Step |
Hyp |
Ref |
Expression |
1 |
|
gexcl.1 |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
2 |
|
gexcl.2 |
⊢ 𝐸 = ( gEx ‘ 𝐺 ) |
3 |
|
eqid |
⊢ ( .g ‘ 𝐺 ) = ( .g ‘ 𝐺 ) |
4 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
5 |
|
eqid |
⊢ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 ( .g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) } = { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 ( .g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) } |
6 |
1 3 4 2 5
|
gexlem1 |
⊢ ( 𝐺 ∈ 𝑉 → ( ( 𝐸 = 0 ∧ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 ( .g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) } = ∅ ) ∨ 𝐸 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 ( .g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) } ) ) |
7 |
|
simpl |
⊢ ( ( 𝐸 = 0 ∧ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 ( .g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) } = ∅ ) → 𝐸 = 0 ) |
8 |
|
elrabi |
⊢ ( 𝐸 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 ( .g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) } → 𝐸 ∈ ℕ ) |
9 |
7 8
|
orim12i |
⊢ ( ( ( 𝐸 = 0 ∧ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 ( .g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) } = ∅ ) ∨ 𝐸 ∈ { 𝑦 ∈ ℕ ∣ ∀ 𝑥 ∈ 𝑋 ( 𝑦 ( .g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) } ) → ( 𝐸 = 0 ∨ 𝐸 ∈ ℕ ) ) |
10 |
6 9
|
syl |
⊢ ( 𝐺 ∈ 𝑉 → ( 𝐸 = 0 ∨ 𝐸 ∈ ℕ ) ) |
11 |
10
|
orcomd |
⊢ ( 𝐺 ∈ 𝑉 → ( 𝐸 ∈ ℕ ∨ 𝐸 = 0 ) ) |
12 |
|
elnn0 |
⊢ ( 𝐸 ∈ ℕ0 ↔ ( 𝐸 ∈ ℕ ∨ 𝐸 = 0 ) ) |
13 |
11 12
|
sylibr |
⊢ ( 𝐺 ∈ 𝑉 → 𝐸 ∈ ℕ0 ) |