Step |
Hyp |
Ref |
Expression |
1 |
|
odcl.1 |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
2 |
|
odcl.2 |
⊢ 𝑂 = ( od ‘ 𝐺 ) |
3 |
|
eqid |
⊢ ( .g ‘ 𝐺 ) = ( .g ‘ 𝐺 ) |
4 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
5 |
|
eqid |
⊢ { 𝑦 ∈ ℕ ∣ ( 𝑦 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) } = { 𝑦 ∈ ℕ ∣ ( 𝑦 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) } |
6 |
1 3 4 2 5
|
odlem1 |
⊢ ( 𝐴 ∈ 𝑋 → ( ( ( 𝑂 ‘ 𝐴 ) = 0 ∧ { 𝑦 ∈ ℕ ∣ ( 𝑦 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) } = ∅ ) ∨ ( 𝑂 ‘ 𝐴 ) ∈ { 𝑦 ∈ ℕ ∣ ( 𝑦 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) } ) ) |
7 |
|
simpl |
⊢ ( ( ( 𝑂 ‘ 𝐴 ) = 0 ∧ { 𝑦 ∈ ℕ ∣ ( 𝑦 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) } = ∅ ) → ( 𝑂 ‘ 𝐴 ) = 0 ) |
8 |
|
elrabi |
⊢ ( ( 𝑂 ‘ 𝐴 ) ∈ { 𝑦 ∈ ℕ ∣ ( 𝑦 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) } → ( 𝑂 ‘ 𝐴 ) ∈ ℕ ) |
9 |
7 8
|
orim12i |
⊢ ( ( ( ( 𝑂 ‘ 𝐴 ) = 0 ∧ { 𝑦 ∈ ℕ ∣ ( 𝑦 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) } = ∅ ) ∨ ( 𝑂 ‘ 𝐴 ) ∈ { 𝑦 ∈ ℕ ∣ ( 𝑦 ( .g ‘ 𝐺 ) 𝐴 ) = ( 0g ‘ 𝐺 ) } ) → ( ( 𝑂 ‘ 𝐴 ) = 0 ∨ ( 𝑂 ‘ 𝐴 ) ∈ ℕ ) ) |
10 |
6 9
|
syl |
⊢ ( 𝐴 ∈ 𝑋 → ( ( 𝑂 ‘ 𝐴 ) = 0 ∨ ( 𝑂 ‘ 𝐴 ) ∈ ℕ ) ) |
11 |
10
|
orcomd |
⊢ ( 𝐴 ∈ 𝑋 → ( ( 𝑂 ‘ 𝐴 ) ∈ ℕ ∨ ( 𝑂 ‘ 𝐴 ) = 0 ) ) |
12 |
|
elnn0 |
⊢ ( ( 𝑂 ‘ 𝐴 ) ∈ ℕ0 ↔ ( ( 𝑂 ‘ 𝐴 ) ∈ ℕ ∨ ( 𝑂 ‘ 𝐴 ) = 0 ) ) |
13 |
11 12
|
sylibr |
⊢ ( 𝐴 ∈ 𝑋 → ( 𝑂 ‘ 𝐴 ) ∈ ℕ0 ) |