Step |
Hyp |
Ref |
Expression |
1 |
|
gexod.1 |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
2 |
|
gexod.2 |
⊢ 𝐸 = ( gEx ‘ 𝐺 ) |
3 |
|
gexod.3 |
⊢ 𝑂 = ( od ‘ 𝐺 ) |
4 |
|
eqid |
⊢ ( .g ‘ 𝐺 ) = ( .g ‘ 𝐺 ) |
5 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
6 |
1 2 4 5
|
gexdvds |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) → ( 𝐸 ∥ 𝑁 ↔ ∀ 𝑥 ∈ 𝑋 ( 𝑁 ( .g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) ) |
7 |
1 3 4 5
|
oddvds |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝑁 ∈ ℤ ) → ( ( 𝑂 ‘ 𝑥 ) ∥ 𝑁 ↔ ( 𝑁 ( .g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) ) |
8 |
7
|
3expa |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑁 ∈ ℤ ) → ( ( 𝑂 ‘ 𝑥 ) ∥ 𝑁 ↔ ( 𝑁 ( .g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) ) |
9 |
8
|
an32s |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) ∧ 𝑥 ∈ 𝑋 ) → ( ( 𝑂 ‘ 𝑥 ) ∥ 𝑁 ↔ ( 𝑁 ( .g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) ) |
10 |
9
|
ralbidva |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) → ( ∀ 𝑥 ∈ 𝑋 ( 𝑂 ‘ 𝑥 ) ∥ 𝑁 ↔ ∀ 𝑥 ∈ 𝑋 ( 𝑁 ( .g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) ) |
11 |
6 10
|
bitr4d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ) → ( 𝐸 ∥ 𝑁 ↔ ∀ 𝑥 ∈ 𝑋 ( 𝑂 ‘ 𝑥 ) ∥ 𝑁 ) ) |