Step | Hyp | Ref
| Expression |
1 | | df-slw 17774 |
. . 3
⊢ pSyl =
(𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ {ℎ ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾s 𝑘)) ↔ ℎ = 𝑘)}) |
2 | 1 | elmpt2cl 6774 |
. 2
⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp)) |
3 | | simp1 1054 |
. . 3
⊢ ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)) → 𝑃 ∈ ℙ) |
4 | | subgrcl 17422 |
. . . 4
⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) |
5 | 4 | 3ad2ant2 1076 |
. . 3
⊢ ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)) → 𝐺 ∈ Grp) |
6 | 3, 5 | jca 553 |
. 2
⊢ ((𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)) → (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp)) |
7 | | simpr 476 |
. . . . . . . . 9
⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺) |
8 | 7 | fveq2d 6107 |
. . . . . . . 8
⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (SubGrp‘𝑔) = (SubGrp‘𝐺)) |
9 | | simpl 472 |
. . . . . . . . . . . 12
⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → 𝑝 = 𝑃) |
10 | 7 | oveq1d 6564 |
. . . . . . . . . . . 12
⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (𝑔 ↾s 𝑘) = (𝐺 ↾s 𝑘)) |
11 | 9, 10 | breq12d 4596 |
. . . . . . . . . . 11
⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → (𝑝 pGrp (𝑔 ↾s 𝑘) ↔ 𝑃 pGrp (𝐺 ↾s 𝑘))) |
12 | 11 | anbi2d 736 |
. . . . . . . . . 10
⊢ ((𝑝 = 𝑃 ∧ 𝑔 = 𝐺) → ((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾s 𝑘)) ↔ (ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) |
13 | 12 | bibi1d 332 |
. . . . . . . . 9
⊢ ((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 3129 |
. . . . . . . 8
⊢ ((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 3168 |
. . . . . . 7
⊢ ((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 6113 |
. . . . . . . 8
⊢
(SubGrp‘𝐺)
∈ V |
17 | 16 | rabex 4740 |
. . . . . . 7
⊢ {ℎ ∈ (SubGrp‘𝐺) ∣ ∀𝑘 ∈ (SubGrp‘𝐺)((ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ ℎ = 𝑘)} ∈ V |
18 | 15, 1, 17 | ovmpt2a 6689 |
. . . . . 6
⊢ ((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 2673 |
. . . . 5
⊢ ((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 | | sseq1 3589 |
. . . . . . . . 9
⊢ (ℎ = 𝐻 → (ℎ ⊆ 𝑘 ↔ 𝐻 ⊆ 𝑘)) |
21 | 20 | anbi1d 737 |
. . . . . . . 8
⊢ (ℎ = 𝐻 → ((ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) |
22 | | eqeq1 2614 |
. . . . . . . 8
⊢ (ℎ = 𝐻 → (ℎ = 𝑘 ↔ 𝐻 = 𝑘)) |
23 | 21, 22 | bibi12d 334 |
. . . . . . 7
⊢ (ℎ = 𝐻 → (((ℎ ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ ℎ = 𝑘) ↔ ((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘))) |
24 | 23 | ralbidv 2969 |
. . . . . 6
⊢ (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))) |
25 | 24 | elrab 3331 |
. . . . 5
⊢ (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))) |
26 | 19, 25 | syl6bb 275 |
. . . 4
⊢ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)))) |
27 | | simpl 472 |
. . . . 5
⊢ ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp) → 𝑃 ∈
ℙ) |
28 | 27 | biantrurd 528 |
. . . 4
⊢ ((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))))) |
29 | 26, 28 | bitrd 267 |
. . 3
⊢ ((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))))) |
30 | | 3anass 1035 |
. . 3
⊢ ((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)))) |
31 | 29, 30 | syl6bbr 277 |
. 2
⊢ ((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)))) |
32 | 2, 6, 31 | pm5.21nii 367 |
1
⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘))) |