Step |
Hyp |
Ref |
Expression |
1 |
|
finodsubmsubg.o |
⊢ 𝑂 = ( od ‘ 𝐺 ) |
2 |
|
finodsubmsubg.g |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
3 |
|
finodsubmsubg.s |
⊢ ( 𝜑 → 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ) |
4 |
|
finodsubmsubg.1 |
⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝑆 ( 𝑂 ‘ 𝑎 ) ∈ ℕ ) |
5 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
6 |
|
eqid |
⊢ ( .g ‘ 𝐺 ) = ( .g ‘ 𝐺 ) |
7 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
8 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → 𝐺 ∈ Grp ) |
9 |
5
|
submss |
⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) → 𝑆 ⊆ ( Base ‘ 𝐺 ) ) |
10 |
3 9
|
syl |
⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝐺 ) ) |
11 |
10
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → 𝑎 ∈ ( Base ‘ 𝐺 ) ) |
12 |
5 1 6 7 8 11
|
odm1inv |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( ( ( 𝑂 ‘ 𝑎 ) − 1 ) ( .g ‘ 𝐺 ) 𝑎 ) = ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ) |
13 |
12
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ ( 𝑂 ‘ 𝑎 ) ∈ ℕ ) → ( ( ( 𝑂 ‘ 𝑎 ) − 1 ) ( .g ‘ 𝐺 ) 𝑎 ) = ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ) |
14 |
|
eqid |
⊢ ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) = ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) |
15 |
|
eqid |
⊢ ( .g ‘ ( 𝐺 ↾s 𝑆 ) ) = ( .g ‘ ( 𝐺 ↾s 𝑆 ) ) |
16 |
|
eqid |
⊢ ( 𝐺 ↾s 𝑆 ) = ( 𝐺 ↾s 𝑆 ) |
17 |
16
|
submmnd |
⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) → ( 𝐺 ↾s 𝑆 ) ∈ Mnd ) |
18 |
3 17
|
syl |
⊢ ( 𝜑 → ( 𝐺 ↾s 𝑆 ) ∈ Mnd ) |
19 |
18
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ ( 𝑂 ‘ 𝑎 ) ∈ ℕ ) → ( 𝐺 ↾s 𝑆 ) ∈ Mnd ) |
20 |
|
nnm1nn0 |
⊢ ( ( 𝑂 ‘ 𝑎 ) ∈ ℕ → ( ( 𝑂 ‘ 𝑎 ) − 1 ) ∈ ℕ0 ) |
21 |
20
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ ( 𝑂 ‘ 𝑎 ) ∈ ℕ ) → ( ( 𝑂 ‘ 𝑎 ) − 1 ) ∈ ℕ0 ) |
22 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ ( 𝑂 ‘ 𝑎 ) ∈ ℕ ) → 𝑎 ∈ 𝑆 ) |
23 |
16 5
|
ressbas2 |
⊢ ( 𝑆 ⊆ ( Base ‘ 𝐺 ) → 𝑆 = ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) |
24 |
10 23
|
syl |
⊢ ( 𝜑 → 𝑆 = ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) |
25 |
24
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ ( 𝑂 ‘ 𝑎 ) ∈ ℕ ) → 𝑆 = ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) |
26 |
22 25
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ ( 𝑂 ‘ 𝑎 ) ∈ ℕ ) → 𝑎 ∈ ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) |
27 |
14 15 19 21 26
|
mulgnn0cld |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ ( 𝑂 ‘ 𝑎 ) ∈ ℕ ) → ( ( ( 𝑂 ‘ 𝑎 ) − 1 ) ( .g ‘ ( 𝐺 ↾s 𝑆 ) ) 𝑎 ) ∈ ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) |
28 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ ( 𝑂 ‘ 𝑎 ) ∈ ℕ ) → 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ) |
29 |
6 16 15
|
submmulg |
⊢ ( ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ∧ ( ( 𝑂 ‘ 𝑎 ) − 1 ) ∈ ℕ0 ∧ 𝑎 ∈ 𝑆 ) → ( ( ( 𝑂 ‘ 𝑎 ) − 1 ) ( .g ‘ 𝐺 ) 𝑎 ) = ( ( ( 𝑂 ‘ 𝑎 ) − 1 ) ( .g ‘ ( 𝐺 ↾s 𝑆 ) ) 𝑎 ) ) |
30 |
28 21 22 29
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ ( 𝑂 ‘ 𝑎 ) ∈ ℕ ) → ( ( ( 𝑂 ‘ 𝑎 ) − 1 ) ( .g ‘ 𝐺 ) 𝑎 ) = ( ( ( 𝑂 ‘ 𝑎 ) − 1 ) ( .g ‘ ( 𝐺 ↾s 𝑆 ) ) 𝑎 ) ) |
31 |
27 30 25
|
3eltr4d |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ ( 𝑂 ‘ 𝑎 ) ∈ ℕ ) → ( ( ( 𝑂 ‘ 𝑎 ) − 1 ) ( .g ‘ 𝐺 ) 𝑎 ) ∈ 𝑆 ) |
32 |
13 31
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ ( 𝑂 ‘ 𝑎 ) ∈ ℕ ) → ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ∈ 𝑆 ) |
33 |
32
|
ex |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( ( 𝑂 ‘ 𝑎 ) ∈ ℕ → ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ∈ 𝑆 ) ) |
34 |
33
|
ralimdva |
⊢ ( 𝜑 → ( ∀ 𝑎 ∈ 𝑆 ( 𝑂 ‘ 𝑎 ) ∈ ℕ → ∀ 𝑎 ∈ 𝑆 ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ∈ 𝑆 ) ) |
35 |
4 34
|
mpd |
⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝑆 ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ∈ 𝑆 ) |
36 |
7
|
issubg3 |
⊢ ( 𝐺 ∈ Grp → ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ↔ ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ∧ ∀ 𝑎 ∈ 𝑆 ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ∈ 𝑆 ) ) ) |
37 |
2 36
|
syl |
⊢ ( 𝜑 → ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ↔ ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ∧ ∀ 𝑎 ∈ 𝑆 ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ∈ 𝑆 ) ) ) |
38 |
3 35 37
|
mpbir2and |
⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |