Step |
Hyp |
Ref |
Expression |
1 |
|
df-slw |
⊢ pSyl = ( 𝑝 ∈ ℙ , 𝑔 ∈ Grp ↦ { ℎ ∈ ( SubGrp ‘ 𝑔 ) ∣ ∀ 𝑘 ∈ ( SubGrp ‘ 𝑔 ) ( ( ℎ ⊆ 𝑘 ∧ 𝑝 pGrp ( 𝑔 ↾s 𝑘 ) ) ↔ ℎ = 𝑘 ) } ) |
2 |
1
|
elmpocl |
⊢ ( 𝐻 ∈ ( 𝑃 pSyl 𝐺 ) → ( 𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ) ) |
3 |
|
simp1 |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐻 ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ) → 𝑃 ∈ ℙ ) |
4 |
|
subgrcl |
⊢ ( 𝐻 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp ) |
5 |
4
|
3ad2ant2 |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐻 ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ) → 𝐺 ∈ Grp ) |
6 |
3 5
|
jca |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐻 ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ) → ( 𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ) ) |
7 |
|
simpr |
⊢ ( ( 𝑝 = 𝑃 ∧ 𝑔 = 𝐺 ) → 𝑔 = 𝐺 ) |
8 |
7
|
fveq2d |
⊢ ( ( 𝑝 = 𝑃 ∧ 𝑔 = 𝐺 ) → ( SubGrp ‘ 𝑔 ) = ( SubGrp ‘ 𝐺 ) ) |
9 |
|
simpl |
⊢ ( ( 𝑝 = 𝑃 ∧ 𝑔 = 𝐺 ) → 𝑝 = 𝑃 ) |
10 |
7
|
oveq1d |
⊢ ( ( 𝑝 = 𝑃 ∧ 𝑔 = 𝐺 ) → ( 𝑔 ↾s 𝑘 ) = ( 𝐺 ↾s 𝑘 ) ) |
11 |
9 10
|
breq12d |
⊢ ( ( 𝑝 = 𝑃 ∧ 𝑔 = 𝐺 ) → ( 𝑝 pGrp ( 𝑔 ↾s 𝑘 ) ↔ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ) |
12 |
11
|
anbi2d |
⊢ ( ( 𝑝 = 𝑃 ∧ 𝑔 = 𝐺 ) → ( ( ℎ ⊆ 𝑘 ∧ 𝑝 pGrp ( 𝑔 ↾s 𝑘 ) ) ↔ ( ℎ ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ) ) |
13 |
12
|
bibi1d |
⊢ ( ( 𝑝 = 𝑃 ∧ 𝑔 = 𝐺 ) → ( ( ( ℎ ⊆ 𝑘 ∧ 𝑝 pGrp ( 𝑔 ↾s 𝑘 ) ) ↔ ℎ = 𝑘 ) ↔ ( ( ℎ ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ ℎ = 𝑘 ) ) ) |
14 |
8 13
|
raleqbidv |
⊢ ( ( 𝑝 = 𝑃 ∧ 𝑔 = 𝐺 ) → ( ∀ 𝑘 ∈ ( SubGrp ‘ 𝑔 ) ( ( ℎ ⊆ 𝑘 ∧ 𝑝 pGrp ( 𝑔 ↾s 𝑘 ) ) ↔ ℎ = 𝑘 ) ↔ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( ℎ ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ ℎ = 𝑘 ) ) ) |
15 |
8 14
|
rabeqbidv |
⊢ ( ( 𝑝 = 𝑃 ∧ 𝑔 = 𝐺 ) → { ℎ ∈ ( SubGrp ‘ 𝑔 ) ∣ ∀ 𝑘 ∈ ( SubGrp ‘ 𝑔 ) ( ( ℎ ⊆ 𝑘 ∧ 𝑝 pGrp ( 𝑔 ↾s 𝑘 ) ) ↔ ℎ = 𝑘 ) } = { ℎ ∈ ( SubGrp ‘ 𝐺 ) ∣ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( ℎ ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ ℎ = 𝑘 ) } ) |
16 |
|
fvex |
⊢ ( SubGrp ‘ 𝐺 ) ∈ V |
17 |
16
|
rabex |
⊢ { ℎ ∈ ( SubGrp ‘ 𝐺 ) ∣ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( ℎ ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ ℎ = 𝑘 ) } ∈ V |
18 |
15 1 17
|
ovmpoa |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ) → ( 𝑃 pSyl 𝐺 ) = { ℎ ∈ ( SubGrp ‘ 𝐺 ) ∣ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( ℎ ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ ℎ = 𝑘 ) } ) |
19 |
18
|
eleq2d |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ) → ( 𝐻 ∈ ( 𝑃 pSyl 𝐺 ) ↔ 𝐻 ∈ { ℎ ∈ ( SubGrp ‘ 𝐺 ) ∣ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( ℎ ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ ℎ = 𝑘 ) } ) ) |
20 |
|
cleq1lem |
⊢ ( ℎ = 𝐻 → ( ( ℎ ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ) ) |
21 |
|
eqeq1 |
⊢ ( ℎ = 𝐻 → ( ℎ = 𝑘 ↔ 𝐻 = 𝑘 ) ) |
22 |
20 21
|
bibi12d |
⊢ ( ℎ = 𝐻 → ( ( ( ℎ ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ ℎ = 𝑘 ) ↔ ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ) ) |
23 |
22
|
ralbidv |
⊢ ( ℎ = 𝐻 → ( ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( ℎ ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ ℎ = 𝑘 ) ↔ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ) ) |
24 |
23
|
elrab |
⊢ ( 𝐻 ∈ { ℎ ∈ ( SubGrp ‘ 𝐺 ) ∣ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( ℎ ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ ℎ = 𝑘 ) } ↔ ( 𝐻 ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ) ) |
25 |
19 24
|
bitrdi |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ) → ( 𝐻 ∈ ( 𝑃 pSyl 𝐺 ) ↔ ( 𝐻 ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ) ) ) |
26 |
|
simpl |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ) → 𝑃 ∈ ℙ ) |
27 |
26
|
biantrurd |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ) → ( ( 𝐻 ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ) ↔ ( 𝑃 ∈ ℙ ∧ ( 𝐻 ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ) ) ) ) |
28 |
25 27
|
bitrd |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ) → ( 𝐻 ∈ ( 𝑃 pSyl 𝐺 ) ↔ ( 𝑃 ∈ ℙ ∧ ( 𝐻 ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ) ) ) ) |
29 |
|
3anass |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐻 ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ) ↔ ( 𝑃 ∈ ℙ ∧ ( 𝐻 ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ) ) ) |
30 |
28 29
|
bitr4di |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ) → ( 𝐻 ∈ ( 𝑃 pSyl 𝐺 ) ↔ ( 𝑃 ∈ ℙ ∧ 𝐻 ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ) ) ) |
31 |
2 6 30
|
pm5.21nii |
⊢ ( 𝐻 ∈ ( 𝑃 pSyl 𝐺 ) ↔ ( 𝑃 ∈ ℙ ∧ 𝐻 ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp ( 𝐺 ↾s 𝑘 ) ) ↔ 𝐻 = 𝑘 ) ) ) |