MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sylow2alem2 Structured version   Visualization version   GIF version

Theorem sylow2alem2 17954
Description: Lemma for sylow2a 17955. 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 8205 . . . . 5 (𝑌 ∈ Fin ↔ 𝒫 𝑌 ∈ Fin)
31, 2sylib 208 . . . 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 17662 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → Er 𝑌)
84, 7syl 17 . . . . 5 (𝜑 Er 𝑌)
98qsss 7753 . . . 4 (𝜑 → (𝑌 / ) ⊆ 𝒫 𝑌)
10 ssfi 8124 . . . 4 ((𝒫 𝑌 ∈ Fin ∧ (𝑌 / ) ⊆ 𝒫 𝑌) → (𝑌 / ) ∈ Fin)
113, 9, 10syl2anc 692 . . 3 (𝜑 → (𝑌 / ) ∈ Fin)
12 diffi 8136 . . 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 17646 . . . . . . 7 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
164, 15syl 17 . . . . . 6 (𝜑𝐺 ∈ Grp)
17 sylow2a.f . . . . . 6 (𝜑𝑋 ∈ Fin)
186pgpfi 17941 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛))))
1916, 17, 18syl2anc 692 . . . . 5 (𝜑 → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛))))
2014, 19mpbid 222 . . . 4 (𝜑 → (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛)))
2120simpld 475 . . 3 (𝜑𝑃 ∈ ℙ)
22 prmz 15313 . . 3 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
2321, 22syl 17 . 2 (𝜑𝑃 ∈ ℤ)
24 eldifi 3710 . . . . 5 (𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍) → 𝑧 ∈ (𝑌 / ))
251adantr 481 . . . . . 6 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑌 ∈ Fin)
269sselda 3583 . . . . . . 7 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧 ∈ 𝒫 𝑌)
2726elpwid 4141 . . . . . 6 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧𝑌)
28 ssfi 8124 . . . . . 6 ((𝑌 ∈ Fin ∧ 𝑧𝑌) → 𝑧 ∈ Fin)
2925, 27, 28syl2anc 692 . . . . 5 ((𝜑𝑧 ∈ (𝑌 / )) → 𝑧 ∈ Fin)
3024, 29sylan2 491 . . . 4 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → 𝑧 ∈ Fin)
31 hashcl 13087 . . . 4 (𝑧 ∈ Fin → (#‘𝑧) ∈ ℕ0)
3230, 31syl 17 . . 3 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → (#‘𝑧) ∈ ℕ0)
3332nn0zd 11424 . 2 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → (#‘𝑧) ∈ ℤ)
34 eldif 3565 . . 3 (𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍) ↔ (𝑧 ∈ (𝑌 / ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍))
35 eqid 2621 . . . . 5 (𝑌 / ) = (𝑌 / )
36 sseq1 3605 . . . . . . . 8 ([𝑤] = 𝑧 → ([𝑤] 𝑍𝑧𝑍))
37 selpw 4137 . . . . . . . 8 (𝑧 ∈ 𝒫 𝑍𝑧𝑍)
3836, 37syl6bbr 278 . . . . . . 7 ([𝑤] = 𝑧 → ([𝑤] 𝑍𝑧 ∈ 𝒫 𝑍))
3938notbid 308 . . . . . 6 ([𝑤] = 𝑧 → (¬ [𝑤] 𝑍 ↔ ¬ 𝑧 ∈ 𝒫 𝑍))
40 fveq2 6148 . . . . . . 7 ([𝑤] = 𝑧 → (#‘[𝑤] ) = (#‘𝑧))
4140breq2d 4625 . . . . . 6 ([𝑤] = 𝑧 → (𝑃 ∥ (#‘[𝑤] ) ↔ 𝑃 ∥ (#‘𝑧)))
4239, 41imbi12d 334 . . . . 5 ([𝑤] = 𝑧 → ((¬ [𝑤] 𝑍𝑃 ∥ (#‘[𝑤] )) ↔ (¬ 𝑧 ∈ 𝒫 𝑍𝑃 ∥ (#‘𝑧))))
4321adantr 481 . . . . . . . . . 10 ((𝜑𝑤𝑌) → 𝑃 ∈ ℙ)
448adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → Er 𝑌)
45 simpr 477 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → 𝑤𝑌)
4644, 45erref 7707 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → 𝑤 𝑤)
47 vex 3189 . . . . . . . . . . . . . 14 𝑤 ∈ V
4847, 47elec 7731 . . . . . . . . . . . . 13 (𝑤 ∈ [𝑤] 𝑤 𝑤)
4946, 48sylibr 224 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → 𝑤 ∈ [𝑤] )
50 ne0i 3897 . . . . . . . . . . . 12 (𝑤 ∈ [𝑤] → [𝑤] ≠ ∅)
5149, 50syl 17 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → [𝑤] ≠ ∅)
528ecss 7733 . . . . . . . . . . . . . 14 (𝜑 → [𝑤] 𝑌)
53 ssfi 8124 . . . . . . . . . . . . . 14 ((𝑌 ∈ Fin ∧ [𝑤] 𝑌) → [𝑤] ∈ Fin)
541, 52, 53syl2anc 692 . . . . . . . . . . . . 13 (𝜑 → [𝑤] ∈ Fin)
5554adantr 481 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → [𝑤] ∈ Fin)
56 hashnncl 13097 . . . . . . . . . . . 12 ([𝑤] ∈ Fin → ((#‘[𝑤] ) ∈ ℕ ↔ [𝑤] ≠ ∅))
5755, 56syl 17 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → ((#‘[𝑤] ) ∈ ℕ ↔ [𝑤] ≠ ∅))
5851, 57mpbird 247 . . . . . . . . . 10 ((𝜑𝑤𝑌) → (#‘[𝑤] ) ∈ ℕ)
59 pceq0 15499 . . . . . . . . . 10 ((𝑃 ∈ ℙ ∧ (#‘[𝑤] ) ∈ ℕ) → ((𝑃 pCnt (#‘[𝑤] )) = 0 ↔ ¬ 𝑃 ∥ (#‘[𝑤] )))
6043, 58, 59syl2anc 692 . . . . . . . . 9 ((𝜑𝑤𝑌) → ((𝑃 pCnt (#‘[𝑤] )) = 0 ↔ ¬ 𝑃 ∥ (#‘[𝑤] )))
61 oveq2 6612 . . . . . . . . . 10 ((𝑃 pCnt (#‘[𝑤] )) = 0 → (𝑃↑(𝑃 pCnt (#‘[𝑤] ))) = (𝑃↑0))
62 hashcl 13087 . . . . . . . . . . . . . . . . . . . . . 22 ([𝑤] ∈ Fin → (#‘[𝑤] ) ∈ ℕ0)
6354, 62syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (#‘[𝑤] ) ∈ ℕ0)
6463nn0zd 11424 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (#‘[𝑤] ) ∈ ℤ)
65 ssrab2 3666 . . . . . . . . . . . . . . . . . . . . . . 23 {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ⊆ 𝑋
66 ssfi 8124 . . . . . . . . . . . . . . . . . . . . . . 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 13087 . . . . . . . . . . . . . . . . . . . . . 22 ({𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} ∈ Fin → (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℕ0)
6967, 68syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℕ0)
7069nn0zd 11424 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℤ)
71 dvdsmul1 14927 . . . . . . . . . . . . . . . . . . . 20 (((#‘[𝑤] ) ∈ ℤ ∧ (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) ∈ ℤ) → (#‘[𝑤] ) ∥ ((#‘[𝑤] ) · (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
7264, 70, 71syl2anc 692 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (#‘[𝑤] ) ∥ ((#‘[𝑤] ) · (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
7372adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝑌) → (#‘[𝑤] ) ∥ ((#‘[𝑤] ) · (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
744adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝑌) → ∈ (𝐺 GrpAct 𝑌))
7517adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝑌) → 𝑋 ∈ Fin)
76 eqid 2621 . . . . . . . . . . . . . . . . . . . 20 {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤} = {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}
77 eqid 2621 . . . . . . . . . . . . . . . . . . . 20 (𝐺 ~QG {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤}) = (𝐺 ~QG {𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})
786, 76, 77, 5orbsta2 17668 . . . . . . . . . . . . . . . . . . 19 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑤𝑌) ∧ 𝑋 ∈ Fin) → (#‘𝑋) = ((#‘[𝑤] ) · (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
7974, 45, 75, 78syl21anc 1322 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝑌) → (#‘𝑋) = ((#‘[𝑤] ) · (#‘{𝑣𝑋 ∣ (𝑣 𝑤) = 𝑤})))
8073, 79breqtrrd 4641 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → (#‘[𝑤] ) ∥ (#‘𝑋))
8120simprd 479 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛))
8281adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛))
83 breq2 4617 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑋) = (𝑃𝑛) → ((#‘[𝑤] ) ∥ (#‘𝑋) ↔ (#‘[𝑤] ) ∥ (𝑃𝑛)))
8483biimpcd 239 . . . . . . . . . . . . . . . . . 18 ((#‘[𝑤] ) ∥ (#‘𝑋) → ((#‘𝑋) = (𝑃𝑛) → (#‘[𝑤] ) ∥ (𝑃𝑛)))
8584reximdv 3010 . . . . . . . . . . . . . . . . 17 ((#‘[𝑤] ) ∥ (#‘𝑋) → (∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛) → ∃𝑛 ∈ ℕ0 (#‘[𝑤] ) ∥ (𝑃𝑛)))
8680, 82, 85sylc 65 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝑌) → ∃𝑛 ∈ ℕ0 (#‘[𝑤] ) ∥ (𝑃𝑛))
87 pcprmpw2 15510 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ ℙ ∧ (#‘[𝑤] ) ∈ ℕ) → (∃𝑛 ∈ ℕ0 (#‘[𝑤] ) ∥ (𝑃𝑛) ↔ (#‘[𝑤] ) = (𝑃↑(𝑃 pCnt (#‘[𝑤] )))))
8843, 58, 87syl2anc 692 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝑌) → (∃𝑛 ∈ ℕ0 (#‘[𝑤] ) ∥ (𝑃𝑛) ↔ (#‘[𝑤] ) = (𝑃↑(𝑃 pCnt (#‘[𝑤] )))))
8986, 88mpbid 222 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝑌) → (#‘[𝑤] ) = (𝑃↑(𝑃 pCnt (#‘[𝑤] ))))
9089eqcomd 2627 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → (𝑃↑(𝑃 pCnt (#‘[𝑤] ))) = (#‘[𝑤] ))
9123adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤𝑌) → 𝑃 ∈ ℤ)
9291zcnd 11427 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝑌) → 𝑃 ∈ ℂ)
9392exp0d 12942 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝑌) → (𝑃↑0) = 1)
94 hash1 13132 . . . . . . . . . . . . . . 15 (#‘1𝑜) = 1
9593, 94syl6eqr 2673 . . . . . . . . . . . . . 14 ((𝜑𝑤𝑌) → (𝑃↑0) = (#‘1𝑜))
9690, 95eqeq12d 2636 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (#‘[𝑤] ))) = (𝑃↑0) ↔ (#‘[𝑤] ) = (#‘1𝑜)))
97 df1o2 7517 . . . . . . . . . . . . . . 15 1𝑜 = {∅}
98 snfi 7982 . . . . . . . . . . . . . . 15 {∅} ∈ Fin
9997, 98eqeltri 2694 . . . . . . . . . . . . . 14 1𝑜 ∈ Fin
100 hashen 13075 . . . . . . . . . . . . . 14 (([𝑤] ∈ Fin ∧ 1𝑜 ∈ Fin) → ((#‘[𝑤] ) = (#‘1𝑜) ↔ [𝑤] ≈ 1𝑜))
10155, 99, 100sylancl 693 . . . . . . . . . . . . 13 ((𝜑𝑤𝑌) → ((#‘[𝑤] ) = (#‘1𝑜) ↔ [𝑤] ≈ 1𝑜))
10296, 101bitrd 268 . . . . . . . . . . . 12 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (#‘[𝑤] ))) = (𝑃↑0) ↔ [𝑤] ≈ 1𝑜))
103 en1b 7968 . . . . . . . . . . . 12 ([𝑤] ≈ 1𝑜 ↔ [𝑤] = { [𝑤] })
104102, 103syl6bb 276 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (#‘[𝑤] ))) = (𝑃↑0) ↔ [𝑤] = { [𝑤] }))
10545adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤𝑌)
1064ad2antrr 761 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ∈ (𝐺 GrpAct 𝑌))
1076gaf 17649 . . . . . . . . . . . . . . . . . . . 20 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
108106, 107syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → :(𝑋 × 𝑌)⟶𝑌)
109 simprl 793 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑋)
110108, 109, 105fovrnd 6759 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ 𝑌)
111 eqid 2621 . . . . . . . . . . . . . . . . . . 19 ( 𝑤) = ( 𝑤)
112 oveq1 6611 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝑘 𝑤) = ( 𝑤))
113112eqeq1d 2623 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝑘 𝑤) = ( 𝑤) ↔ ( 𝑤) = ( 𝑤)))
114113rspcev 3295 . . . . . . . . . . . . . . . . . . 19 ((𝑋 ∧ ( 𝑤) = ( 𝑤)) → ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤))
115109, 111, 114sylancl 693 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤))
1165gaorb 17661 . . . . . . . . . . . . . . . . . 18 (𝑤 ( 𝑤) ↔ (𝑤𝑌 ∧ ( 𝑤) ∈ 𝑌 ∧ ∃𝑘𝑋 (𝑘 𝑤) = ( 𝑤)))
117105, 110, 115, 116syl3anbrc 1244 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ( 𝑤))
118 ovex 6632 . . . . . . . . . . . . . . . . . 18 ( 𝑤) ∈ V
119118, 47elec 7731 . . . . . . . . . . . . . . . . 17 (( 𝑤) ∈ [𝑤] 𝑤 ( 𝑤))
120117, 119sylibr 224 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ [𝑤] )
121 simprr 795 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → [𝑤] = { [𝑤] })
122120, 121eleqtrd 2700 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) ∈ { [𝑤] })
123118elsn 4163 . . . . . . . . . . . . . . 15 (( 𝑤) ∈ { [𝑤] } ↔ ( 𝑤) = [𝑤] )
124122, 123sylib 208 . . . . . . . . . . . . . 14 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) = [𝑤] )
12549adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ∈ [𝑤] )
126125, 121eleqtrd 2700 . . . . . . . . . . . . . . 15 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 ∈ { [𝑤] })
12747elsn 4163 . . . . . . . . . . . . . . 15 (𝑤 ∈ { [𝑤] } ↔ 𝑤 = [𝑤] )
128126, 127sylib 208 . . . . . . . . . . . . . 14 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → 𝑤 = [𝑤] )
129124, 128eqtr4d 2658 . . . . . . . . . . . . 13 (((𝜑𝑤𝑌) ∧ (𝑋 ∧ [𝑤] = { [𝑤] })) → ( 𝑤) = 𝑤)
130129expr 642 . . . . . . . . . . . 12 (((𝜑𝑤𝑌) ∧ 𝑋) → ([𝑤] = { [𝑤] } → ( 𝑤) = 𝑤))
131130ralrimdva 2963 . . . . . . . . . . 11 ((𝜑𝑤𝑌) → ([𝑤] = { [𝑤] } → ∀𝑋 ( 𝑤) = 𝑤))
132104, 131sylbid 230 . . . . . . . . . 10 ((𝜑𝑤𝑌) → ((𝑃↑(𝑃 pCnt (#‘[𝑤] ))) = (𝑃↑0) → ∀𝑋 ( 𝑤) = 𝑤))
13361, 132syl5 34 . . . . . . . . 9 ((𝜑𝑤𝑌) → ((𝑃 pCnt (#‘[𝑤] )) = 0 → ∀𝑋 ( 𝑤) = 𝑤))
13460, 133sylbird 250 . . . . . . . 8 ((𝜑𝑤𝑌) → (¬ 𝑃 ∥ (#‘[𝑤] ) → ∀𝑋 ( 𝑤) = 𝑤))
135 oveq2 6612 . . . . . . . . . . . . 13 (𝑢 = 𝑤 → ( 𝑢) = ( 𝑤))
136 id 22 . . . . . . . . . . . . 13 (𝑢 = 𝑤𝑢 = 𝑤)
137135, 136eqeq12d 2636 . . . . . . . . . . . 12 (𝑢 = 𝑤 → (( 𝑢) = 𝑢 ↔ ( 𝑤) = 𝑤))
138137ralbidv 2980 . . . . . . . . . . 11 (𝑢 = 𝑤 → (∀𝑋 ( 𝑢) = 𝑢 ↔ ∀𝑋 ( 𝑤) = 𝑤))
139 sylow2a.z . . . . . . . . . . 11 𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}
140138, 139elrab2 3348 . . . . . . . . . 10 (𝑤𝑍 ↔ (𝑤𝑌 ∧ ∀𝑋 ( 𝑤) = 𝑤))
141140baib 943 . . . . . . . . 9 (𝑤𝑌 → (𝑤𝑍 ↔ ∀𝑋 ( 𝑤) = 𝑤))
142141adantl 482 . . . . . . . 8 ((𝜑𝑤𝑌) → (𝑤𝑍 ↔ ∀𝑋 ( 𝑤) = 𝑤))
143134, 142sylibrd 249 . . . . . . 7 ((𝜑𝑤𝑌) → (¬ 𝑃 ∥ (#‘[𝑤] ) → 𝑤𝑍))
1446, 4, 14, 17, 1, 139, 5sylow2alem1 17953 . . . . . . . . . 10 ((𝜑𝑤𝑍) → [𝑤] = {𝑤})
145 simpr 477 . . . . . . . . . . 11 ((𝜑𝑤𝑍) → 𝑤𝑍)
146145snssd 4309 . . . . . . . . . 10 ((𝜑𝑤𝑍) → {𝑤} ⊆ 𝑍)
147144, 146eqsstrd 3618 . . . . . . . . 9 ((𝜑𝑤𝑍) → [𝑤] 𝑍)
148147ex 450 . . . . . . . 8 (𝜑 → (𝑤𝑍 → [𝑤] 𝑍))
149148adantr 481 . . . . . . 7 ((𝜑𝑤𝑌) → (𝑤𝑍 → [𝑤] 𝑍))
150143, 149syld 47 . . . . . 6 ((𝜑𝑤𝑌) → (¬ 𝑃 ∥ (#‘[𝑤] ) → [𝑤] 𝑍))
151150con1d 139 . . . . 5 ((𝜑𝑤𝑌) → (¬ [𝑤] 𝑍𝑃 ∥ (#‘[𝑤] )))
15235, 42, 151ectocld 7759 . . . 4 ((𝜑𝑧 ∈ (𝑌 / )) → (¬ 𝑧 ∈ 𝒫 𝑍𝑃 ∥ (#‘𝑧)))
153152impr 648 . . 3 ((𝜑 ∧ (𝑧 ∈ (𝑌 / ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍)) → 𝑃 ∥ (#‘𝑧))
15434, 153sylan2b 492 . 2 ((𝜑𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)) → 𝑃 ∥ (#‘𝑧))
15513, 23, 33, 154fsumdvds 14954 1 (𝜑𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)(#‘𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  cdif 3552  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148  {cpr 4150   cuni 4402   class class class wbr 4613  {copab 4672   × cxp 5072  wf 5843  cfv 5847  (class class class)co 6604  1𝑜c1o 7498   Er wer 7684  [cec 7685   / cqs 7686  cen 7896  Fincfn 7899  0cc0 9880  1c1 9881   · cmul 9885  cn 10964  0cn0 11236  cz 11321  cexp 12800  #chash 13057  Σcsu 14350  cdvds 14907  cprime 15309   pCnt cpc 15465  Basecbs 15781  Grpcgrp 17343   ~QG cqg 17511   GrpAct cga 17643   pGrp cpgp 17867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-disj 4584  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-er 7687  df-ec 7689  df-qs 7693  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-acn 8712  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-fz 12269  df-fzo 12407  df-fl 12533  df-mod 12609  df-seq 12742  df-exp 12801  df-fac 13001  df-bc 13030  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-sum 14351  df-dvds 14908  df-gcd 15141  df-prm 15310  df-pc 15466  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-grp 17346  df-minusg 17347  df-sbg 17348  df-mulg 17462  df-subg 17512  df-eqg 17514  df-ga 17644  df-od 17869  df-pgp 17871
This theorem is referenced by:  sylow2a  17955
  Copyright terms: Public domain W3C validator