Step |
Hyp |
Ref |
Expression |
1 |
|
odadd1.1 |
⊢ 𝑂 = ( od ‘ 𝐺 ) |
2 |
|
odadd1.2 |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
3 |
|
odadd1.3 |
⊢ + = ( +g ‘ 𝐺 ) |
4 |
|
simpl1 |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → 𝐺 ∈ Abel ) |
5 |
|
ablgrp |
⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp ) |
6 |
4 5
|
syl |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → 𝐺 ∈ Grp ) |
7 |
|
simpl2 |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → 𝐴 ∈ 𝑋 ) |
8 |
|
simpl3 |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → 𝐵 ∈ 𝑋 ) |
9 |
2 3
|
grpcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 + 𝐵 ) ∈ 𝑋 ) |
10 |
6 7 8 9
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( 𝐴 + 𝐵 ) ∈ 𝑋 ) |
11 |
2 1
|
odcl |
⊢ ( ( 𝐴 + 𝐵 ) ∈ 𝑋 → ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℕ0 ) |
12 |
10 11
|
syl |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℕ0 ) |
13 |
2 1
|
odcl |
⊢ ( 𝐴 ∈ 𝑋 → ( 𝑂 ‘ 𝐴 ) ∈ ℕ0 ) |
14 |
7 13
|
syl |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( 𝑂 ‘ 𝐴 ) ∈ ℕ0 ) |
15 |
2 1
|
odcl |
⊢ ( 𝐵 ∈ 𝑋 → ( 𝑂 ‘ 𝐵 ) ∈ ℕ0 ) |
16 |
8 15
|
syl |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( 𝑂 ‘ 𝐵 ) ∈ ℕ0 ) |
17 |
14 16
|
nn0mulcld |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ∈ ℕ0 ) |
18 |
|
simpr |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) |
19 |
18
|
oveq2d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) = ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · 1 ) ) |
20 |
12
|
nn0cnd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℂ ) |
21 |
20
|
mulid1d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · 1 ) = ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ) |
22 |
19 21
|
eqtrd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) = ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ) |
23 |
1 2 3
|
odadd1 |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ) |
24 |
23
|
adantr |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ) ∥ ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ) |
25 |
22 24
|
eqbrtrrd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∥ ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ) |
26 |
1 2 3
|
odadd2 |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ) ) |
27 |
26
|
adantr |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ∥ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ) ) |
28 |
18
|
oveq1d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) = ( 1 ↑ 2 ) ) |
29 |
|
sq1 |
⊢ ( 1 ↑ 2 ) = 1 |
30 |
28 29
|
eqtrdi |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) = 1 ) |
31 |
30
|
oveq2d |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ) = ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · 1 ) ) |
32 |
31 21
|
eqtrd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) · ( ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) ↑ 2 ) ) = ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ) |
33 |
27 32
|
breqtrd |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ∥ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ) |
34 |
|
dvdseq |
⊢ ( ( ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∈ ℕ0 ∧ ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ∈ ℕ0 ) ∧ ( ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ∥ ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ∧ ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ∥ ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) ) ) → ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) = ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ) |
35 |
12 17 25 33 34
|
syl22anc |
⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ( 𝑂 ‘ 𝐴 ) gcd ( 𝑂 ‘ 𝐵 ) ) = 1 ) → ( 𝑂 ‘ ( 𝐴 + 𝐵 ) ) = ( ( 𝑂 ‘ 𝐴 ) · ( 𝑂 ‘ 𝐵 ) ) ) |