135
|
mulcan2ad |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) gcd ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) = 1 ) |
137 |
|
coprmdvds2 |
⊢ ( ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∈ ℤ ∧ ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∈ ℤ ∧ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℤ ) ∧ ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) gcd ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) = 1 ) → ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∧ ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ) → ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) ∥ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ) ) |
138 |
41 46 55 136 137
|
syl31anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∧ ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ) → ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) ∥ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ) ) |
139 |
99 127 138
|
mp2and |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) ∥ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ) |
140 |
41 46
|
zmulcld |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) ∈ ℤ ) |
141 |
|
zsqcl |
⊢ ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ∈ ℤ → ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ∈ ℤ ) |
142 |
37 141
|
syl |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ∈ ℤ ) |
143 |
|
dvdsmulc |
⊢ ( ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) ∈ ℤ ∧ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℤ ∧ ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ∈ ℤ ) → ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) ∥ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) → ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ) ) ) |
144 |
140 55 142 143
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) ∥ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) → ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ) ) ) |
145 |
139 144
|
mpd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( ( ( 𝑂 ‘ 𝐴 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) · ( ( 𝑂 ‘ 𝐵 ) / ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ) ) |
146 |
54 145
|
eqbrtrrd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ≠ 0 ) → ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ) ) |
147 |
27 146
|
pm2.61dane |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ) ) |