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Theorem sylow2alem2 18417
Description: Lemma for sylow2a 18418. All the orbits which are not for fixed points have size 𝐺 ∣ / ∣ 𝐺𝑥 (where 𝐺𝑥 is the stabilizer subgroup) and thus are powers of 𝑃. And since they are all nontrivial (because any orbit which is a singleton is a fixed point), they all divide 𝑃, and so does the sum of all of them. (Contributed by Mario Carneiro, 17-Jan-2015.)
Hypotheses
Ref Expression
sylow2a.x 𝑋 = (Base‘𝐺)
sylow2a.m (𝜑 ∈ (𝐺 GrpAct 𝑌))
sylow2a.p (𝜑𝑃 pGrp 𝐺)
sylow2a.f (𝜑𝑋 ∈ Fin)
sylow2a.y (𝜑𝑌 ∈ Fin)
sylow2a.z 𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}
sylow2a.r = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
Assertion
Ref Expression
sylow2alem2 (𝜑𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)(♯‘𝑧))
Distinct variable groups:   𝑧,,   𝑔,,𝑢,𝑥,𝑦   𝑔,𝐺,𝑥,𝑦   𝑧,𝑃   ,𝑔,,𝑢,𝑥,𝑦   𝑔,𝑋,,𝑢,𝑥,𝑦   𝑧,𝑍   𝜑,,𝑧   𝑧,𝑔,𝑌,,𝑢,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑢,𝑔)   𝑃(𝑥,𝑦,𝑢,𝑔,)   (𝑧)   (𝑥,𝑦,𝑢,𝑔)   𝐺(𝑧,𝑢,)   𝑋(𝑧)   𝑍(𝑥,𝑦,𝑢,𝑔,)

Proof of Theorem sylow2alem2
Dummy variables 𝑘 𝑛 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sylow2a.y . . . . 5 (𝜑𝑌 ∈ Fin)
2 pwfi 8549 . . . . 5 (𝑌 ∈ Fin ↔ 𝒫 𝑌 ∈ Fin)
31, 2sylib 210 . . . 4 (𝜑 → 𝒫 𝑌 ∈ Fin)
4 sylow2a.m . . . . . 6 (𝜑 ∈ (𝐺 GrpAct 𝑌))
5 sylow2a.r . . . . . . 7 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
6 sylow2a.x . . . . . . 7 𝑋 = (Base‘𝐺)
75, 6gaorber 18124 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → Er 𝑌)
84, 7syl 17 . . . . 5 (𝜑 Er 𝑌)
98qsss 8091 . . . 4 (𝜑 → (𝑌 / ) ⊆ 𝒫 𝑌)
103, 9ssfid 8471 . . 3 (𝜑 → (𝑌 / ) ∈ Fin)
11 diffi 8480 . . 3 ((𝑌 / ) ∈ Fin → ((𝑌 / ) ∖ 𝒫 𝑍) ∈ Fin)
1210, 11syl 17 . 2 (𝜑 → ((𝑌 / ) ∖ 𝒫 𝑍) ∈ Fin)
13 sylow2a.p . . . . 5 (𝜑𝑃 pGrp 𝐺)
14 gagrp 18108 . . . . . . 7 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
154, 14syl 17 . . . . . 6 (𝜑𝐺 ∈ Grp)
16 sylow2a.f . . . . . 6 (𝜑𝑋 ∈ Fin)
176pgpfi 18404 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛))))
1815, 16, 17syl2anc 579 . . . . 5 (𝜑 → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛))))
1913, 18mpbid 224 . . . 4 (𝜑 → (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛)))
2019simpld 490 . . 3 (𝜑𝑃 ∈ ℙ)
21 prmz 15794 . . 3 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
2220, 21syl 17 . 2 (𝜑𝑃 ∈ ℤ)
23 eldifi 3955 . . . . 5 (𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍) → 𝑧 ∈ (𝑌 / ))
241adantr 474 . . . . . 6 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑌 ∈ Fin)
259sselda 3821 . . . . . . 7 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧 ∈ 𝒫 𝑌)
2625elpwid 4391 . . . . . 6 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧𝑌)
2724, 26ssfid 8471 . . . . 5 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧 ∈ Fin)
2823, 27sylan2 586 . . . 4 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → 𝑧 ∈ Fin)
29 hashcl 13462 . . . 4 (𝑧 ∈ Fin → (♯‘𝑧) ∈ ℕ0)
3028, 29syl 17 . . 3 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → (♯‘𝑧) ∈ ℕ0)
3130nn0zd 11832 . 2 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → (♯‘𝑧) ∈ ℤ)
32 eldif 3802 . . 3 (𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍) ↔ (𝑧 ∈ (𝑌 / ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍))
33 eqid 2778 . . . . 5 (𝑌 / ) = (𝑌 / )
34 sseq1 3845 . . . . . . . 8 ([𝑤] = 𝑧 → ([𝑤] 𝑍𝑧𝑍))
35 selpw 4386 . . . . . . . 8 (𝑧 ∈ 𝒫 𝑍𝑧𝑍)
3634, 35syl6bbr 281 . . . . . . 7 ([𝑤] = 𝑧 → ([𝑤] 𝑍𝑧 ∈ 𝒫 𝑍))
3736notbid 310 . . . . . 6 ([𝑤] = 𝑧 → (¬ [𝑤] 𝑍 ↔ ¬ 𝑧 ∈ 𝒫 𝑍))
38 fveq2 6446 . . . . . . 7 ([𝑤] = 𝑧 → (♯‘[𝑤] ) = (♯‘𝑧))
3938breq2d 4898 . . . . . 6 ([𝑤] = 𝑧 → (𝑃 ∥ (♯‘[𝑤] ) ↔ 𝑃 ∥ (♯‘𝑧)))
4037, 39imbi12d 336 . . . . 5 ([𝑤] = 𝑧 → ((¬ [𝑤] 𝑍𝑃 ∥ (♯‘[𝑤] )) ↔ (¬ 𝑧 ∈ 𝒫 𝑍𝑃 ∥ (♯‘𝑧))))
4120adantr 474 . . . . . . . . . 10 ((𝜑𝑤𝑌) → 𝑃 ∈ ℙ)
428adantr 474 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → Er 𝑌)
43 simpr 479 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → 𝑤𝑌)
4442, 43erref 8046 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → 𝑤 𝑤)
45 vex 3401 . . . . . . . . . . . . . 14 𝑤 ∈ V
4645, 45elec 8068 . . . . . . . . . . . . 13 (𝑤 ∈ [𝑤] 𝑤 𝑤)
4744, 46sylibr 226 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → 𝑤 ∈ [𝑤] )
4847ne0d 4150 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → [𝑤] ≠ ∅)
498ecss 8070 . . . . . . . . . . . . . 14 (𝜑 → [𝑤] 𝑌)
501, 49ssfid 8471 . . . . . . . . . . . . 13 (𝜑 → [𝑤] ∈ Fin)
5150adantr 474 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → [𝑤] ∈ Fin)
52 hashnncl 13472 . . . . . . . . . . . 12 ([𝑤] ∈ Fin → ((♯‘[𝑤] ) ∈ ℕ ↔ [𝑤] ≠ ∅))
5351, 52syl 17 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → ((♯‘[𝑤] ) ∈ ℕ ↔ [𝑤] ≠ ∅))
5448, 53mpbird 249 . . . . . . . . . 10 ((𝜑𝑤𝑌) → (♯‘[𝑤] ) ∈ ℕ)
55 pceq0 15979 . . . . . . . . . 10 ((𝑃 ∈ ℙ ∧ (♯‘[𝑤] ) ∈ ℕ) → ((𝑃 pCnt (♯‘[𝑤] )) = 0 ↔ ¬ 𝑃 ∥ (♯‘[𝑤] )))
5641, 54, 55syl2anc 579 . . . . . . . . 9 ((𝜑𝑤𝑌) → ((𝑃 pCnt (♯‘[𝑤] )) = 0 ↔ ¬ 𝑃 ∥ (♯‘[𝑤] )))
57 oveq2 6930 . . . . . . . . . 10 ((𝑃 pCnt (♯‘[𝑤] )) = 0 → (𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (𝑃↑0))
58 hashcl 13462 . . . . . . . . . . . . . . . . . . . . . 22 ([𝑤] ∈ Fin → (♯‘[𝑤] ) ∈ ℕ0)
5950, 58syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘[𝑤] ) ∈ ℕ0)
6059nn0zd 11832 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (♯‘[𝑤] ) ∈ ℤ)
61 ssrab2 3908 . . . . . . . . . . . . . . . . . . . . . . 23 {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ⊆ 𝑋
62 ssfi 8468 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑋 ∈ Fin ∧ {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ⊆ 𝑋) → {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ∈ Fin)
6316, 61, 62sylancl 580 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ∈ Fin)
64 hashcl 13462 . . . . . . . . . . . . . . . . . . . . . 22 ({𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ∈ Fin → (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℕ0)
6563, 64syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℕ0)
6665nn0zd 11832 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℤ)
67 dvdsmul1 15410 . . . . . . . . . . . . . . . . . . . 20 (((♯‘[𝑤] ) ∈ ℤ ∧ (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℤ) → (♯‘[𝑤] ) ∥ ((♯‘[𝑤] ) · (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
6860, 66, 67syl2anc 579 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (♯‘[𝑤] ) ∥ ((♯‘[𝑤] ) · (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
6968adantr 474 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝑌) → (♯‘[𝑤] ) ∥ ((♯‘[𝑤] ) · (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
704adantr 474 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝑌) → ∈ (𝐺 GrpAct 𝑌))
7116adantr 474 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝑌) → 𝑋 ∈ Fin)
72 eqid 2778 . . . . . . . . . . . . . . . . . . . 20 {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} = {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}
73 eqid 2778 . . . . . . . . . . . . . . . . . . . 20 (𝐺 ~QG {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) = (𝐺 ~QG {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})
746, 72, 73, 5orbsta2 18130 . . . . . . . . . . . . . . . . . . 19 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑤𝑌) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘[𝑤] ) · (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
7570, 43, 71, 74syl21anc 828 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝑌) → (♯‘𝑋) = ((♯‘[𝑤] ) · (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
7669, 75breqtrrd 4914 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → (♯‘[𝑤] ) ∥ (♯‘𝑋))
7719simprd 491 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛))
7877adantr 474 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛))
79 breq2 4890 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑋) = (𝑃𝑛) → ((♯‘[𝑤] ) ∥ (♯‘𝑋) ↔ (♯‘[𝑤] ) ∥ (𝑃𝑛)))
8079biimpcd 241 . . . . . . . . . . . . . . . . . 18 ((♯‘[𝑤] ) ∥ (♯‘𝑋) → ((♯‘𝑋) = (𝑃𝑛) → (♯‘[𝑤] ) ∥ (𝑃𝑛)))
8180reximdv 3197 . . . . . . . . . . . . . . . . 17 ((♯‘[𝑤] ) ∥ (♯‘𝑋) → (∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛) → ∃𝑛 ∈ ℕ0 (♯‘[𝑤] ) ∥ (𝑃𝑛)))
8276, 78, 81sylc 65 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝑌) → ∃𝑛 ∈ ℕ0 (♯‘[𝑤] ) ∥ (𝑃𝑛))
83 pcprmpw2 15990 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ ℙ ∧ (♯‘[𝑤] ) ∈ ℕ) → (∃𝑛 ∈ ℕ0 (♯‘[𝑤] ) ∥ (𝑃𝑛) ↔ (♯‘[𝑤] ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] )))))
8441, 54, 83syl2anc 579 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝑌) → (∃𝑛 ∈ ℕ0 (♯‘[𝑤] ) ∥ (𝑃𝑛) ↔ (♯‘[𝑤] ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] )))))
8582, 84mpbid 224 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝑌) → (♯‘[𝑤] ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] ))))
8685eqcomd 2784 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → (𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (♯‘[𝑤] ))
8722adantr 474 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → 𝑃 ∈ ℤ)
8887zcnd 11835 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝑌) → 𝑃 ∈ ℂ)
8988exp0d 13321 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝑌) → (𝑃↑0) = 1)
90 hash1 13506 . . . . . . . . . . . . . . 15 (♯‘1o) = 1
9189, 90syl6eqr 2832 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → (𝑃↑0) = (♯‘1o))
9286, 91eqeq12d 2793 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (𝑃↑0) ↔ (♯‘[𝑤] ) = (♯‘1o)))
93 df1o2 7856 . . . . . . . . . . . . . . 15 1o = {∅}
94 snfi 8326 . . . . . . . . . . . . . . 15 {∅} ∈ Fin
9593, 94eqeltri 2855 . . . . . . . . . . . . . 14 1o ∈ Fin
96 hashen 13452 . . . . . . . . . . . . . 14 (([𝑤] ∈ Fin ∧ 1o ∈ Fin) → ((♯‘[𝑤] ) = (♯‘1o) ↔ [𝑤] ≈ 1o))
9751, 95, 96sylancl 580 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → ((♯‘[𝑤] ) = (♯‘1o) ↔ [𝑤] ≈ 1o))
9892, 97bitrd 271 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (𝑃↑0) ↔ [𝑤] ≈ 1o))
99 en1b 8309 . . . . . . . . . . . 12 ([𝑤] ≈ 1o ↔ [𝑤] = { [𝑤] })
10098, 99syl6bb 279 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (𝑃↑0) ↔ [𝑤] = { [𝑤] }))
10143adantr 474 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤𝑌)
1024ad2antrr 716 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ∈ (𝐺 GrpAct 𝑌))
1036gaf 18111 . . . . . . . . . . . . . . . . . . . 20 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
104102, 103syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → :(𝑋 × 𝑌)⟶𝑌)
105 simprl 761 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑋)
106104, 105, 101fovrnd 7083 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ 𝑌)
107 eqid 2778 . . . . . . . . . . . . . . . . . . 19 ( 𝑤) = ( 𝑤)
108 oveq1 6929 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝑘 𝑤) = ( 𝑤))
109108eqeq1d 2780 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝑘 𝑤) = ( 𝑤) ↔ ( 𝑤) = ( 𝑤)))
110109rspcev 3511 . . . . . . . . . . . . . . . . . . 19 ((𝑋 ∧ ( 𝑤) = ( 𝑤)) → ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤))
111105, 107, 110sylancl 580 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤))
1125gaorb 18123 . . . . . . . . . . . . . . . . . 18 (𝑤 ( 𝑤) ↔ (𝑤𝑌 ∧ ( 𝑤) ∈ 𝑌 ∧ ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤)))
113101, 106, 111, 112syl3anbrc 1400 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ( 𝑤))
114 ovex 6954 . . . . . . . . . . . . . . . . . 18 ( 𝑤) ∈ V
115114, 45elec 8068 . . . . . . . . . . . . . . . . 17 (( 𝑤) ∈ [𝑤] 𝑤 ( 𝑤))
116113, 115sylibr 226 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ [𝑤] )
117 simprr 763 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → [𝑤] = { [𝑤] })
118116, 117eleqtrd 2861 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ { [𝑤] })
119114elsn 4413 . . . . . . . . . . . . . . 15 (( 𝑤) ∈ { [𝑤] } ↔ ( 𝑤) = [𝑤] )
120118, 119sylib 210 . . . . . . . . . . . . . 14 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) = [𝑤] )
12147adantr 474 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ∈ [𝑤] )
122121, 117eleqtrd 2861 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ∈ { [𝑤] })
12345elsn 4413 . . . . . . . . . . . . . . 15 (𝑤 ∈ { [𝑤] } ↔ 𝑤 = [𝑤] )
124122, 123sylib 210 . . . . . . . . . . . . . 14 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 = [𝑤] )
125120, 124eqtr4d 2817 . . . . . . . . . . . . 13 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) = 𝑤)
126125expr 450 . . . . . . . . . . . 12 (((𝜑𝑤𝑌) ∧ 𝑋) → ([𝑤] = { [𝑤] } → ( 𝑤) = 𝑤))
127126ralrimdva 3151 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → ([𝑤] = { [𝑤] } → ∀𝑋 ( 𝑤) = 𝑤))
128100, 127sylbid 232 . . . . . . . . . 10 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (𝑃↑0) → ∀𝑋 ( 𝑤) = 𝑤))
12957, 128syl5 34 . . . . . . . . 9 ((𝜑𝑤𝑌) → ((𝑃 pCnt (♯‘[𝑤] )) = 0 → ∀𝑋 ( 𝑤) = 𝑤))
13056, 129sylbird 252 . . . . . . . 8 ((𝜑𝑤𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ) → ∀𝑋 ( 𝑤) = 𝑤))
131 oveq2 6930 . . . . . . . . . . . . 13 (𝑢 = 𝑤 → ( 𝑢) = ( 𝑤))
132 id 22 . . . . . . . . . . . . 13 (𝑢 = 𝑤𝑢 = 𝑤)
133131, 132eqeq12d 2793 . . . . . . . . . . . 12 (𝑢 = 𝑤 → (( 𝑢) = 𝑢 ↔ ( 𝑤) = 𝑤))
134133ralbidv 3168 . . . . . . . . . . 11 (𝑢 = 𝑤 → (∀𝑋 ( 𝑢) = 𝑢 ↔ ∀𝑋 ( 𝑤) = 𝑤))
135 sylow2a.z . . . . . . . . . . 11 𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}
136134, 135elrab2 3576 . . . . . . . . . 10 (𝑤𝑍 ↔ (𝑤𝑌 ∧ ∀𝑋 ( 𝑤) = 𝑤))
137136baib 531 . . . . . . . . 9 (𝑤𝑌 → (𝑤𝑍 ↔ ∀𝑋 ( 𝑤) = 𝑤))
138137adantl 475 . . . . . . . 8 ((𝜑𝑤𝑌) → (𝑤𝑍 ↔ ∀𝑋 ( 𝑤) = 𝑤))
139130, 138sylibrd 251 . . . . . . 7 ((𝜑𝑤𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ) → 𝑤𝑍))
1406, 4, 13, 16, 1, 135, 5sylow2alem1 18416 . . . . . . . . . 10 ((𝜑𝑤𝑍) → [𝑤] = {𝑤})
141 simpr 479 . . . . . . . . . . 11 ((𝜑𝑤𝑍) → 𝑤𝑍)
142141snssd 4571 . . . . . . . . . 10 ((𝜑𝑤𝑍) → {𝑤} ⊆ 𝑍)
143140, 142eqsstrd 3858 . . . . . . . . 9 ((𝜑𝑤𝑍) → [𝑤] 𝑍)
144143ex 403 . . . . . . . 8 (𝜑 → (𝑤𝑍 → [𝑤] 𝑍))
145144adantr 474 . . . . . . 7 ((𝜑𝑤𝑌) → (𝑤𝑍 → [𝑤] 𝑍))
146139, 145syld 47 . . . . . 6 ((𝜑𝑤𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ) → [𝑤] 𝑍))
147146con1d 142 . . . . 5 ((𝜑𝑤𝑌) → (¬ [𝑤] 𝑍𝑃 ∥ (♯‘[𝑤] )))
14833, 40, 147ectocld 8097 . . . 4 ((𝜑𝑧 ∈ (𝑌 / )) → (¬ 𝑧 ∈ 𝒫 𝑍𝑃 ∥ (♯‘𝑧)))
149148impr 448 . . 3 ((𝜑 ∧ (𝑧 ∈ (𝑌 / ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍)) → 𝑃 ∥ (♯‘𝑧))
15032, 149sylan2b 587 . 2 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → 𝑃 ∥ (♯‘𝑧))
15112, 22, 31, 150fsumdvds 15437 1 (𝜑𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)(♯‘𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  wne 2969  wral 3090  wrex 3091  {crab 3094  cdif 3789  wss 3792  c0 4141  𝒫 cpw 4379  {csn 4398  {cpr 4400   cuni 4671   class class class wbr 4886  {copab 4948   × cxp 5353  wf 6131  cfv 6135  (class class class)co 6922  1oc1o 7836   Er wer 8023  [cec 8024   / cqs 8025  cen 8238  Fincfn 8241  0cc0 10272  1c1 10273   · cmul 10277  cn 11374  0cn0 11642  cz 11728  cexp 13178  chash 13435  Σcsu 14824  cdvds 15387  cprime 15790   pCnt cpc 15945  Basecbs 16255  Grpcgrp 17809   ~QG cqg 17974   GrpAct cga 18105   pGrp cpgp 18330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-disj 4855  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-omul 7848  df-er 8026  df-ec 8028  df-qs 8032  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-sup 8636  df-inf 8637  df-oi 8704  df-card 9098  df-acn 9101  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-xnn0 11715  df-z 11729  df-uz 11993  df-q 12096  df-rp 12138  df-fz 12644  df-fzo 12785  df-fl 12912  df-mod 12988  df-seq 13120  df-exp 13179  df-fac 13379  df-bc 13408  df-hash 13436  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-clim 14627  df-sum 14825  df-dvds 15388  df-gcd 15623  df-prm 15791  df-pc 15946  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-0g 16488  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-submnd 17722  df-grp 17812  df-minusg 17813  df-sbg 17814  df-mulg 17928  df-subg 17975  df-eqg 17977  df-ga 18106  df-od 18332  df-pgp 18334
This theorem is referenced by:  sylow2a  18418
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