Proof of Theorem sylow3lem4
| Step | Hyp | Ref
| Expression |
| 1 | | sylow3.x |
. . 3
⊢ 𝑋 = (Base‘𝐺) |
| 2 | | sylow3.g |
. . 3
⊢ (𝜑 → 𝐺 ∈ Grp) |
| 3 | | sylow3.xf |
. . 3
⊢ (𝜑 → 𝑋 ∈ Fin) |
| 4 | | sylow3.p |
. . 3
⊢ (𝜑 → 𝑃 ∈ ℙ) |
| 5 | | sylow3lem1.a |
. . 3
⊢ + =
(+g‘𝐺) |
| 6 | | sylow3lem1.d |
. . 3
⊢ − =
(-g‘𝐺) |
| 7 | | sylow3lem1.m |
. . 3
⊢ ⊕ =
(𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) |
| 8 | | sylow3lem2.k |
. . 3
⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) |
| 9 | | sylow3lem2.h |
. . 3
⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐾) = 𝐾} |
| 10 | | sylow3lem2.n |
. . 3
⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)} |
| 11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | sylow3lem3 19647 |
. 2
⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) = (♯‘(𝑋 / (𝐺 ~QG 𝑁)))) |
| 12 | | slwsubg 19628 |
. . . . . . . . . 10
⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺)) |
| 13 | 8, 12 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) |
| 14 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (𝐺 ↾s 𝑁) = (𝐺 ↾s 𝑁) |
| 15 | 10, 1, 5, 14 | nmznsg 19186 |
. . . . . . . . . 10
⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ∈ (NrmSGrp‘(𝐺 ↾s 𝑁))) |
| 16 | | nsgsubg 19176 |
. . . . . . . . . 10
⊢ (𝐾 ∈ (NrmSGrp‘(𝐺 ↾s 𝑁)) → 𝐾 ∈ (SubGrp‘(𝐺 ↾s 𝑁))) |
| 17 | 15, 16 | syl 17 |
. . . . . . . . 9
⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ∈ (SubGrp‘(𝐺 ↾s 𝑁))) |
| 18 | 13, 17 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ (SubGrp‘(𝐺 ↾s 𝑁))) |
| 19 | 10, 1, 5 | nmzsubg 19183 |
. . . . . . . . . . 11
⊢ (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺)) |
| 20 | 2, 19 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺)) |
| 21 | 14 | subgbas 19148 |
. . . . . . . . . 10
⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘(𝐺 ↾s 𝑁))) |
| 22 | 20, 21 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 = (Base‘(𝐺 ↾s 𝑁))) |
| 23 | 1 | subgss 19145 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ 𝑋) |
| 24 | 20, 23 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ⊆ 𝑋) |
| 25 | 3, 24 | ssfid 9301 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ Fin) |
| 26 | 22, 25 | eqeltrrd 2842 |
. . . . . . . 8
⊢ (𝜑 → (Base‘(𝐺 ↾s 𝑁)) ∈ Fin) |
| 27 | | eqid 2737 |
. . . . . . . . 9
⊢
(Base‘(𝐺
↾s 𝑁)) =
(Base‘(𝐺
↾s 𝑁)) |
| 28 | 27 | lagsubg 19213 |
. . . . . . . 8
⊢ ((𝐾 ∈ (SubGrp‘(𝐺 ↾s 𝑁)) ∧ (Base‘(𝐺 ↾s 𝑁)) ∈ Fin) →
(♯‘𝐾) ∥
(♯‘(Base‘(𝐺 ↾s 𝑁)))) |
| 29 | 18, 26, 28 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → (♯‘𝐾) ∥
(♯‘(Base‘(𝐺 ↾s 𝑁)))) |
| 30 | 22 | fveq2d 6910 |
. . . . . . 7
⊢ (𝜑 → (♯‘𝑁) =
(♯‘(Base‘(𝐺 ↾s 𝑁)))) |
| 31 | 29, 30 | breqtrrd 5171 |
. . . . . 6
⊢ (𝜑 → (♯‘𝐾) ∥ (♯‘𝑁)) |
| 32 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(0g‘𝐺) = (0g‘𝐺) |
| 33 | 32 | subg0cl 19152 |
. . . . . . . . . . 11
⊢ (𝐾 ∈ (SubGrp‘𝐺) →
(0g‘𝐺)
∈ 𝐾) |
| 34 | 13, 33 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (0g‘𝐺) ∈ 𝐾) |
| 35 | 34 | ne0d 4342 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ≠ ∅) |
| 36 | 1 | subgss 19145 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋) |
| 37 | 13, 36 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐾 ⊆ 𝑋) |
| 38 | 3, 37 | ssfid 9301 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ Fin) |
| 39 | | hashnncl 14405 |
. . . . . . . . . 10
⊢ (𝐾 ∈ Fin →
((♯‘𝐾) ∈
ℕ ↔ 𝐾 ≠
∅)) |
| 40 | 38, 39 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((♯‘𝐾) ∈ ℕ ↔ 𝐾 ≠ ∅)) |
| 41 | 35, 40 | mpbird 257 |
. . . . . . . 8
⊢ (𝜑 → (♯‘𝐾) ∈
ℕ) |
| 42 | 41 | nnzd 12640 |
. . . . . . 7
⊢ (𝜑 → (♯‘𝐾) ∈
ℤ) |
| 43 | | hashcl 14395 |
. . . . . . . . 9
⊢ (𝑁 ∈ Fin →
(♯‘𝑁) ∈
ℕ0) |
| 44 | 25, 43 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (♯‘𝑁) ∈
ℕ0) |
| 45 | 44 | nn0zd 12639 |
. . . . . . 7
⊢ (𝜑 → (♯‘𝑁) ∈
ℤ) |
| 46 | | pwfi 9357 |
. . . . . . . . . . 11
⊢ (𝑋 ∈ Fin ↔ 𝒫
𝑋 ∈
Fin) |
| 47 | 3, 46 | sylib 218 |
. . . . . . . . . 10
⊢ (𝜑 → 𝒫 𝑋 ∈ Fin) |
| 48 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (𝐺 ~QG 𝑁) = (𝐺 ~QG 𝑁) |
| 49 | 1, 48 | eqger 19196 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝑁) Er 𝑋) |
| 50 | 20, 49 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐺 ~QG 𝑁) Er 𝑋) |
| 51 | 50 | qsss 8818 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑋 / (𝐺 ~QG 𝑁)) ⊆ 𝒫 𝑋) |
| 52 | 47, 51 | ssfid 9301 |
. . . . . . . . 9
⊢ (𝜑 → (𝑋 / (𝐺 ~QG 𝑁)) ∈ Fin) |
| 53 | | hashcl 14395 |
. . . . . . . . 9
⊢ ((𝑋 / (𝐺 ~QG 𝑁)) ∈ Fin → (♯‘(𝑋 / (𝐺 ~QG 𝑁))) ∈
ℕ0) |
| 54 | 52, 53 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (♯‘(𝑋 / (𝐺 ~QG 𝑁))) ∈
ℕ0) |
| 55 | 54 | nn0zd 12639 |
. . . . . . 7
⊢ (𝜑 → (♯‘(𝑋 / (𝐺 ~QG 𝑁))) ∈ ℤ) |
| 56 | | dvdscmul 16320 |
. . . . . . 7
⊢
(((♯‘𝐾)
∈ ℤ ∧ (♯‘𝑁) ∈ ℤ ∧ (♯‘(𝑋 / (𝐺 ~QG 𝑁))) ∈ ℤ) →
((♯‘𝐾) ∥
(♯‘𝑁) →
((♯‘(𝑋 /
(𝐺 ~QG
𝑁))) ·
(♯‘𝐾)) ∥
((♯‘(𝑋 /
(𝐺 ~QG
𝑁))) ·
(♯‘𝑁)))) |
| 57 | 42, 45, 55, 56 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → ((♯‘𝐾) ∥ (♯‘𝑁) → ((♯‘(𝑋 / (𝐺 ~QG 𝑁))) · (♯‘𝐾)) ∥
((♯‘(𝑋 /
(𝐺 ~QG
𝑁))) ·
(♯‘𝑁)))) |
| 58 | 31, 57 | mpd 15 |
. . . . 5
⊢ (𝜑 → ((♯‘(𝑋 / (𝐺 ~QG 𝑁))) · (♯‘𝐾)) ∥
((♯‘(𝑋 /
(𝐺 ~QG
𝑁))) ·
(♯‘𝑁))) |
| 59 | | hashcl 14395 |
. . . . . . . . 9
⊢ (𝑋 ∈ Fin →
(♯‘𝑋) ∈
ℕ0) |
| 60 | 3, 59 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (♯‘𝑋) ∈
ℕ0) |
| 61 | 60 | nn0cnd 12589 |
. . . . . . 7
⊢ (𝜑 → (♯‘𝑋) ∈
ℂ) |
| 62 | 41 | nncnd 12282 |
. . . . . . 7
⊢ (𝜑 → (♯‘𝐾) ∈
ℂ) |
| 63 | 41 | nnne0d 12316 |
. . . . . . 7
⊢ (𝜑 → (♯‘𝐾) ≠ 0) |
| 64 | 61, 62, 63 | divcan1d 12044 |
. . . . . 6
⊢ (𝜑 → (((♯‘𝑋) / (♯‘𝐾)) · (♯‘𝐾)) = (♯‘𝑋)) |
| 65 | 1, 48, 20, 3 | lagsubg2 19212 |
. . . . . 6
⊢ (𝜑 → (♯‘𝑋) = ((♯‘(𝑋 / (𝐺 ~QG 𝑁))) · (♯‘𝑁))) |
| 66 | 64, 65 | eqtrd 2777 |
. . . . 5
⊢ (𝜑 → (((♯‘𝑋) / (♯‘𝐾)) · (♯‘𝐾)) = ((♯‘(𝑋 / (𝐺 ~QG 𝑁))) · (♯‘𝑁))) |
| 67 | 58, 66 | breqtrrd 5171 |
. . . 4
⊢ (𝜑 → ((♯‘(𝑋 / (𝐺 ~QG 𝑁))) · (♯‘𝐾)) ∥
(((♯‘𝑋) /
(♯‘𝐾)) ·
(♯‘𝐾))) |
| 68 | 1 | lagsubg 19213 |
. . . . . . 7
⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (♯‘𝐾) ∥ (♯‘𝑋)) |
| 69 | 13, 3, 68 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (♯‘𝐾) ∥ (♯‘𝑋)) |
| 70 | 60 | nn0zd 12639 |
. . . . . . 7
⊢ (𝜑 → (♯‘𝑋) ∈
ℤ) |
| 71 | | dvdsval2 16293 |
. . . . . . 7
⊢
(((♯‘𝐾)
∈ ℤ ∧ (♯‘𝐾) ≠ 0 ∧ (♯‘𝑋) ∈ ℤ) →
((♯‘𝐾) ∥
(♯‘𝑋) ↔
((♯‘𝑋) /
(♯‘𝐾)) ∈
ℤ)) |
| 72 | 42, 63, 70, 71 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → ((♯‘𝐾) ∥ (♯‘𝑋) ↔ ((♯‘𝑋) / (♯‘𝐾)) ∈
ℤ)) |
| 73 | 69, 72 | mpbid 232 |
. . . . 5
⊢ (𝜑 → ((♯‘𝑋) / (♯‘𝐾)) ∈
ℤ) |
| 74 | | dvdsmulcr 16323 |
. . . . 5
⊢
(((♯‘(𝑋
/ (𝐺
~QG 𝑁)))
∈ ℤ ∧ ((♯‘𝑋) / (♯‘𝐾)) ∈ ℤ ∧
((♯‘𝐾) ∈
ℤ ∧ (♯‘𝐾) ≠ 0)) → (((♯‘(𝑋 / (𝐺 ~QG 𝑁))) · (♯‘𝐾)) ∥
(((♯‘𝑋) /
(♯‘𝐾)) ·
(♯‘𝐾)) ↔
(♯‘(𝑋 /
(𝐺 ~QG
𝑁))) ∥
((♯‘𝑋) /
(♯‘𝐾)))) |
| 75 | 55, 73, 42, 63, 74 | syl112anc 1376 |
. . . 4
⊢ (𝜑 → (((♯‘(𝑋 / (𝐺 ~QG 𝑁))) · (♯‘𝐾)) ∥
(((♯‘𝑋) /
(♯‘𝐾)) ·
(♯‘𝐾)) ↔
(♯‘(𝑋 /
(𝐺 ~QG
𝑁))) ∥
((♯‘𝑋) /
(♯‘𝐾)))) |
| 76 | 67, 75 | mpbid 232 |
. . 3
⊢ (𝜑 → (♯‘(𝑋 / (𝐺 ~QG 𝑁))) ∥ ((♯‘𝑋) / (♯‘𝐾))) |
| 77 | 1, 3, 8 | slwhash 19642 |
. . . 4
⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) |
| 78 | 77 | oveq2d 7447 |
. . 3
⊢ (𝜑 → ((♯‘𝑋) / (♯‘𝐾)) = ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋))))) |
| 79 | 76, 78 | breqtrd 5169 |
. 2
⊢ (𝜑 → (♯‘(𝑋 / (𝐺 ~QG 𝑁))) ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋))))) |
| 80 | 11, 79 | eqbrtrd 5165 |
1
⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋))))) |