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Theorem sylow2alem2 19400
Description: Lemma for sylow2a 19401. 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 9122 . . . . 5 (𝑌 ∈ Fin ↔ 𝒫 𝑌 ∈ Fin)
31, 2sylib 217 . . . 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 19088 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → Er 𝑌)
84, 7syl 17 . . . . 5 (𝜑 Er 𝑌)
98qsss 8717 . . . 4 (𝜑 → (𝑌 / ) ⊆ 𝒫 𝑌)
103, 9ssfid 9211 . . 3 (𝜑 → (𝑌 / ) ∈ Fin)
11 diffi 9123 . . 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 19072 . . . . . . 7 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
154, 14syl 17 . . . . . 6 (𝜑𝐺 ∈ Grp)
16 sylow2a.f . . . . . 6 (𝜑𝑋 ∈ Fin)
176pgpfi 19387 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛))))
1815, 16, 17syl2anc 584 . . . . 5 (𝜑 → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛))))
1913, 18mpbid 231 . . . 4 (𝜑 → (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛)))
2019simpld 495 . . 3 (𝜑𝑃 ∈ ℙ)
21 prmz 16551 . . 3 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
2220, 21syl 17 . 2 (𝜑𝑃 ∈ ℤ)
23 eldifi 4086 . . . . 5 (𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍) → 𝑧 ∈ (𝑌 / ))
241adantr 481 . . . . . 6 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑌 ∈ Fin)
259sselda 3944 . . . . . . 7 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧 ∈ 𝒫 𝑌)
2625elpwid 4569 . . . . . 6 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧𝑌)
2724, 26ssfid 9211 . . . . 5 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧 ∈ Fin)
2823, 27sylan2 593 . . . 4 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → 𝑧 ∈ Fin)
29 hashcl 14256 . . . 4 (𝑧 ∈ Fin → (♯‘𝑧) ∈ ℕ0)
3028, 29syl 17 . . 3 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → (♯‘𝑧) ∈ ℕ0)
3130nn0zd 12525 . 2 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → (♯‘𝑧) ∈ ℤ)
32 eldif 3920 . . 3 (𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍) ↔ (𝑧 ∈ (𝑌 / ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍))
33 eqid 2736 . . . . 5 (𝑌 / ) = (𝑌 / )
34 sseq1 3969 . . . . . . . 8 ([𝑤] = 𝑧 → ([𝑤] 𝑍𝑧𝑍))
35 velpw 4565 . . . . . . . 8 (𝑧 ∈ 𝒫 𝑍𝑧𝑍)
3634, 35bitr4di 288 . . . . . . 7 ([𝑤] = 𝑧 → ([𝑤] 𝑍𝑧 ∈ 𝒫 𝑍))
3736notbid 317 . . . . . 6 ([𝑤] = 𝑧 → (¬ [𝑤] 𝑍 ↔ ¬ 𝑧 ∈ 𝒫 𝑍))
38 fveq2 6842 . . . . . . 7 ([𝑤] = 𝑧 → (♯‘[𝑤] ) = (♯‘𝑧))
3938breq2d 5117 . . . . . 6 ([𝑤] = 𝑧 → (𝑃 ∥ (♯‘[𝑤] ) ↔ 𝑃 ∥ (♯‘𝑧)))
4037, 39imbi12d 344 . . . . 5 ([𝑤] = 𝑧 → ((¬ [𝑤] 𝑍𝑃 ∥ (♯‘[𝑤] )) ↔ (¬ 𝑧 ∈ 𝒫 𝑍𝑃 ∥ (♯‘𝑧))))
4120adantr 481 . . . . . . . . . 10 ((𝜑𝑤𝑌) → 𝑃 ∈ ℙ)
428adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → Er 𝑌)
43 simpr 485 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → 𝑤𝑌)
4442, 43erref 8668 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → 𝑤 𝑤)
45 vex 3449 . . . . . . . . . . . . . 14 𝑤 ∈ V
4645, 45elec 8692 . . . . . . . . . . . . 13 (𝑤 ∈ [𝑤] 𝑤 𝑤)
4744, 46sylibr 233 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → 𝑤 ∈ [𝑤] )
4847ne0d 4295 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → [𝑤] ≠ ∅)
498ecss 8694 . . . . . . . . . . . . . 14 (𝜑 → [𝑤] 𝑌)
501, 49ssfid 9211 . . . . . . . . . . . . 13 (𝜑 → [𝑤] ∈ Fin)
5150adantr 481 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → [𝑤] ∈ Fin)
52 hashnncl 14266 . . . . . . . . . . . 12 ([𝑤] ∈ Fin → ((♯‘[𝑤] ) ∈ ℕ ↔ [𝑤] ≠ ∅))
5351, 52syl 17 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → ((♯‘[𝑤] ) ∈ ℕ ↔ [𝑤] ≠ ∅))
5448, 53mpbird 256 . . . . . . . . . 10 ((𝜑𝑤𝑌) → (♯‘[𝑤] ) ∈ ℕ)
55 pceq0 16743 . . . . . . . . . 10 ((𝑃 ∈ ℙ ∧ (♯‘[𝑤] ) ∈ ℕ) → ((𝑃 pCnt (♯‘[𝑤] )) = 0 ↔ ¬ 𝑃 ∥ (♯‘[𝑤] )))
5641, 54, 55syl2anc 584 . . . . . . . . 9 ((𝜑𝑤𝑌) → ((𝑃 pCnt (♯‘[𝑤] )) = 0 ↔ ¬ 𝑃 ∥ (♯‘[𝑤] )))
57 oveq2 7365 . . . . . . . . . 10 ((𝑃 pCnt (♯‘[𝑤] )) = 0 → (𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (𝑃↑0))
58 hashcl 14256 . . . . . . . . . . . . . . . . . . . . . 22 ([𝑤] ∈ Fin → (♯‘[𝑤] ) ∈ ℕ0)
5950, 58syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘[𝑤] ) ∈ ℕ0)
6059nn0zd 12525 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (♯‘[𝑤] ) ∈ ℤ)
61 ssrab2 4037 . . . . . . . . . . . . . . . . . . . . . . 23 {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ⊆ 𝑋
62 ssfi 9117 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑋 ∈ Fin ∧ {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ⊆ 𝑋) → {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ∈ Fin)
6316, 61, 62sylancl 586 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ∈ Fin)
64 hashcl 14256 . . . . . . . . . . . . . . . . . . . . . 22 ({𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ∈ Fin → (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℕ0)
6563, 64syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℕ0)
6665nn0zd 12525 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℤ)
67 dvdsmul1 16160 . . . . . . . . . . . . . . . . . . . 20 (((♯‘[𝑤] ) ∈ ℤ ∧ (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℤ) → (♯‘[𝑤] ) ∥ ((♯‘[𝑤] ) · (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
6860, 66, 67syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (♯‘[𝑤] ) ∥ ((♯‘[𝑤] ) · (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
6968adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝑌) → (♯‘[𝑤] ) ∥ ((♯‘[𝑤] ) · (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
704adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝑌) → ∈ (𝐺 GrpAct 𝑌))
7116adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝑌) → 𝑋 ∈ Fin)
72 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} = {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}
73 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (𝐺 ~QG {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) = (𝐺 ~QG {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})
746, 72, 73, 5orbsta2 19094 . . . . . . . . . . . . . . . . . . 19 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑤𝑌) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘[𝑤] ) · (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
7570, 43, 71, 74syl21anc 836 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝑌) → (♯‘𝑋) = ((♯‘[𝑤] ) · (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
7669, 75breqtrrd 5133 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → (♯‘[𝑤] ) ∥ (♯‘𝑋))
7719simprd 496 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛))
7877adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛))
79 breq2 5109 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑋) = (𝑃𝑛) → ((♯‘[𝑤] ) ∥ (♯‘𝑋) ↔ (♯‘[𝑤] ) ∥ (𝑃𝑛)))
8079biimpcd 248 . . . . . . . . . . . . . . . . . 18 ((♯‘[𝑤] ) ∥ (♯‘𝑋) → ((♯‘𝑋) = (𝑃𝑛) → (♯‘[𝑤] ) ∥ (𝑃𝑛)))
8180reximdv 3167 . . . . . . . . . . . . . . . . 17 ((♯‘[𝑤] ) ∥ (♯‘𝑋) → (∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛) → ∃𝑛 ∈ ℕ0 (♯‘[𝑤] ) ∥ (𝑃𝑛)))
8276, 78, 81sylc 65 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝑌) → ∃𝑛 ∈ ℕ0 (♯‘[𝑤] ) ∥ (𝑃𝑛))
83 pcprmpw2 16754 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ ℙ ∧ (♯‘[𝑤] ) ∈ ℕ) → (∃𝑛 ∈ ℕ0 (♯‘[𝑤] ) ∥ (𝑃𝑛) ↔ (♯‘[𝑤] ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] )))))
8441, 54, 83syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝑌) → (∃𝑛 ∈ ℕ0 (♯‘[𝑤] ) ∥ (𝑃𝑛) ↔ (♯‘[𝑤] ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] )))))
8582, 84mpbid 231 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝑌) → (♯‘[𝑤] ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] ))))
8685eqcomd 2742 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → (𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (♯‘[𝑤] ))
8722adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → 𝑃 ∈ ℤ)
8887zcnd 12608 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝑌) → 𝑃 ∈ ℂ)
8988exp0d 14045 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝑌) → (𝑃↑0) = 1)
90 hash1 14304 . . . . . . . . . . . . . . 15 (♯‘1o) = 1
9189, 90eqtr4di 2794 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → (𝑃↑0) = (♯‘1o))
9286, 91eqeq12d 2752 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (𝑃↑0) ↔ (♯‘[𝑤] ) = (♯‘1o)))
93 df1o2 8419 . . . . . . . . . . . . . . 15 1o = {∅}
94 snfi 8988 . . . . . . . . . . . . . . 15 {∅} ∈ Fin
9593, 94eqeltri 2834 . . . . . . . . . . . . . 14 1o ∈ Fin
96 hashen 14247 . . . . . . . . . . . . . 14 (([𝑤] ∈ Fin ∧ 1o ∈ Fin) → ((♯‘[𝑤] ) = (♯‘1o) ↔ [𝑤] ≈ 1o))
9751, 95, 96sylancl 586 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → ((♯‘[𝑤] ) = (♯‘1o) ↔ [𝑤] ≈ 1o))
9892, 97bitrd 278 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (𝑃↑0) ↔ [𝑤] ≈ 1o))
99 en1b 8967 . . . . . . . . . . . 12 ([𝑤] ≈ 1o ↔ [𝑤] = { [𝑤] })
10098, 99bitrdi 286 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (𝑃↑0) ↔ [𝑤] = { [𝑤] }))
10143adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤𝑌)
1024ad2antrr 724 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ∈ (𝐺 GrpAct 𝑌))
1036gaf 19075 . . . . . . . . . . . . . . . . . . . 20 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
104102, 103syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → :(𝑋 × 𝑌)⟶𝑌)
105 simprl 769 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑋)
106104, 105, 101fovcdmd 7526 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ 𝑌)
107 eqid 2736 . . . . . . . . . . . . . . . . . . 19 ( 𝑤) = ( 𝑤)
108 oveq1 7364 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝑘 𝑤) = ( 𝑤))
109108eqeq1d 2738 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝑘 𝑤) = ( 𝑤) ↔ ( 𝑤) = ( 𝑤)))
110109rspcev 3581 . . . . . . . . . . . . . . . . . . 19 ((𝑋 ∧ ( 𝑤) = ( 𝑤)) → ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤))
111105, 107, 110sylancl 586 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤))
1125gaorb 19087 . . . . . . . . . . . . . . . . . 18 (𝑤 ( 𝑤) ↔ (𝑤𝑌 ∧ ( 𝑤) ∈ 𝑌 ∧ ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤)))
113101, 106, 111, 112syl3anbrc 1343 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ( 𝑤))
114 ovex 7390 . . . . . . . . . . . . . . . . . 18 ( 𝑤) ∈ V
115114, 45elec 8692 . . . . . . . . . . . . . . . . 17 (( 𝑤) ∈ [𝑤] 𝑤 ( 𝑤))
116113, 115sylibr 233 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ [𝑤] )
117 simprr 771 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → [𝑤] = { [𝑤] })
118116, 117eleqtrd 2840 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ { [𝑤] })
119114elsn 4601 . . . . . . . . . . . . . . 15 (( 𝑤) ∈ { [𝑤] } ↔ ( 𝑤) = [𝑤] )
120118, 119sylib 217 . . . . . . . . . . . . . 14 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) = [𝑤] )
12147adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ∈ [𝑤] )
122121, 117eleqtrd 2840 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ∈ { [𝑤] })
12345elsn 4601 . . . . . . . . . . . . . . 15 (𝑤 ∈ { [𝑤] } ↔ 𝑤 = [𝑤] )
124122, 123sylib 217 . . . . . . . . . . . . . 14 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 = [𝑤] )
125120, 124eqtr4d 2779 . . . . . . . . . . . . 13 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) = 𝑤)
126125expr 457 . . . . . . . . . . . 12 (((𝜑𝑤𝑌) ∧ 𝑋) → ([𝑤] = { [𝑤] } → ( 𝑤) = 𝑤))
127126ralrimdva 3151 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → ([𝑤] = { [𝑤] } → ∀𝑋 ( 𝑤) = 𝑤))
128100, 127sylbid 239 . . . . . . . . . 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 259 . . . . . . . 8 ((𝜑𝑤𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ) → ∀𝑋 ( 𝑤) = 𝑤))
131 oveq2 7365 . . . . . . . . . . . . 13 (𝑢 = 𝑤 → ( 𝑢) = ( 𝑤))
132 id 22 . . . . . . . . . . . . 13 (𝑢 = 𝑤𝑢 = 𝑤)
133131, 132eqeq12d 2752 . . . . . . . . . . . 12 (𝑢 = 𝑤 → (( 𝑢) = 𝑢 ↔ ( 𝑤) = 𝑤))
134133ralbidv 3174 . . . . . . . . . . 11 (𝑢 = 𝑤 → (∀𝑋 ( 𝑢) = 𝑢 ↔ ∀𝑋 ( 𝑤) = 𝑤))
135 sylow2a.z . . . . . . . . . . 11 𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}
136134, 135elrab2 3648 . . . . . . . . . 10 (𝑤𝑍 ↔ (𝑤𝑌 ∧ ∀𝑋 ( 𝑤) = 𝑤))
137136baib 536 . . . . . . . . 9 (𝑤𝑌 → (𝑤𝑍 ↔ ∀𝑋 ( 𝑤) = 𝑤))
138137adantl 482 . . . . . . . 8 ((𝜑𝑤𝑌) → (𝑤𝑍 ↔ ∀𝑋 ( 𝑤) = 𝑤))
139130, 138sylibrd 258 . . . . . . 7 ((𝜑𝑤𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ) → 𝑤𝑍))
1406, 4, 13, 16, 1, 135, 5sylow2alem1 19399 . . . . . . . . . 10 ((𝜑𝑤𝑍) → [𝑤] = {𝑤})
141 simpr 485 . . . . . . . . . . 11 ((𝜑𝑤𝑍) → 𝑤𝑍)
142141snssd 4769 . . . . . . . . . 10 ((𝜑𝑤𝑍) → {𝑤} ⊆ 𝑍)
143140, 142eqsstrd 3982 . . . . . . . . 9 ((𝜑𝑤𝑍) → [𝑤] 𝑍)
144143ex 413 . . . . . . . 8 (𝜑 → (𝑤𝑍 → [𝑤] 𝑍))
145144adantr 481 . . . . . . 7 ((𝜑𝑤𝑌) → (𝑤𝑍 → [𝑤] 𝑍))
146139, 145syld 47 . . . . . 6 ((𝜑𝑤𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ) → [𝑤] 𝑍))
147146con1d 145 . . . . 5 ((𝜑𝑤𝑌) → (¬ [𝑤] 𝑍𝑃 ∥ (♯‘[𝑤] )))
14833, 40, 147ectocld 8723 . . . 4 ((𝜑𝑧 ∈ (𝑌 / )) → (¬ 𝑧 ∈ 𝒫 𝑍𝑃 ∥ (♯‘𝑧)))
149148impr 455 . . 3 ((𝜑 ∧ (𝑧 ∈ (𝑌 / ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍)) → 𝑃 ∥ (♯‘𝑧))
15032, 149sylan2b 594 . 2 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → 𝑃 ∥ (♯‘𝑧))
15112, 22, 31, 150fsumdvds 16190 1 (𝜑𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)(♯‘𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  {crab 3407  cdif 3907  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586  {cpr 4588   cuni 4865   class class class wbr 5105  {copab 5167   × cxp 5631  wf 6492  cfv 6496  (class class class)co 7357  1oc1o 8405   Er wer 8645  [cec 8646   / cqs 8647  cen 8880  Fincfn 8883  0cc0 11051  1c1 11052   · cmul 11056  cn 12153  0cn0 12413  cz 12499  cexp 13967  chash 14230  Σcsu 15570  cdvds 16136  cprime 16547   pCnt cpc 16708  Basecbs 17083  Grpcgrp 18748   ~QG cqg 18924   GrpAct cga 19069   pGrp cpgp 19308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-omul 8417  df-er 8648  df-ec 8650  df-qs 8654  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-dvds 16137  df-gcd 16375  df-prm 16548  df-pc 16709  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-eqg 18927  df-ga 19070  df-od 19310  df-pgp 19312
This theorem is referenced by:  sylow2a  19401
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