Step | Hyp | Ref
| Expression |
1 | | simpr 476 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → 𝐻 ∈ (𝑃 pSyl 𝐺)) |
2 | | slwsubg 17848 |
. . . 4
⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺)) |
3 | 1, 2 | syl 17 |
. . 3
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → 𝐻 ∈ (SubGrp‘𝐺)) |
4 | | fislw.1 |
. . . 4
⊢ 𝑋 = (Base‘𝐺) |
5 | | simpl2 1058 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → 𝑋 ∈ Fin) |
6 | 4, 5, 1 | slwhash 17862 |
. . 3
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋)))) |
7 | 3, 6 | jca 553 |
. 2
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) |
8 | | simpl3 1059 |
. . 3
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝑃 ∈ ℙ) |
9 | | simprl 790 |
. . 3
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐻 ∈ (SubGrp‘𝐺)) |
10 | | simpl2 1058 |
. . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝑋 ∈ Fin) |
11 | 10 | adantr 480 |
. . . . . . . 8
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → 𝑋 ∈ Fin) |
12 | | simprl 790 |
. . . . . . . . 9
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → 𝑘 ∈ (SubGrp‘𝐺)) |
13 | 4 | subgss 17418 |
. . . . . . . . 9
⊢ (𝑘 ∈ (SubGrp‘𝐺) → 𝑘 ⊆ 𝑋) |
14 | 12, 13 | syl 17 |
. . . . . . . 8
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → 𝑘 ⊆ 𝑋) |
15 | | ssfi 8065 |
. . . . . . . 8
⊢ ((𝑋 ∈ Fin ∧ 𝑘 ⊆ 𝑋) → 𝑘 ∈ Fin) |
16 | 11, 14, 15 | syl2anc 691 |
. . . . . . 7
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → 𝑘 ∈ Fin) |
17 | | simprrl 800 |
. . . . . . 7
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → 𝐻 ⊆ 𝑘) |
18 | | ssdomg 7887 |
. . . . . . . . 9
⊢ (𝑘 ∈ Fin → (𝐻 ⊆ 𝑘 → 𝐻 ≼ 𝑘)) |
19 | 16, 17, 18 | sylc 63 |
. . . . . . . 8
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → 𝐻 ≼ 𝑘) |
20 | | simprrr 801 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → 𝑃 pGrp (𝐺 ↾s 𝑘)) |
21 | | eqid 2610 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐺 ↾s 𝑘) = (𝐺 ↾s 𝑘) |
22 | 21 | subggrp 17420 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑘) ∈ Grp) |
23 | 12, 22 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (𝐺 ↾s 𝑘) ∈ Grp) |
24 | 21 | subgbas 17421 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (SubGrp‘𝐺) → 𝑘 = (Base‘(𝐺 ↾s 𝑘))) |
25 | 12, 24 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → 𝑘 = (Base‘(𝐺 ↾s 𝑘))) |
26 | 25, 16 | eqeltrrd 2689 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (Base‘(𝐺 ↾s 𝑘)) ∈ Fin) |
27 | | eqid 2610 |
. . . . . . . . . . . . . . . . 17
⊢
(Base‘(𝐺
↾s 𝑘)) =
(Base‘(𝐺
↾s 𝑘)) |
28 | 27 | pgpfi 17843 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐺 ↾s 𝑘) ∈ Grp ∧
(Base‘(𝐺
↾s 𝑘))
∈ Fin) → (𝑃 pGrp
(𝐺 ↾s
𝑘) ↔ (𝑃 ∈ ℙ ∧
∃𝑛 ∈
ℕ0 (#‘(Base‘(𝐺 ↾s 𝑘))) = (𝑃↑𝑛)))) |
29 | 23, 26, 28 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (𝑃 pGrp (𝐺 ↾s 𝑘) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0
(#‘(Base‘(𝐺
↾s 𝑘))) =
(𝑃↑𝑛)))) |
30 | 20, 29 | mpbid 221 |
. . . . . . . . . . . . . 14
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0
(#‘(Base‘(𝐺
↾s 𝑘))) =
(𝑃↑𝑛))) |
31 | 30 | simpld 474 |
. . . . . . . . . . . . 13
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → 𝑃 ∈ ℙ) |
32 | | prmnn 15226 |
. . . . . . . . . . . . 13
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℕ) |
33 | 31, 32 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → 𝑃 ∈ ℕ) |
34 | 33 | nnred 10912 |
. . . . . . . . . . 11
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → 𝑃 ∈ ℝ) |
35 | 33 | nnge1d 10940 |
. . . . . . . . . . 11
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → 1 ≤ 𝑃) |
36 | | eqid 2610 |
. . . . . . . . . . . . . . . . . 18
⊢
(0g‘𝐺) = (0g‘𝐺) |
37 | 36 | subg0cl 17425 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (SubGrp‘𝐺) →
(0g‘𝐺)
∈ 𝑘) |
38 | 12, 37 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (0g‘𝐺) ∈ 𝑘) |
39 | | ne0i 3880 |
. . . . . . . . . . . . . . . 16
⊢
((0g‘𝐺) ∈ 𝑘 → 𝑘 ≠ ∅) |
40 | 38, 39 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → 𝑘 ≠ ∅) |
41 | | hashnncl 13018 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ Fin →
((#‘𝑘) ∈ ℕ
↔ 𝑘 ≠
∅)) |
42 | 16, 41 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → ((#‘𝑘) ∈ ℕ ↔ 𝑘 ≠ ∅)) |
43 | 40, 42 | mpbird 246 |
. . . . . . . . . . . . . 14
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (#‘𝑘) ∈ ℕ) |
44 | 31, 43 | pccld 15393 |
. . . . . . . . . . . . 13
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (𝑃 pCnt (#‘𝑘)) ∈
ℕ0) |
45 | 44 | nn0zd 11356 |
. . . . . . . . . . . 12
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (𝑃 pCnt (#‘𝑘)) ∈ ℤ) |
46 | | simpl1 1057 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐺 ∈ Grp) |
47 | 4 | grpbn0 17274 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅) |
48 | 46, 47 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝑋 ≠ ∅) |
49 | | hashnncl 13018 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑋 ∈ Fin →
((#‘𝑋) ∈ ℕ
↔ 𝑋 ≠
∅)) |
50 | 10, 49 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅)) |
51 | 48, 50 | mpbird 246 |
. . . . . . . . . . . . . . 15
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (#‘𝑋) ∈ ℕ) |
52 | 8, 51 | pccld 15393 |
. . . . . . . . . . . . . 14
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (𝑃 pCnt (#‘𝑋)) ∈
ℕ0) |
53 | 52 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (𝑃 pCnt (#‘𝑋)) ∈
ℕ0) |
54 | 53 | nn0zd 11356 |
. . . . . . . . . . . 12
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (𝑃 pCnt (#‘𝑋)) ∈ ℤ) |
55 | 4 | lagsubg 17479 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (#‘𝑘) ∥ (#‘𝑋)) |
56 | 12, 11, 55 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (#‘𝑘) ∥ (#‘𝑋)) |
57 | 43 | nnzd 11357 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (#‘𝑘) ∈ ℤ) |
58 | 51 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (#‘𝑋) ∈ ℕ) |
59 | 58 | nnzd 11357 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (#‘𝑋) ∈ ℤ) |
60 | | pc2dvds 15421 |
. . . . . . . . . . . . . . 15
⊢
(((#‘𝑘) ∈
ℤ ∧ (#‘𝑋)
∈ ℤ) → ((#‘𝑘) ∥ (#‘𝑋) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (#‘𝑘)) ≤ (𝑝 pCnt (#‘𝑋)))) |
61 | 57, 59, 60 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → ((#‘𝑘) ∥ (#‘𝑋) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (#‘𝑘)) ≤ (𝑝 pCnt (#‘𝑋)))) |
62 | 56, 61 | mpbid 221 |
. . . . . . . . . . . . 13
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → ∀𝑝 ∈ ℙ (𝑝 pCnt (#‘𝑘)) ≤ (𝑝 pCnt (#‘𝑋))) |
63 | | oveq1 6556 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 = 𝑃 → (𝑝 pCnt (#‘𝑘)) = (𝑃 pCnt (#‘𝑘))) |
64 | | oveq1 6556 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 = 𝑃 → (𝑝 pCnt (#‘𝑋)) = (𝑃 pCnt (#‘𝑋))) |
65 | 63, 64 | breq12d 4596 |
. . . . . . . . . . . . . 14
⊢ (𝑝 = 𝑃 → ((𝑝 pCnt (#‘𝑘)) ≤ (𝑝 pCnt (#‘𝑋)) ↔ (𝑃 pCnt (#‘𝑘)) ≤ (𝑃 pCnt (#‘𝑋)))) |
66 | 65 | rspcv 3278 |
. . . . . . . . . . . . 13
⊢ (𝑃 ∈ ℙ →
(∀𝑝 ∈ ℙ
(𝑝 pCnt (#‘𝑘)) ≤ (𝑝 pCnt (#‘𝑋)) → (𝑃 pCnt (#‘𝑘)) ≤ (𝑃 pCnt (#‘𝑋)))) |
67 | 31, 62, 66 | sylc 63 |
. . . . . . . . . . . 12
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (𝑃 pCnt (#‘𝑘)) ≤ (𝑃 pCnt (#‘𝑋))) |
68 | | eluz2 11569 |
. . . . . . . . . . . 12
⊢ ((𝑃 pCnt (#‘𝑋)) ∈
(ℤ≥‘(𝑃 pCnt (#‘𝑘))) ↔ ((𝑃 pCnt (#‘𝑘)) ∈ ℤ ∧ (𝑃 pCnt (#‘𝑋)) ∈ ℤ ∧ (𝑃 pCnt (#‘𝑘)) ≤ (𝑃 pCnt (#‘𝑋)))) |
69 | 45, 54, 67, 68 | syl3anbrc 1239 |
. . . . . . . . . . 11
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (𝑃 pCnt (#‘𝑋)) ∈
(ℤ≥‘(𝑃 pCnt (#‘𝑘)))) |
70 | 34, 35, 69 | leexp2ad 12903 |
. . . . . . . . . 10
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (𝑃↑(𝑃 pCnt (#‘𝑘))) ≤ (𝑃↑(𝑃 pCnt (#‘𝑋)))) |
71 | 30 | simprd 478 |
. . . . . . . . . . . 12
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → ∃𝑛 ∈ ℕ0
(#‘(Base‘(𝐺
↾s 𝑘))) =
(𝑃↑𝑛)) |
72 | 25 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (#‘𝑘) = (#‘(Base‘(𝐺 ↾s 𝑘)))) |
73 | 72 | eqeq1d 2612 |
. . . . . . . . . . . . 13
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → ((#‘𝑘) = (𝑃↑𝑛) ↔ (#‘(Base‘(𝐺 ↾s 𝑘))) = (𝑃↑𝑛))) |
74 | 73 | rexbidv 3034 |
. . . . . . . . . . . 12
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (∃𝑛 ∈ ℕ0 (#‘𝑘) = (𝑃↑𝑛) ↔ ∃𝑛 ∈ ℕ0
(#‘(Base‘(𝐺
↾s 𝑘))) =
(𝑃↑𝑛))) |
75 | 71, 74 | mpbird 246 |
. . . . . . . . . . 11
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → ∃𝑛 ∈ ℕ0 (#‘𝑘) = (𝑃↑𝑛)) |
76 | | pcprmpw 15425 |
. . . . . . . . . . . 12
⊢ ((𝑃 ∈ ℙ ∧
(#‘𝑘) ∈ ℕ)
→ (∃𝑛 ∈
ℕ0 (#‘𝑘) = (𝑃↑𝑛) ↔ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑘))))) |
77 | 31, 43, 76 | syl2anc 691 |
. . . . . . . . . . 11
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (∃𝑛 ∈ ℕ0 (#‘𝑘) = (𝑃↑𝑛) ↔ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑘))))) |
78 | 75, 77 | mpbid 221 |
. . . . . . . . . 10
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑘)))) |
79 | | simplrr 797 |
. . . . . . . . . 10
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋)))) |
80 | 70, 78, 79 | 3brtr4d 4615 |
. . . . . . . . 9
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → (#‘𝑘) ≤ (#‘𝐻)) |
81 | 4 | subgss 17418 |
. . . . . . . . . . . . 13
⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋) |
82 | 81 | ad2antrl 760 |
. . . . . . . . . . . 12
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐻 ⊆ 𝑋) |
83 | | ssfi 8065 |
. . . . . . . . . . . 12
⊢ ((𝑋 ∈ Fin ∧ 𝐻 ⊆ 𝑋) → 𝐻 ∈ Fin) |
84 | 10, 82, 83 | syl2anc 691 |
. . . . . . . . . . 11
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐻 ∈ Fin) |
85 | 84 | adantr 480 |
. . . . . . . . . 10
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → 𝐻 ∈ Fin) |
86 | | hashdom 13029 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ Fin ∧ 𝐻 ∈ Fin) →
((#‘𝑘) ≤
(#‘𝐻) ↔ 𝑘 ≼ 𝐻)) |
87 | 16, 85, 86 | syl2anc 691 |
. . . . . . . . 9
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → ((#‘𝑘) ≤ (#‘𝐻) ↔ 𝑘 ≼ 𝐻)) |
88 | 80, 87 | mpbid 221 |
. . . . . . . 8
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → 𝑘 ≼ 𝐻) |
89 | | sbth 7965 |
. . . . . . . 8
⊢ ((𝐻 ≼ 𝑘 ∧ 𝑘 ≼ 𝐻) → 𝐻 ≈ 𝑘) |
90 | 19, 88, 89 | syl2anc 691 |
. . . . . . 7
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → 𝐻 ≈ 𝑘) |
91 | | fisseneq 8056 |
. . . . . . 7
⊢ ((𝑘 ∈ Fin ∧ 𝐻 ⊆ 𝑘 ∧ 𝐻 ≈ 𝑘) → 𝐻 = 𝑘) |
92 | 16, 17, 90, 91 | syl3anc 1318 |
. . . . . 6
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) → 𝐻 = 𝑘) |
93 | 92 | expr 641 |
. . . . 5
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) → 𝐻 = 𝑘)) |
94 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢ (𝐺 ↾s 𝐻) = (𝐺 ↾s 𝐻) |
95 | 94 | subgbas 17421 |
. . . . . . . . . . . 12
⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 = (Base‘(𝐺 ↾s 𝐻))) |
96 | 95 | ad2antrl 760 |
. . . . . . . . . . 11
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐻 = (Base‘(𝐺 ↾s 𝐻))) |
97 | 96 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (#‘𝐻) = (#‘(Base‘(𝐺 ↾s 𝐻)))) |
98 | | simprr 792 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋)))) |
99 | 97, 98 | eqtr3d 2646 |
. . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (#‘(Base‘(𝐺 ↾s 𝐻))) = (𝑃↑(𝑃 pCnt (#‘𝑋)))) |
100 | | oveq2 6557 |
. . . . . . . . . . 11
⊢ (𝑛 = (𝑃 pCnt (#‘𝑋)) → (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt (#‘𝑋)))) |
101 | 100 | eqeq2d 2620 |
. . . . . . . . . 10
⊢ (𝑛 = (𝑃 pCnt (#‘𝑋)) → ((#‘(Base‘(𝐺 ↾s 𝐻))) = (𝑃↑𝑛) ↔ (#‘(Base‘(𝐺 ↾s 𝐻))) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) |
102 | 101 | rspcev 3282 |
. . . . . . . . 9
⊢ (((𝑃 pCnt (#‘𝑋)) ∈ ℕ0 ∧
(#‘(Base‘(𝐺
↾s 𝐻))) =
(𝑃↑(𝑃 pCnt (#‘𝑋)))) → ∃𝑛 ∈ ℕ0
(#‘(Base‘(𝐺
↾s 𝐻))) =
(𝑃↑𝑛)) |
103 | 52, 99, 102 | syl2anc 691 |
. . . . . . . 8
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → ∃𝑛 ∈ ℕ0
(#‘(Base‘(𝐺
↾s 𝐻))) =
(𝑃↑𝑛)) |
104 | 94 | subggrp 17420 |
. . . . . . . . . 10
⊢ (𝐻 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝐻) ∈ Grp) |
105 | 104 | ad2antrl 760 |
. . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (𝐺 ↾s 𝐻) ∈ Grp) |
106 | 96, 84 | eqeltrrd 2689 |
. . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (Base‘(𝐺 ↾s 𝐻)) ∈ Fin) |
107 | | eqid 2610 |
. . . . . . . . . 10
⊢
(Base‘(𝐺
↾s 𝐻)) =
(Base‘(𝐺
↾s 𝐻)) |
108 | 107 | pgpfi 17843 |
. . . . . . . . 9
⊢ (((𝐺 ↾s 𝐻) ∈ Grp ∧
(Base‘(𝐺
↾s 𝐻))
∈ Fin) → (𝑃 pGrp
(𝐺 ↾s
𝐻) ↔ (𝑃 ∈ ℙ ∧
∃𝑛 ∈
ℕ0 (#‘(Base‘(𝐺 ↾s 𝐻))) = (𝑃↑𝑛)))) |
109 | 105, 106,
108 | syl2anc 691 |
. . . . . . . 8
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (𝑃 pGrp (𝐺 ↾s 𝐻) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0
(#‘(Base‘(𝐺
↾s 𝐻))) =
(𝑃↑𝑛)))) |
110 | 8, 103, 109 | mpbir2and 959 |
. . . . . . 7
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝑃 pGrp (𝐺 ↾s 𝐻)) |
111 | 110 | adantr 480 |
. . . . . 6
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺 ↾s 𝐻)) |
112 | | oveq2 6557 |
. . . . . . . 8
⊢ (𝐻 = 𝑘 → (𝐺 ↾s 𝐻) = (𝐺 ↾s 𝑘)) |
113 | 112 | breq2d 4595 |
. . . . . . 7
⊢ (𝐻 = 𝑘 → (𝑃 pGrp (𝐺 ↾s 𝐻) ↔ 𝑃 pGrp (𝐺 ↾s 𝑘))) |
114 | | eqimss 3620 |
. . . . . . . 8
⊢ (𝐻 = 𝑘 → 𝐻 ⊆ 𝑘) |
115 | 114 | biantrurd 528 |
. . . . . . 7
⊢ (𝐻 = 𝑘 → (𝑃 pGrp (𝐺 ↾s 𝑘) ↔ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) |
116 | 113, 115 | bitrd 267 |
. . . . . 6
⊢ (𝐻 = 𝑘 → (𝑃 pGrp (𝐺 ↾s 𝐻) ↔ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) |
117 | 111, 116 | syl5ibcom 234 |
. . . . 5
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → (𝐻 = 𝑘 → (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)))) |
118 | 93, 117 | impbid 201 |
. . . 4
⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)) |
119 | 118 | ralrimiva 2949 |
. . 3
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)) |
120 | | isslw 17846 |
. . 3
⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘))) |
121 | 8, 9, 119, 120 | syl3anbrc 1239 |
. 2
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐻 ∈ (𝑃 pSyl 𝐺)) |
122 | 7, 121 | impbida 873 |
1
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋)))))) |