Step |
Hyp |
Ref |
Expression |
1 |
|
iscyg.1 |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
iscyg.2 |
⊢ · = ( .g ‘ 𝐺 ) |
3 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( Base ‘ 𝑔 ) = ( Base ‘ 𝐺 ) ) |
4 |
3 1
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( Base ‘ 𝑔 ) = 𝐵 ) |
5 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( .g ‘ 𝑔 ) = ( .g ‘ 𝐺 ) ) |
6 |
5 2
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( .g ‘ 𝑔 ) = · ) |
7 |
6
|
oveqd |
⊢ ( 𝑔 = 𝐺 → ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) = ( 𝑛 · 𝑥 ) ) |
8 |
7
|
mpteq2dv |
⊢ ( 𝑔 = 𝐺 → ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) ) = ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) ) |
9 |
8
|
rneqd |
⊢ ( 𝑔 = 𝐺 → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) ) = ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) ) |
10 |
9 4
|
eqeq12d |
⊢ ( 𝑔 = 𝐺 → ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) ) = ( Base ‘ 𝑔 ) ↔ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 ) ) |
11 |
4 10
|
rexeqbidv |
⊢ ( 𝑔 = 𝐺 → ( ∃ 𝑥 ∈ ( Base ‘ 𝑔 ) ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) ) = ( Base ‘ 𝑔 ) ↔ ∃ 𝑥 ∈ 𝐵 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 ) ) |
12 |
|
df-cyg |
⊢ CycGrp = { 𝑔 ∈ Grp ∣ ∃ 𝑥 ∈ ( Base ‘ 𝑔 ) ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) ) = ( Base ‘ 𝑔 ) } |
13 |
11 12
|
elrab2 |
⊢ ( 𝐺 ∈ CycGrp ↔ ( 𝐺 ∈ Grp ∧ ∃ 𝑥 ∈ 𝐵 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 ) ) |