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Theorem sylow2alem2 19528
Description: Lemma for sylow2a 19529. 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 9174 . . . . 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 19214 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → Er 𝑌)
84, 7syl 17 . . . . 5 (𝜑 Er 𝑌)
98qsss 8768 . . . 4 (𝜑 → (𝑌 / ) ⊆ 𝒫 𝑌)
103, 9ssfid 9263 . . 3 (𝜑 → (𝑌 / ) ∈ Fin)
11 diffi 9175 . . 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 19198 . . . . . . 7 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
154, 14syl 17 . . . . . 6 (𝜑𝐺 ∈ Grp)
16 sylow2a.f . . . . . 6 (𝜑𝑋 ∈ Fin)
176pgpfi 19515 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛))))
1815, 16, 17syl2anc 583 . . . . 5 (𝜑 → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛))))
1913, 18mpbid 231 . . . 4 (𝜑 → (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛)))
2019simpld 494 . . 3 (𝜑𝑃 ∈ ℙ)
21 prmz 16609 . . 3 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
2220, 21syl 17 . 2 (𝜑𝑃 ∈ ℤ)
23 eldifi 4118 . . . . 5 (𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍) → 𝑧 ∈ (𝑌 / ))
241adantr 480 . . . . . 6 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑌 ∈ Fin)
259sselda 3974 . . . . . . 7 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧 ∈ 𝒫 𝑌)
2625elpwid 4603 . . . . . 6 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧𝑌)
2724, 26ssfid 9263 . . . . 5 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧 ∈ Fin)
2823, 27sylan2 592 . . . 4 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → 𝑧 ∈ Fin)
29 hashcl 14313 . . . 4 (𝑧 ∈ Fin → (♯‘𝑧) ∈ ℕ0)
3028, 29syl 17 . . 3 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → (♯‘𝑧) ∈ ℕ0)
3130nn0zd 12581 . 2 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → (♯‘𝑧) ∈ ℤ)
32 eldif 3950 . . 3 (𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍) ↔ (𝑧 ∈ (𝑌 / ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍))
33 eqid 2724 . . . . 5 (𝑌 / ) = (𝑌 / )
34 sseq1 3999 . . . . . . . 8 ([𝑤] = 𝑧 → ([𝑤] 𝑍𝑧𝑍))
35 velpw 4599 . . . . . . . 8 (𝑧 ∈ 𝒫 𝑍𝑧𝑍)
3634, 35bitr4di 289 . . . . . . 7 ([𝑤] = 𝑧 → ([𝑤] 𝑍𝑧 ∈ 𝒫 𝑍))
3736notbid 318 . . . . . 6 ([𝑤] = 𝑧 → (¬ [𝑤] 𝑍 ↔ ¬ 𝑧 ∈ 𝒫 𝑍))
38 fveq2 6881 . . . . . . 7 ([𝑤] = 𝑧 → (♯‘[𝑤] ) = (♯‘𝑧))
3938breq2d 5150 . . . . . 6 ([𝑤] = 𝑧 → (𝑃 ∥ (♯‘[𝑤] ) ↔ 𝑃 ∥ (♯‘𝑧)))
4037, 39imbi12d 344 . . . . 5 ([𝑤] = 𝑧 → ((¬ [𝑤] 𝑍𝑃 ∥ (♯‘[𝑤] )) ↔ (¬ 𝑧 ∈ 𝒫 𝑍𝑃 ∥ (♯‘𝑧))))
4120adantr 480 . . . . . . . . . 10 ((𝜑𝑤𝑌) → 𝑃 ∈ ℙ)
428adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → Er 𝑌)
43 simpr 484 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → 𝑤𝑌)
4442, 43erref 8719 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → 𝑤 𝑤)
45 vex 3470 . . . . . . . . . . . . . 14 𝑤 ∈ V
4645, 45elec 8743 . . . . . . . . . . . . 13 (𝑤 ∈ [𝑤] 𝑤 𝑤)
4744, 46sylibr 233 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → 𝑤 ∈ [𝑤] )
4847ne0d 4327 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → [𝑤] ≠ ∅)
498ecss 8745 . . . . . . . . . . . . . 14 (𝜑 → [𝑤] 𝑌)
501, 49ssfid 9263 . . . . . . . . . . . . 13 (𝜑 → [𝑤] ∈ Fin)
5150adantr 480 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → [𝑤] ∈ Fin)
52 hashnncl 14323 . . . . . . . . . . . 12 ([𝑤] ∈ Fin → ((♯‘[𝑤] ) ∈ ℕ ↔ [𝑤] ≠ ∅))
5351, 52syl 17 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → ((♯‘[𝑤] ) ∈ ℕ ↔ [𝑤] ≠ ∅))
5448, 53mpbird 257 . . . . . . . . . 10 ((𝜑𝑤𝑌) → (♯‘[𝑤] ) ∈ ℕ)
55 pceq0 16803 . . . . . . . . . 10 ((𝑃 ∈ ℙ ∧ (♯‘[𝑤] ) ∈ ℕ) → ((𝑃 pCnt (♯‘[𝑤] )) = 0 ↔ ¬ 𝑃 ∥ (♯‘[𝑤] )))
5641, 54, 55syl2anc 583 . . . . . . . . 9 ((𝜑𝑤𝑌) → ((𝑃 pCnt (♯‘[𝑤] )) = 0 ↔ ¬ 𝑃 ∥ (♯‘[𝑤] )))
57 oveq2 7409 . . . . . . . . . 10 ((𝑃 pCnt (♯‘[𝑤] )) = 0 → (𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (𝑃↑0))
58 hashcl 14313 . . . . . . . . . . . . . . . . . . . . . 22 ([𝑤] ∈ Fin → (♯‘[𝑤] ) ∈ ℕ0)
5950, 58syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘[𝑤] ) ∈ ℕ0)
6059nn0zd 12581 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (♯‘[𝑤] ) ∈ ℤ)
61 ssrab2 4069 . . . . . . . . . . . . . . . . . . . . . . 23 {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ⊆ 𝑋
62 ssfi 9169 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑋 ∈ Fin ∧ {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ⊆ 𝑋) → {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ∈ Fin)
6316, 61, 62sylancl 585 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ∈ Fin)
64 hashcl 14313 . . . . . . . . . . . . . . . . . . . . . 22 ({𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ∈ Fin → (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℕ0)
6563, 64syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℕ0)
6665nn0zd 12581 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℤ)
67 dvdsmul1 16218 . . . . . . . . . . . . . . . . . . . 20 (((♯‘[𝑤] ) ∈ ℤ ∧ (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℤ) → (♯‘[𝑤] ) ∥ ((♯‘[𝑤] ) · (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
6860, 66, 67syl2anc 583 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (♯‘[𝑤] ) ∥ ((♯‘[𝑤] ) · (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
6968adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝑌) → (♯‘[𝑤] ) ∥ ((♯‘[𝑤] ) · (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
704adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝑌) → ∈ (𝐺 GrpAct 𝑌))
7116adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝑌) → 𝑋 ∈ Fin)
72 eqid 2724 . . . . . . . . . . . . . . . . . . . 20 {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} = {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}
73 eqid 2724 . . . . . . . . . . . . . . . . . . . 20 (𝐺 ~QG {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) = (𝐺 ~QG {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})
746, 72, 73, 5orbsta2 19220 . . . . . . . . . . . . . . . . . . 19 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑤𝑌) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘[𝑤] ) · (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
7570, 43, 71, 74syl21anc 835 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝑌) → (♯‘𝑋) = ((♯‘[𝑤] ) · (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
7669, 75breqtrrd 5166 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → (♯‘[𝑤] ) ∥ (♯‘𝑋))
7719simprd 495 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛))
7877adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛))
79 breq2 5142 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑋) = (𝑃𝑛) → ((♯‘[𝑤] ) ∥ (♯‘𝑋) ↔ (♯‘[𝑤] ) ∥ (𝑃𝑛)))
8079biimpcd 248 . . . . . . . . . . . . . . . . . 18 ((♯‘[𝑤] ) ∥ (♯‘𝑋) → ((♯‘𝑋) = (𝑃𝑛) → (♯‘[𝑤] ) ∥ (𝑃𝑛)))
8180reximdv 3162 . . . . . . . . . . . . . . . . 17 ((♯‘[𝑤] ) ∥ (♯‘𝑋) → (∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛) → ∃𝑛 ∈ ℕ0 (♯‘[𝑤] ) ∥ (𝑃𝑛)))
8276, 78, 81sylc 65 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝑌) → ∃𝑛 ∈ ℕ0 (♯‘[𝑤] ) ∥ (𝑃𝑛))
83 pcprmpw2 16814 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ ℙ ∧ (♯‘[𝑤] ) ∈ ℕ) → (∃𝑛 ∈ ℕ0 (♯‘[𝑤] ) ∥ (𝑃𝑛) ↔ (♯‘[𝑤] ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] )))))
8441, 54, 83syl2anc 583 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝑌) → (∃𝑛 ∈ ℕ0 (♯‘[𝑤] ) ∥ (𝑃𝑛) ↔ (♯‘[𝑤] ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] )))))
8582, 84mpbid 231 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝑌) → (♯‘[𝑤] ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] ))))
8685eqcomd 2730 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → (𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (♯‘[𝑤] ))
8722adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → 𝑃 ∈ ℤ)
8887zcnd 12664 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝑌) → 𝑃 ∈ ℂ)
8988exp0d 14102 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝑌) → (𝑃↑0) = 1)
90 hash1 14361 . . . . . . . . . . . . . . 15 (♯‘1o) = 1
9189, 90eqtr4di 2782 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → (𝑃↑0) = (♯‘1o))
9286, 91eqeq12d 2740 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (𝑃↑0) ↔ (♯‘[𝑤] ) = (♯‘1o)))
93 df1o2 8468 . . . . . . . . . . . . . . 15 1o = {∅}
94 snfi 9040 . . . . . . . . . . . . . . 15 {∅} ∈ Fin
9593, 94eqeltri 2821 . . . . . . . . . . . . . 14 1o ∈ Fin
96 hashen 14304 . . . . . . . . . . . . . 14 (([𝑤] ∈ Fin ∧ 1o ∈ Fin) → ((♯‘[𝑤] ) = (♯‘1o) ↔ [𝑤] ≈ 1o))
9751, 95, 96sylancl 585 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → ((♯‘[𝑤] ) = (♯‘1o) ↔ [𝑤] ≈ 1o))
9892, 97bitrd 279 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (𝑃↑0) ↔ [𝑤] ≈ 1o))
99 en1b 9019 . . . . . . . . . . . 12 ([𝑤] ≈ 1o ↔ [𝑤] = { [𝑤] })
10098, 99bitrdi 287 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (𝑃↑0) ↔ [𝑤] = { [𝑤] }))
10143adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤𝑌)
1024ad2antrr 723 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ∈ (𝐺 GrpAct 𝑌))
1036gaf 19201 . . . . . . . . . . . . . . . . . . . 20 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
104102, 103syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → :(𝑋 × 𝑌)⟶𝑌)
105 simprl 768 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑋)
106104, 105, 101fovcdmd 7572 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ 𝑌)
107 eqid 2724 . . . . . . . . . . . . . . . . . . 19 ( 𝑤) = ( 𝑤)
108 oveq1 7408 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝑘 𝑤) = ( 𝑤))
109108eqeq1d 2726 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝑘 𝑤) = ( 𝑤) ↔ ( 𝑤) = ( 𝑤)))
110109rspcev 3604 . . . . . . . . . . . . . . . . . . 19 ((𝑋 ∧ ( 𝑤) = ( 𝑤)) → ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤))
111105, 107, 110sylancl 585 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤))
1125gaorb 19213 . . . . . . . . . . . . . . . . . 18 (𝑤 ( 𝑤) ↔ (𝑤𝑌 ∧ ( 𝑤) ∈ 𝑌 ∧ ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤)))
113101, 106, 111, 112syl3anbrc 1340 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ( 𝑤))
114 ovex 7434 . . . . . . . . . . . . . . . . . 18 ( 𝑤) ∈ V
115114, 45elec 8743 . . . . . . . . . . . . . . . . 17 (( 𝑤) ∈ [𝑤] 𝑤 ( 𝑤))
116113, 115sylibr 233 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ [𝑤] )
117 simprr 770 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → [𝑤] = { [𝑤] })
118116, 117eleqtrd 2827 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ { [𝑤] })
119114elsn 4635 . . . . . . . . . . . . . . 15 (( 𝑤) ∈ { [𝑤] } ↔ ( 𝑤) = [𝑤] )
120118, 119sylib 217 . . . . . . . . . . . . . 14 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) = [𝑤] )
12147adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ∈ [𝑤] )
122121, 117eleqtrd 2827 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ∈ { [𝑤] })
12345elsn 4635 . . . . . . . . . . . . . . 15 (𝑤 ∈ { [𝑤] } ↔ 𝑤 = [𝑤] )
124122, 123sylib 217 . . . . . . . . . . . . . 14 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 = [𝑤] )
125120, 124eqtr4d 2767 . . . . . . . . . . . . 13 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) = 𝑤)
126125expr 456 . . . . . . . . . . . 12 (((𝜑𝑤𝑌) ∧ 𝑋) → ([𝑤] = { [𝑤] } → ( 𝑤) = 𝑤))
127126ralrimdva 3146 . . . . . . . . . . 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 260 . . . . . . . 8 ((𝜑𝑤𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ) → ∀𝑋 ( 𝑤) = 𝑤))
131 oveq2 7409 . . . . . . . . . . . . 13 (𝑢 = 𝑤 → ( 𝑢) = ( 𝑤))
132 id 22 . . . . . . . . . . . . 13 (𝑢 = 𝑤𝑢 = 𝑤)
133131, 132eqeq12d 2740 . . . . . . . . . . . 12 (𝑢 = 𝑤 → (( 𝑢) = 𝑢 ↔ ( 𝑤) = 𝑤))
134133ralbidv 3169 . . . . . . . . . . 11 (𝑢 = 𝑤 → (∀𝑋 ( 𝑢) = 𝑢 ↔ ∀𝑋 ( 𝑤) = 𝑤))
135 sylow2a.z . . . . . . . . . . 11 𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}
136134, 135elrab2 3678 . . . . . . . . . 10 (𝑤𝑍 ↔ (𝑤𝑌 ∧ ∀𝑋 ( 𝑤) = 𝑤))
137136baib 535 . . . . . . . . 9 (𝑤𝑌 → (𝑤𝑍 ↔ ∀𝑋 ( 𝑤) = 𝑤))
138137adantl 481 . . . . . . . 8 ((𝜑𝑤𝑌) → (𝑤𝑍 ↔ ∀𝑋 ( 𝑤) = 𝑤))
139130, 138sylibrd 259 . . . . . . 7 ((𝜑𝑤𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ) → 𝑤𝑍))
1406, 4, 13, 16, 1, 135, 5sylow2alem1 19527 . . . . . . . . . 10 ((𝜑𝑤𝑍) → [𝑤] = {𝑤})
141 simpr 484 . . . . . . . . . . 11 ((𝜑𝑤𝑍) → 𝑤𝑍)
142141snssd 4804 . . . . . . . . . 10 ((𝜑𝑤𝑍) → {𝑤} ⊆ 𝑍)
143140, 142eqsstrd 4012 . . . . . . . . 9 ((𝜑𝑤𝑍) → [𝑤] 𝑍)
144143ex 412 . . . . . . . 8 (𝜑 → (𝑤𝑍 → [𝑤] 𝑍))
145144adantr 480 . . . . . . 7 ((𝜑𝑤𝑌) → (𝑤𝑍 → [𝑤] 𝑍))
146139, 145syld 47 . . . . . 6 ((𝜑𝑤𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ) → [𝑤] 𝑍))
147146con1d 145 . . . . 5 ((𝜑𝑤𝑌) → (¬ [𝑤] 𝑍𝑃 ∥ (♯‘[𝑤] )))
14833, 40, 147ectocld 8774 . . . 4 ((𝜑𝑧 ∈ (𝑌 / )) → (¬ 𝑧 ∈ 𝒫 𝑍𝑃 ∥ (♯‘𝑧)))
149148impr 454 . . 3 ((𝜑 ∧ (𝑧 ∈ (𝑌 / ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍)) → 𝑃 ∥ (♯‘𝑧))
15032, 149sylan2b 593 . 2 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → 𝑃 ∥ (♯‘𝑧))
15112, 22, 31, 150fsumdvds 16248 1 (𝜑𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)(♯‘𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wne 2932  wral 3053  wrex 3062  {crab 3424  cdif 3937  wss 3940  c0 4314  𝒫 cpw 4594  {csn 4620  {cpr 4622   cuni 4899   class class class wbr 5138  {copab 5200   × cxp 5664  wf 6529  cfv 6533  (class class class)co 7401  1oc1o 8454   Er wer 8696  [cec 8697   / cqs 8698  cen 8932  Fincfn 8935  0cc0 11106  1c1 11107   · cmul 11111  cn 12209  0cn0 12469  cz 12555  cexp 14024  chash 14287  Σcsu 15629  cdvds 16194  cprime 16605   pCnt cpc 16768  Basecbs 17143  Grpcgrp 18853   ~QG cqg 19039   GrpAct cga 19195   pGrp cpgp 19436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-disj 5104  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-oadd 8465  df-omul 8466  df-er 8699  df-ec 8701  df-qs 8705  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-inf 9434  df-oi 9501  df-dju 9892  df-card 9930  df-acn 9933  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-xnn0 12542  df-z 12556  df-uz 12820  df-q 12930  df-rp 12972  df-fz 13482  df-fzo 13625  df-fl 13754  df-mod 13832  df-seq 13964  df-exp 14025  df-fac 14231  df-bc 14260  df-hash 14288  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-clim 15429  df-sum 15630  df-dvds 16195  df-gcd 16433  df-prm 16606  df-pc 16769  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-0g 17386  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-submnd 18704  df-grp 18856  df-minusg 18857  df-sbg 18858  df-mulg 18986  df-subg 19040  df-eqg 19042  df-ga 19196  df-od 19438  df-pgp 19440
This theorem is referenced by:  sylow2a  19529
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