Step |
Hyp |
Ref |
Expression |
1 |
|
cycsubgcyg2.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
cycsubgcyg2.k |
⊢ 𝐾 = ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) |
3 |
|
eqid |
⊢ ( .g ‘ 𝐺 ) = ( .g ‘ 𝐺 ) |
4 |
|
eqid |
⊢ ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) 𝐴 ) ) = ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) 𝐴 ) ) |
5 |
1 3 4 2
|
cycsubg2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵 ) → ( 𝐾 ‘ { 𝐴 } ) = ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) 𝐴 ) ) ) |
6 |
5
|
oveq2d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵 ) → ( 𝐺 ↾s ( 𝐾 ‘ { 𝐴 } ) ) = ( 𝐺 ↾s ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) 𝐴 ) ) ) ) |
7 |
|
eqid |
⊢ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) 𝐴 ) ) = ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) 𝐴 ) ) |
8 |
1 3 7
|
cycsubgcyg |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵 ) → ( 𝐺 ↾s ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) 𝐴 ) ) ) ∈ CycGrp ) |
9 |
6 8
|
eqeltrd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵 ) → ( 𝐺 ↾s ( 𝐾 ‘ { 𝐴 } ) ) ∈ CycGrp ) |