Step |
Hyp |
Ref |
Expression |
1 |
|
iscyg.1 |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
iscyg.2 |
⊢ · = ( .g ‘ 𝐺 ) |
3 |
|
iscygd.3 |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
4 |
|
iscygd.4 |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
5 |
|
iscygd.5 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 · 𝑋 ) ) |
6 |
5
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐵 ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 · 𝑋 ) ) |
7 |
|
eqid |
⊢ { 𝑥 ∈ 𝐵 ∣ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 } = { 𝑥 ∈ 𝐵 ∣ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 } |
8 |
1 2 7
|
iscyggen2 |
⊢ ( 𝐺 ∈ Grp → ( 𝑋 ∈ { 𝑥 ∈ 𝐵 ∣ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 } ↔ ( 𝑋 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 · 𝑋 ) ) ) ) |
9 |
3 8
|
syl |
⊢ ( 𝜑 → ( 𝑋 ∈ { 𝑥 ∈ 𝐵 ∣ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 } ↔ ( 𝑋 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 · 𝑋 ) ) ) ) |
10 |
4 6 9
|
mpbir2and |
⊢ ( 𝜑 → 𝑋 ∈ { 𝑥 ∈ 𝐵 ∣ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 } ) |
11 |
10
|
ne0d |
⊢ ( 𝜑 → { 𝑥 ∈ 𝐵 ∣ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 } ≠ ∅ ) |
12 |
1 2 7
|
iscyg2 |
⊢ ( 𝐺 ∈ CycGrp ↔ ( 𝐺 ∈ Grp ∧ { 𝑥 ∈ 𝐵 ∣ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 } ≠ ∅ ) ) |
13 |
3 11 12
|
sylanbrc |
⊢ ( 𝜑 → 𝐺 ∈ CycGrp ) |