Step |
Hyp |
Ref |
Expression |
1 |
|
odsubdvds.1 |
⊢ 𝑂 = ( od ‘ 𝐺 ) |
2 |
|
eqid |
⊢ ( 𝐺 ↾s 𝑆 ) = ( 𝐺 ↾s 𝑆 ) |
3 |
2
|
subggrp |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐺 ↾s 𝑆 ) ∈ Grp ) |
4 |
3
|
3ad2ant1 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆 ) → ( 𝐺 ↾s 𝑆 ) ∈ Grp ) |
5 |
2
|
subgbas |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝑆 = ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) |
6 |
5
|
3ad2ant1 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆 ) → 𝑆 = ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) |
7 |
|
simp2 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆 ) → 𝑆 ∈ Fin ) |
8 |
6 7
|
eqeltrrd |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆 ) → ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ∈ Fin ) |
9 |
|
simp3 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆 ) → 𝐴 ∈ 𝑆 ) |
10 |
9 6
|
eleqtrd |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆 ) → 𝐴 ∈ ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) |
11 |
|
eqid |
⊢ ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) = ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) |
12 |
|
eqid |
⊢ ( od ‘ ( 𝐺 ↾s 𝑆 ) ) = ( od ‘ ( 𝐺 ↾s 𝑆 ) ) |
13 |
11 12
|
oddvds2 |
⊢ ( ( ( 𝐺 ↾s 𝑆 ) ∈ Grp ∧ ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ∈ Fin ∧ 𝐴 ∈ ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) → ( ( od ‘ ( 𝐺 ↾s 𝑆 ) ) ‘ 𝐴 ) ∥ ( ♯ ‘ ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) ) |
14 |
4 8 10 13
|
syl3anc |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆 ) → ( ( od ‘ ( 𝐺 ↾s 𝑆 ) ) ‘ 𝐴 ) ∥ ( ♯ ‘ ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) ) |
15 |
2 1 12
|
subgod |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ 𝑆 ) → ( 𝑂 ‘ 𝐴 ) = ( ( od ‘ ( 𝐺 ↾s 𝑆 ) ) ‘ 𝐴 ) ) |
16 |
15
|
3adant2 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆 ) → ( 𝑂 ‘ 𝐴 ) = ( ( od ‘ ( 𝐺 ↾s 𝑆 ) ) ‘ 𝐴 ) ) |
17 |
6
|
fveq2d |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆 ) → ( ♯ ‘ 𝑆 ) = ( ♯ ‘ ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) ) |
18 |
14 16 17
|
3brtr4d |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆 ) → ( 𝑂 ‘ 𝐴 ) ∥ ( ♯ ‘ 𝑆 ) ) |