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Theorem sylow2alem2 17856
 Description: Lemma for sylow2a 17857. 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 8144 . . . . 5 (𝑌 ∈ Fin ↔ 𝒫 𝑌 ∈ Fin)
31, 2sylib 207 . . . 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 17564 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → Er 𝑌)
84, 7syl 17 . . . . 5 (𝜑 Er 𝑌)
98qsss 7695 . . . 4 (𝜑 → (𝑌 / ) ⊆ 𝒫 𝑌)
10 ssfi 8065 . . . 4 ((𝒫 𝑌 ∈ Fin ∧ (𝑌 / ) ⊆ 𝒫 𝑌) → (𝑌 / ) ∈ Fin)
113, 9, 10syl2anc 691 . . 3 (𝜑 → (𝑌 / ) ∈ Fin)
12 diffi 8077 . . 3 ((𝑌 / ) ∈ Fin → ((𝑌 / ) ∖ 𝒫 𝑍) ∈ Fin)
1311, 12syl 17 . 2 (𝜑 → ((𝑌 / ) ∖ 𝒫 𝑍) ∈ Fin)
14 sylow2a.p . . . . 5 (𝜑𝑃 pGrp 𝐺)
15 gagrp 17548 . . . . . . 7 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
164, 15syl 17 . . . . . 6 (𝜑𝐺 ∈ Grp)
17 sylow2a.f . . . . . 6 (𝜑𝑋 ∈ Fin)
186pgpfi 17843 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛))))
1916, 17, 18syl2anc 691 . . . . 5 (𝜑 → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛))))
2014, 19mpbid 221 . . . 4 (𝜑 → (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛)))
2120simpld 474 . . 3 (𝜑𝑃 ∈ ℙ)
22 prmz 15227 . . 3 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
2321, 22syl 17 . 2 (𝜑𝑃 ∈ ℤ)
24 eldifi 3694 . . . . 5 (𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍) → 𝑧 ∈ (𝑌 / ))
251adantr 480 . . . . . 6 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑌 ∈ Fin)
269sselda 3568 . . . . . . 7 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧 ∈ 𝒫 𝑌)
2726elpwid 4118 . . . . . 6 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧𝑌)
28 ssfi 8065 . . . . . 6 ((𝑌 ∈ Fin ∧ 𝑧𝑌) → 𝑧 ∈ Fin)
2925, 27, 28syl2anc 691 . . . . 5 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧 ∈ Fin)
3024, 29sylan2 490 . . . 4 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → 𝑧 ∈ Fin)
31 hashcl 13009 . . . 4 (𝑧 ∈ Fin → (#‘𝑧) ∈ ℕ0)
3230, 31syl 17 . . 3 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → (#‘𝑧) ∈ ℕ0)
3332nn0zd 11356 . 2 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → (#‘𝑧) ∈ ℤ)
34 eldif 3550 . . 3 (𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍) ↔ (𝑧 ∈ (𝑌 / ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍))
35 eqid 2610 . . . . 5 (𝑌 / ) = (𝑌 / )
36 sseq1 3589 . . . . . . . 8 ([𝑤] = 𝑧 → ([𝑤] 𝑍𝑧𝑍))
37 selpw 4115 . . . . . . . 8 (𝑧 ∈ 𝒫 𝑍𝑧𝑍)
3836, 37syl6bbr 277 . . . . . . 7 ([𝑤] = 𝑧 → ([𝑤] 𝑍𝑧 ∈ 𝒫 𝑍))
3938notbid 307 . . . . . 6 ([𝑤] = 𝑧 → (¬ [𝑤] 𝑍 ↔ ¬ 𝑧 ∈ 𝒫 𝑍))
40 fveq2 6103 . . . . . . 7 ([𝑤] = 𝑧 → (#‘[𝑤] ) = (#‘𝑧))
4140breq2d 4595 . . . . . 6 ([𝑤] = 𝑧 → (𝑃 ∥ (#‘[𝑤] ) ↔ 𝑃 ∥ (#‘𝑧)))
4239, 41imbi12d 333 . . . . 5 ([𝑤] = 𝑧 → ((¬ [𝑤] 𝑍𝑃 ∥ (#‘[𝑤] )) ↔ (¬ 𝑧 ∈ 𝒫 𝑍𝑃 ∥ (#‘𝑧))))
4321adantr 480 . . . . . . . . . 10 ((𝜑𝑤𝑌) → 𝑃 ∈ ℙ)
448adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → Er 𝑌)
45 simpr 476 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → 𝑤𝑌)
4644, 45erref 7649 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → 𝑤 𝑤)
47 vex 3176 . . . . . . . . . . . . . 14 𝑤 ∈ V
4847, 47elec 7673 . . . . . . . . . . . . 13 (𝑤 ∈ [𝑤] 𝑤 𝑤)
4946, 48sylibr 223 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → 𝑤 ∈ [𝑤] )
50 ne0i 3880 . . . . . . . . . . . 12 (𝑤 ∈ [𝑤] → [𝑤] ≠ ∅)
5149, 50syl 17 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → [𝑤] ≠ ∅)
528ecss 7675 . . . . . . . . . . . . . 14 (𝜑 → [𝑤] 𝑌)
53 ssfi 8065 . . . . . . . . . . . . . 14 ((𝑌 ∈ Fin ∧ [𝑤] 𝑌) → [𝑤] ∈ Fin)
541, 52, 53syl2anc 691 . . . . . . . . . . . . 13 (𝜑 → [𝑤] ∈ Fin)
5554adantr 480 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → [𝑤] ∈ Fin)
56 hashnncl 13018 . . . . . . . . . . . 12 ([𝑤] ∈ Fin → ((#‘[𝑤] ) ∈ ℕ ↔ [𝑤] ≠ ∅))
5755, 56syl 17 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → ((#‘[𝑤] ) ∈ ℕ ↔ [𝑤] ≠ ∅))
5851, 57mpbird 246 . . . . . . . . . 10 ((𝜑𝑤𝑌) → (#‘[𝑤] ) ∈ ℕ)
59 pceq0 15413 . . . . . . . . . 10 ((𝑃 ∈ ℙ ∧ (#‘[𝑤] ) ∈ ℕ) → ((𝑃 pCnt (#‘[𝑤] )) = 0 ↔ ¬ 𝑃 ∥ (#‘[𝑤] )))
6043, 58, 59syl2anc 691 . . . . . . . . 9 ((𝜑𝑤𝑌) → ((𝑃 pCnt (#‘[𝑤] )) = 0 ↔ ¬ 𝑃 ∥ (#‘[𝑤] )))
61 oveq2 6557 . . . . . . . . . 10 ((𝑃 pCnt (#‘[𝑤] )) = 0 → (𝑃↑(𝑃 pCnt (#‘[𝑤] ))) = (𝑃↑0))
62 hashcl 13009 . . . . . . . . . . . . . . . . . . . . . 22 ([𝑤] ∈ Fin → (#‘[𝑤] ) ∈ ℕ0)
6354, 62syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (#‘[𝑤] ) ∈ ℕ0)
6463nn0zd 11356 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (#‘[𝑤] ) ∈ ℤ)
65 ssrab2 3650 . . . . . . . . . . . . . . . . . . . . . . 23 {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ⊆ 𝑋
66 ssfi 8065 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑋 ∈ Fin ∧ {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ⊆ 𝑋) → {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ∈ Fin)
6717, 65, 66sylancl 693 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ∈ Fin)
68 hashcl 13009 . . . . . . . . . . . . . . . . . . . . . 22 ({𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ∈ Fin → (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℕ0)
6967, 68syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℕ0)
7069nn0zd 11356 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℤ)
71 dvdsmul1 14841 . . . . . . . . . . . . . . . . . . . 20 (((#‘[𝑤] ) ∈ ℤ ∧ (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℤ) → (#‘[𝑤] ) ∥ ((#‘[𝑤] ) · (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
7264, 70, 71syl2anc 691 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (#‘[𝑤] ) ∥ ((#‘[𝑤] ) · (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
7372adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝑌) → (#‘[𝑤] ) ∥ ((#‘[𝑤] ) · (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
744adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝑌) → ∈ (𝐺 GrpAct 𝑌))
7517adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝑌) → 𝑋 ∈ Fin)
76 eqid 2610 . . . . . . . . . . . . . . . . . . . 20 {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} = {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}
77 eqid 2610 . . . . . . . . . . . . . . . . . . . 20 (𝐺 ~QG {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) = (𝐺 ~QG {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})
786, 76, 77, 5orbsta2 17570 . . . . . . . . . . . . . . . . . . 19 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑤𝑌) ∧ 𝑋 ∈ Fin) → (#‘𝑋) = ((#‘[𝑤] ) · (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
7974, 45, 75, 78syl21anc 1317 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝑌) → (#‘𝑋) = ((#‘[𝑤] ) · (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
8073, 79breqtrrd 4611 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → (#‘[𝑤] ) ∥ (#‘𝑋))
8120simprd 478 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛))
8281adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛))
83 breq2 4587 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑋) = (𝑃𝑛) → ((#‘[𝑤] ) ∥ (#‘𝑋) ↔ (#‘[𝑤] ) ∥ (𝑃𝑛)))
8483biimpcd 238 . . . . . . . . . . . . . . . . . 18 ((#‘[𝑤] ) ∥ (#‘𝑋) → ((#‘𝑋) = (𝑃𝑛) → (#‘[𝑤] ) ∥ (𝑃𝑛)))
8584reximdv 2999 . . . . . . . . . . . . . . . . 17 ((#‘[𝑤] ) ∥ (#‘𝑋) → (∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛) → ∃𝑛 ∈ ℕ0 (#‘[𝑤] ) ∥ (𝑃𝑛)))
8680, 82, 85sylc 63 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝑌) → ∃𝑛 ∈ ℕ0 (#‘[𝑤] ) ∥ (𝑃𝑛))
87 pcprmpw2 15424 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ ℙ ∧ (#‘[𝑤] ) ∈ ℕ) → (∃𝑛 ∈ ℕ0 (#‘[𝑤] ) ∥ (𝑃𝑛) ↔ (#‘[𝑤] ) = (𝑃↑(𝑃 pCnt (#‘[𝑤] )))))
8843, 58, 87syl2anc 691 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝑌) → (∃𝑛 ∈ ℕ0 (#‘[𝑤] ) ∥ (𝑃𝑛) ↔ (#‘[𝑤] ) = (𝑃↑(𝑃 pCnt (#‘[𝑤] )))))
8986, 88mpbid 221 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝑌) → (#‘[𝑤] ) = (𝑃↑(𝑃 pCnt (#‘[𝑤] ))))
9089eqcomd 2616 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → (𝑃↑(𝑃 pCnt (#‘[𝑤] ))) = (#‘[𝑤] ))
9123adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → 𝑃 ∈ ℤ)
9291zcnd 11359 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝑌) → 𝑃 ∈ ℂ)
9392exp0d 12864 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝑌) → (𝑃↑0) = 1)
94 hash1 13053 . . . . . . . . . . . . . . 15 (#‘1𝑜) = 1
9593, 94syl6eqr 2662 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → (𝑃↑0) = (#‘1𝑜))
9690, 95eqeq12d 2625 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (#‘[𝑤] ))) = (𝑃↑0) ↔ (#‘[𝑤] ) = (#‘1𝑜)))
97 df1o2 7459 . . . . . . . . . . . . . . 15 1𝑜 = {∅}
98 snfi 7923 . . . . . . . . . . . . . . 15 {∅} ∈ Fin
9997, 98eqeltri 2684 . . . . . . . . . . . . . 14 1𝑜 ∈ Fin
100 hashen 12997 . . . . . . . . . . . . . 14 (([𝑤] ∈ Fin ∧ 1𝑜 ∈ Fin) → ((#‘[𝑤] ) = (#‘1𝑜) ↔ [𝑤] ≈ 1𝑜))
10155, 99, 100sylancl 693 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → ((#‘[𝑤] ) = (#‘1𝑜) ↔ [𝑤] ≈ 1𝑜))
10296, 101bitrd 267 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (#‘[𝑤] ))) = (𝑃↑0) ↔ [𝑤] ≈ 1𝑜))
103 en1b 7910 . . . . . . . . . . . 12 ([𝑤] ≈ 1𝑜 ↔ [𝑤] = { [𝑤] })
104102, 103syl6bb 275 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (#‘[𝑤] ))) = (𝑃↑0) ↔ [𝑤] = { [𝑤] }))
10545adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤𝑌)
1064ad2antrr 758 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ∈ (𝐺 GrpAct 𝑌))
1076gaf 17551 . . . . . . . . . . . . . . . . . . . 20 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
108106, 107syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → :(𝑋 × 𝑌)⟶𝑌)
109 simprl 790 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑋)
110108, 109, 105fovrnd 6704 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ 𝑌)
111 eqid 2610 . . . . . . . . . . . . . . . . . . 19 ( 𝑤) = ( 𝑤)
112 oveq1 6556 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝑘 𝑤) = ( 𝑤))
113112eqeq1d 2612 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝑘 𝑤) = ( 𝑤) ↔ ( 𝑤) = ( 𝑤)))
114113rspcev 3282 . . . . . . . . . . . . . . . . . . 19 ((𝑋 ∧ ( 𝑤) = ( 𝑤)) → ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤))
115109, 111, 114sylancl 693 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤))
1165gaorb 17563 . . . . . . . . . . . . . . . . . 18 (𝑤 ( 𝑤) ↔ (𝑤𝑌 ∧ ( 𝑤) ∈ 𝑌 ∧ ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤)))
117105, 110, 115, 116syl3anbrc 1239 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ( 𝑤))
118 ovex 6577 . . . . . . . . . . . . . . . . . 18 ( 𝑤) ∈ V
119118, 47elec 7673 . . . . . . . . . . . . . . . . 17 (( 𝑤) ∈ [𝑤] 𝑤 ( 𝑤))
120117, 119sylibr 223 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ [𝑤] )
121 simprr 792 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → [𝑤] = { [𝑤] })
122120, 121eleqtrd 2690 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ { [𝑤] })
123118elsn 4140 . . . . . . . . . . . . . . 15 (( 𝑤) ∈ { [𝑤] } ↔ ( 𝑤) = [𝑤] )
124122, 123sylib 207 . . . . . . . . . . . . . 14 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) = [𝑤] )
12549adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ∈ [𝑤] )
126125, 121eleqtrd 2690 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ∈ { [𝑤] })
12747elsn 4140 . . . . . . . . . . . . . . 15 (𝑤 ∈ { [𝑤] } ↔ 𝑤 = [𝑤] )
128126, 127sylib 207 . . . . . . . . . . . . . 14 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 = [𝑤] )
129124, 128eqtr4d 2647 . . . . . . . . . . . . 13 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) = 𝑤)
130129expr 641 . . . . . . . . . . . 12 (((𝜑𝑤𝑌) ∧ 𝑋) → ([𝑤] = { [𝑤] } → ( 𝑤) = 𝑤))
131130ralrimdva 2952 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → ([𝑤] = { [𝑤] } → ∀𝑋 ( 𝑤) = 𝑤))
132104, 131sylbid 229 . . . . . . . . . 10 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (#‘[𝑤] ))) = (𝑃↑0) → ∀𝑋 ( 𝑤) = 𝑤))
13361, 132syl5 33 . . . . . . . . 9 ((𝜑𝑤𝑌) → ((𝑃 pCnt (#‘[𝑤] )) = 0 → ∀𝑋 ( 𝑤) = 𝑤))
13460, 133sylbird 249 . . . . . . . 8 ((𝜑𝑤𝑌) → (¬ 𝑃 ∥ (#‘[𝑤] ) → ∀𝑋 ( 𝑤) = 𝑤))
135 oveq2 6557 . . . . . . . . . . . . 13 (𝑢 = 𝑤 → ( 𝑢) = ( 𝑤))
136 id 22 . . . . . . . . . . . . 13 (𝑢 = 𝑤𝑢 = 𝑤)
137135, 136eqeq12d 2625 . . . . . . . . . . . 12 (𝑢 = 𝑤 → (( 𝑢) = 𝑢 ↔ ( 𝑤) = 𝑤))
138137ralbidv 2969 . . . . . . . . . . 11 (𝑢 = 𝑤 → (∀𝑋 ( 𝑢) = 𝑢 ↔ ∀𝑋 ( 𝑤) = 𝑤))
139 sylow2a.z . . . . . . . . . . 11 𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}
140138, 139elrab2 3333 . . . . . . . . . 10 (𝑤𝑍 ↔ (𝑤𝑌 ∧ ∀𝑋 ( 𝑤) = 𝑤))
141140baib 942 . . . . . . . . 9 (𝑤𝑌 → (𝑤𝑍 ↔ ∀𝑋 ( 𝑤) = 𝑤))
142141adantl 481 . . . . . . . 8 ((𝜑𝑤𝑌) → (𝑤𝑍 ↔ ∀𝑋 ( 𝑤) = 𝑤))
143134, 142sylibrd 248 . . . . . . 7 ((𝜑𝑤𝑌) → (¬ 𝑃 ∥ (#‘[𝑤] ) → 𝑤𝑍))
1446, 4, 14, 17, 1, 139, 5sylow2alem1 17855 . . . . . . . . . 10 ((𝜑𝑤𝑍) → [𝑤] = {𝑤})
145 simpr 476 . . . . . . . . . . 11 ((𝜑𝑤𝑍) → 𝑤𝑍)
146145snssd 4281 . . . . . . . . . 10 ((𝜑𝑤𝑍) → {𝑤} ⊆ 𝑍)
147144, 146eqsstrd 3602 . . . . . . . . 9 ((𝜑𝑤𝑍) → [𝑤] 𝑍)
148147ex 449 . . . . . . . 8 (𝜑 → (𝑤𝑍 → [𝑤] 𝑍))
149148adantr 480 . . . . . . 7 ((𝜑𝑤𝑌) → (𝑤𝑍 → [𝑤] 𝑍))
150143, 149syld 46 . . . . . 6 ((𝜑𝑤𝑌) → (¬ 𝑃 ∥ (#‘[𝑤] ) → [𝑤] 𝑍))
151150con1d 138 . . . . 5 ((𝜑𝑤𝑌) → (¬ [𝑤] 𝑍𝑃 ∥ (#‘[𝑤] )))
15235, 42, 151ectocld 7701 . . . 4 ((𝜑𝑧 ∈ (𝑌 / )) → (¬ 𝑧 ∈ 𝒫 𝑍𝑃 ∥ (#‘𝑧)))
153152impr 647 . . 3 ((𝜑 ∧ (𝑧 ∈ (𝑌 / ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍)) → 𝑃 ∥ (#‘𝑧))
15434, 153sylan2b 491 . 2 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → 𝑃 ∥ (#‘𝑧))
15513, 23, 33, 154fsumdvds 14868 1 (𝜑𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)(#‘𝑧))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127  ∪ cuni 4372   class class class wbr 4583  {copab 4642   × cxp 5036  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1𝑜c1o 7440   Er wer 7626  [cec 7627   / cqs 7628   ≈ cen 7838  Fincfn 7841  0cc0 9815  1c1 9816   · cmul 9820  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  ↑cexp 12722  #chash 12979  Σcsu 14264   ∥ cdvds 14821  ℙcprime 15223   pCnt cpc 15379  Basecbs 15695  Grpcgrp 17245   ~QG cqg 17413   GrpAct cga 17545   pGrp cpgp 17769 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-dvds 14822  df-gcd 15055  df-prm 15224  df-pc 15380  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-eqg 17416  df-ga 17546  df-od 17771  df-pgp 17773 This theorem is referenced by:  sylow2a  17857
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