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Theorem sylow2alem2 18672
Description: Lemma for sylow2a 18673. 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 8807 . . . . 5 (𝑌 ∈ Fin ↔ 𝒫 𝑌 ∈ Fin)
31, 2sylib 219 . . . 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 18376 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → Er 𝑌)
84, 7syl 17 . . . . 5 (𝜑 Er 𝑌)
98qsss 8347 . . . 4 (𝜑 → (𝑌 / ) ⊆ 𝒫 𝑌)
103, 9ssfid 8729 . . 3 (𝜑 → (𝑌 / ) ∈ Fin)
11 diffi 8738 . . 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 18360 . . . . . . 7 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
154, 14syl 17 . . . . . 6 (𝜑𝐺 ∈ Grp)
16 sylow2a.f . . . . . 6 (𝜑𝑋 ∈ Fin)
176pgpfi 18659 . . . . . 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 233 . . . 4 (𝜑 → (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛)))
2019simpld 495 . . 3 (𝜑𝑃 ∈ ℙ)
21 prmz 16007 . . 3 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
2220, 21syl 17 . 2 (𝜑𝑃 ∈ ℤ)
23 eldifi 4100 . . . . 5 (𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍) → 𝑧 ∈ (𝑌 / ))
241adantr 481 . . . . . 6 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑌 ∈ Fin)
259sselda 3964 . . . . . . 7 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧 ∈ 𝒫 𝑌)
2625elpwid 4549 . . . . . 6 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧𝑌)
2724, 26ssfid 8729 . . . . 5 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧 ∈ Fin)
2823, 27sylan2 592 . . . 4 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → 𝑧 ∈ Fin)
29 hashcl 13705 . . . 4 (𝑧 ∈ Fin → (♯‘𝑧) ∈ ℕ0)
3028, 29syl 17 . . 3 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → (♯‘𝑧) ∈ ℕ0)
3130nn0zd 12073 . 2 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → (♯‘𝑧) ∈ ℤ)
32 eldif 3943 . . 3 (𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍) ↔ (𝑧 ∈ (𝑌 / ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍))
33 eqid 2818 . . . . 5 (𝑌 / ) = (𝑌 / )
34 sseq1 3989 . . . . . . . 8 ([𝑤] = 𝑧 → ([𝑤] 𝑍𝑧𝑍))
35 velpw 4543 . . . . . . . 8 (𝑧 ∈ 𝒫 𝑍𝑧𝑍)
3634, 35syl6bbr 290 . . . . . . 7 ([𝑤] = 𝑧 → ([𝑤] 𝑍𝑧 ∈ 𝒫 𝑍))
3736notbid 319 . . . . . 6 ([𝑤] = 𝑧 → (¬ [𝑤] 𝑍 ↔ ¬ 𝑧 ∈ 𝒫 𝑍))
38 fveq2 6663 . . . . . . 7 ([𝑤] = 𝑧 → (♯‘[𝑤] ) = (♯‘𝑧))
3938breq2d 5069 . . . . . 6 ([𝑤] = 𝑧 → (𝑃 ∥ (♯‘[𝑤] ) ↔ 𝑃 ∥ (♯‘𝑧)))
4037, 39imbi12d 346 . . . . 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 8298 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → 𝑤 𝑤)
45 vex 3495 . . . . . . . . . . . . . 14 𝑤 ∈ V
4645, 45elec 8322 . . . . . . . . . . . . 13 (𝑤 ∈ [𝑤] 𝑤 𝑤)
4744, 46sylibr 235 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → 𝑤 ∈ [𝑤] )
4847ne0d 4298 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → [𝑤] ≠ ∅)
498ecss 8324 . . . . . . . . . . . . . 14 (𝜑 → [𝑤] 𝑌)
501, 49ssfid 8729 . . . . . . . . . . . . 13 (𝜑 → [𝑤] ∈ Fin)
5150adantr 481 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → [𝑤] ∈ Fin)
52 hashnncl 13715 . . . . . . . . . . . 12 ([𝑤] ∈ Fin → ((♯‘[𝑤] ) ∈ ℕ ↔ [𝑤] ≠ ∅))
5351, 52syl 17 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → ((♯‘[𝑤] ) ∈ ℕ ↔ [𝑤] ≠ ∅))
5448, 53mpbird 258 . . . . . . . . . 10 ((𝜑𝑤𝑌) → (♯‘[𝑤] ) ∈ ℕ)
55 pceq0 16195 . . . . . . . . . 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 7153 . . . . . . . . . 10 ((𝑃 pCnt (♯‘[𝑤] )) = 0 → (𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (𝑃↑0))
58 hashcl 13705 . . . . . . . . . . . . . . . . . . . . . 22 ([𝑤] ∈ Fin → (♯‘[𝑤] ) ∈ ℕ0)
5950, 58syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘[𝑤] ) ∈ ℕ0)
6059nn0zd 12073 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (♯‘[𝑤] ) ∈ ℤ)
61 ssrab2 4053 . . . . . . . . . . . . . . . . . . . . . . 23 {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ⊆ 𝑋
62 ssfi 8726 . . . . . . . . . . . . . . . . . . . . . . 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 13705 . . . . . . . . . . . . . . . . . . . . . 22 ({𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ∈ Fin → (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℕ0)
6563, 64syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℕ0)
6665nn0zd 12073 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℤ)
67 dvdsmul1 15619 . . . . . . . . . . . . . . . . . . . 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 2818 . . . . . . . . . . . . . . . . . . . 20 {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} = {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}
73 eqid 2818 . . . . . . . . . . . . . . . . . . . 20 (𝐺 ~QG {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) = (𝐺 ~QG {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})
746, 72, 73, 5orbsta2 18382 . . . . . . . . . . . . . . . . . . 19 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑤𝑌) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘[𝑤] ) · (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
7570, 43, 71, 74syl21anc 833 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝑌) → (♯‘𝑋) = ((♯‘[𝑤] ) · (♯‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
7669, 75breqtrrd 5085 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → (♯‘[𝑤] ) ∥ (♯‘𝑋))
7719simprd 496 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛))
7877adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛))
79 breq2 5061 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑋) = (𝑃𝑛) → ((♯‘[𝑤] ) ∥ (♯‘𝑋) ↔ (♯‘[𝑤] ) ∥ (𝑃𝑛)))
8079biimpcd 250 . . . . . . . . . . . . . . . . . 18 ((♯‘[𝑤] ) ∥ (♯‘𝑋) → ((♯‘𝑋) = (𝑃𝑛) → (♯‘[𝑤] ) ∥ (𝑃𝑛)))
8180reximdv 3270 . . . . . . . . . . . . . . . . 17 ((♯‘[𝑤] ) ∥ (♯‘𝑋) → (∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛) → ∃𝑛 ∈ ℕ0 (♯‘[𝑤] ) ∥ (𝑃𝑛)))
8276, 78, 81sylc 65 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝑌) → ∃𝑛 ∈ ℕ0 (♯‘[𝑤] ) ∥ (𝑃𝑛))
83 pcprmpw2 16206 . . . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝑌) → (♯‘[𝑤] ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] ))))
8685eqcomd 2824 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → (𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (♯‘[𝑤] ))
8722adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → 𝑃 ∈ ℤ)
8887zcnd 12076 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝑌) → 𝑃 ∈ ℂ)
8988exp0d 13492 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝑌) → (𝑃↑0) = 1)
90 hash1 13753 . . . . . . . . . . . . . . 15 (♯‘1o) = 1
9189, 90syl6eqr 2871 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → (𝑃↑0) = (♯‘1o))
9286, 91eqeq12d 2834 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (𝑃↑0) ↔ (♯‘[𝑤] ) = (♯‘1o)))
93 df1o2 8105 . . . . . . . . . . . . . . 15 1o = {∅}
94 snfi 8582 . . . . . . . . . . . . . . 15 {∅} ∈ Fin
9593, 94eqeltri 2906 . . . . . . . . . . . . . 14 1o ∈ Fin
96 hashen 13695 . . . . . . . . . . . . . 14 (([𝑤] ∈ Fin ∧ 1o ∈ Fin) → ((♯‘[𝑤] ) = (♯‘1o) ↔ [𝑤] ≈ 1o))
9751, 95, 96sylancl 586 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → ((♯‘[𝑤] ) = (♯‘1o) ↔ [𝑤] ≈ 1o))
9892, 97bitrd 280 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (𝑃↑0) ↔ [𝑤] ≈ 1o))
99 en1b 8565 . . . . . . . . . . . 12 ([𝑤] ≈ 1o ↔ [𝑤] = { [𝑤] })
10098, 99syl6bb 288 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ))) = (𝑃↑0) ↔ [𝑤] = { [𝑤] }))
10143adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤𝑌)
1024ad2antrr 722 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ∈ (𝐺 GrpAct 𝑌))
1036gaf 18363 . . . . . . . . . . . . . . . . . . . 20 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
104102, 103syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → :(𝑋 × 𝑌)⟶𝑌)
105 simprl 767 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑋)
106104, 105, 101fovrnd 7309 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ 𝑌)
107 eqid 2818 . . . . . . . . . . . . . . . . . . 19 ( 𝑤) = ( 𝑤)
108 oveq1 7152 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝑘 𝑤) = ( 𝑤))
109108eqeq1d 2820 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝑘 𝑤) = ( 𝑤) ↔ ( 𝑤) = ( 𝑤)))
110109rspcev 3620 . . . . . . . . . . . . . . . . . . 19 ((𝑋 ∧ ( 𝑤) = ( 𝑤)) → ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤))
111105, 107, 110sylancl 586 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤))
1125gaorb 18375 . . . . . . . . . . . . . . . . . 18 (𝑤 ( 𝑤) ↔ (𝑤𝑌 ∧ ( 𝑤) ∈ 𝑌 ∧ ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤)))
113101, 106, 111, 112syl3anbrc 1335 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ( 𝑤))
114 ovex 7178 . . . . . . . . . . . . . . . . . 18 ( 𝑤) ∈ V
115114, 45elec 8322 . . . . . . . . . . . . . . . . 17 (( 𝑤) ∈ [𝑤] 𝑤 ( 𝑤))
116113, 115sylibr 235 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ [𝑤] )
117 simprr 769 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → [𝑤] = { [𝑤] })
118116, 117eleqtrd 2912 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ { [𝑤] })
119114elsn 4572 . . . . . . . . . . . . . . 15 (( 𝑤) ∈ { [𝑤] } ↔ ( 𝑤) = [𝑤] )
120118, 119sylib 219 . . . . . . . . . . . . . 14 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) = [𝑤] )
12147adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ∈ [𝑤] )
122121, 117eleqtrd 2912 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ∈ { [𝑤] })
12345elsn 4572 . . . . . . . . . . . . . . 15 (𝑤 ∈ { [𝑤] } ↔ 𝑤 = [𝑤] )
124122, 123sylib 219 . . . . . . . . . . . . . 14 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 = [𝑤] )
125120, 124eqtr4d 2856 . . . . . . . . . . . . 13 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) = 𝑤)
126125expr 457 . . . . . . . . . . . 12 (((𝜑𝑤𝑌) ∧ 𝑋) → ([𝑤] = { [𝑤] } → ( 𝑤) = 𝑤))
127126ralrimdva 3186 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → ([𝑤] = { [𝑤] } → ∀𝑋 ( 𝑤) = 𝑤))
128100, 127sylbid 241 . . . . . . . . . 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 261 . . . . . . . 8 ((𝜑𝑤𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ) → ∀𝑋 ( 𝑤) = 𝑤))
131 oveq2 7153 . . . . . . . . . . . . 13 (𝑢 = 𝑤 → ( 𝑢) = ( 𝑤))
132 id 22 . . . . . . . . . . . . 13 (𝑢 = 𝑤𝑢 = 𝑤)
133131, 132eqeq12d 2834 . . . . . . . . . . . 12 (𝑢 = 𝑤 → (( 𝑢) = 𝑢 ↔ ( 𝑤) = 𝑤))
134133ralbidv 3194 . . . . . . . . . . 11 (𝑢 = 𝑤 → (∀𝑋 ( 𝑢) = 𝑢 ↔ ∀𝑋 ( 𝑤) = 𝑤))
135 sylow2a.z . . . . . . . . . . 11 𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}
136134, 135elrab2 3680 . . . . . . . . . 10 (𝑤𝑍 ↔ (𝑤𝑌 ∧ ∀𝑋 ( 𝑤) = 𝑤))
137136baib 536 . . . . . . . . 9 (𝑤𝑌 → (𝑤𝑍 ↔ ∀𝑋 ( 𝑤) = 𝑤))
138137adantl 482 . . . . . . . 8 ((𝜑𝑤𝑌) → (𝑤𝑍 ↔ ∀𝑋 ( 𝑤) = 𝑤))
139130, 138sylibrd 260 . . . . . . 7 ((𝜑𝑤𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ) → 𝑤𝑍))
1406, 4, 13, 16, 1, 135, 5sylow2alem1 18671 . . . . . . . . . 10 ((𝜑𝑤𝑍) → [𝑤] = {𝑤})
141 simpr 485 . . . . . . . . . . 11 ((𝜑𝑤𝑍) → 𝑤𝑍)
142141snssd 4734 . . . . . . . . . 10 ((𝜑𝑤𝑍) → {𝑤} ⊆ 𝑍)
143140, 142eqsstrd 4002 . . . . . . . . 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 147 . . . . 5 ((𝜑𝑤𝑌) → (¬ [𝑤] 𝑍𝑃 ∥ (♯‘[𝑤] )))
14833, 40, 147ectocld 8353 . . . 4 ((𝜑𝑧 ∈ (𝑌 / )) → (¬ 𝑧 ∈ 𝒫 𝑍𝑃 ∥ (♯‘𝑧)))
149148impr 455 . . 3 ((𝜑 ∧ (𝑧 ∈ (𝑌 / ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍)) → 𝑃 ∥ (♯‘𝑧))
15032, 149sylan2b 593 . 2 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → 𝑃 ∥ (♯‘𝑧))
15112, 22, 31, 150fsumdvds 15646 1 (𝜑𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)(♯‘𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  {crab 3139  cdif 3930  wss 3933  c0 4288  𝒫 cpw 4535  {csn 4557  {cpr 4559   cuni 4830   class class class wbr 5057  {copab 5119   × cxp 5546  wf 6344  cfv 6348  (class class class)co 7145  1oc1o 8084   Er wer 8275  [cec 8276   / cqs 8277  cen 8494  Fincfn 8497  0cc0 10525  1c1 10526   · cmul 10530  cn 11626  0cn0 11885  cz 11969  cexp 13417  chash 13678  Σcsu 15030  cdvds 15595  cprime 16003   pCnt cpc 16161  Basecbs 16471  Grpcgrp 18041   ~QG cqg 18213   GrpAct cga 18357   pGrp cpgp 18583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-disj 5023  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-omul 8096  df-er 8278  df-ec 8280  df-qs 8284  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-acn 9359  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-fz 12881  df-fzo 13022  df-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418  df-fac 13622  df-bc 13651  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-sum 15031  df-dvds 15596  df-gcd 15832  df-prm 16004  df-pc 16162  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-submnd 17945  df-grp 18044  df-minusg 18045  df-sbg 18046  df-mulg 18163  df-subg 18214  df-eqg 18216  df-ga 18358  df-od 18585  df-pgp 18587
This theorem is referenced by:  sylow2a  18673
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