Step |
Hyp |
Ref |
Expression |
1 |
|
subgslw.1 |
⊢ 𝐻 = ( 𝐺 ↾s 𝑆 ) |
2 |
|
slwprm |
⊢ ( 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) → 𝑃 ∈ ℙ ) |
3 |
2
|
3ad2ant2 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) → 𝑃 ∈ ℙ ) |
4 |
|
slwsubg |
⊢ ( 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) → 𝐾 ∈ ( SubGrp ‘ 𝐺 ) ) |
5 |
4
|
3ad2ant2 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) → 𝐾 ∈ ( SubGrp ‘ 𝐺 ) ) |
6 |
|
simp3 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) → 𝐾 ⊆ 𝑆 ) |
7 |
1
|
subsubg |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐾 ∈ ( SubGrp ‘ 𝐻 ) ↔ ( 𝐾 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ) ) |
8 |
7
|
3ad2ant1 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) → ( 𝐾 ∈ ( SubGrp ‘ 𝐻 ) ↔ ( 𝐾 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ) ) |
9 |
5 6 8
|
mpbir2and |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) → 𝐾 ∈ ( SubGrp ‘ 𝐻 ) ) |
10 |
1
|
oveq1i |
⊢ ( 𝐻 ↾s 𝑥 ) = ( ( 𝐺 ↾s 𝑆 ) ↾s 𝑥 ) |
11 |
|
simpl1 |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
12 |
1
|
subsubg |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ↔ ( 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑥 ⊆ 𝑆 ) ) ) |
13 |
12
|
3ad2ant1 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) → ( 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ↔ ( 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑥 ⊆ 𝑆 ) ) ) |
14 |
13
|
simplbda |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → 𝑥 ⊆ 𝑆 ) |
15 |
|
ressabs |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑥 ⊆ 𝑆 ) → ( ( 𝐺 ↾s 𝑆 ) ↾s 𝑥 ) = ( 𝐺 ↾s 𝑥 ) ) |
16 |
11 14 15
|
syl2anc |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → ( ( 𝐺 ↾s 𝑆 ) ↾s 𝑥 ) = ( 𝐺 ↾s 𝑥 ) ) |
17 |
10 16
|
eqtrid |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → ( 𝐻 ↾s 𝑥 ) = ( 𝐺 ↾s 𝑥 ) ) |
18 |
17
|
breq2d |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → ( 𝑃 pGrp ( 𝐻 ↾s 𝑥 ) ↔ 𝑃 pGrp ( 𝐺 ↾s 𝑥 ) ) ) |
19 |
18
|
anbi2d |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → ( ( 𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp ( 𝐻 ↾s 𝑥 ) ) ↔ ( 𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑥 ) ) ) ) |
20 |
|
simpl2 |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ) |
21 |
13
|
simprbda |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ) |
22 |
|
eqid |
⊢ ( 𝐺 ↾s 𝑥 ) = ( 𝐺 ↾s 𝑥 ) |
23 |
22
|
slwispgp |
⊢ ( ( 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑥 ) ) ↔ 𝐾 = 𝑥 ) ) |
24 |
20 21 23
|
syl2anc |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → ( ( 𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑥 ) ) ↔ 𝐾 = 𝑥 ) ) |
25 |
19 24
|
bitrd |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → ( ( 𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp ( 𝐻 ↾s 𝑥 ) ) ↔ 𝐾 = 𝑥 ) ) |
26 |
25
|
ralrimiva |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) → ∀ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ( ( 𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp ( 𝐻 ↾s 𝑥 ) ) ↔ 𝐾 = 𝑥 ) ) |
27 |
|
isslw |
⊢ ( 𝐾 ∈ ( 𝑃 pSyl 𝐻 ) ↔ ( 𝑃 ∈ ℙ ∧ 𝐾 ∈ ( SubGrp ‘ 𝐻 ) ∧ ∀ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ( ( 𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp ( 𝐻 ↾s 𝑥 ) ) ↔ 𝐾 = 𝑥 ) ) ) |
28 |
3 9 26 27
|
syl3anbrc |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) → 𝐾 ∈ ( 𝑃 pSyl 𝐻 ) ) |