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	⊢ (𝜑 → 𝑃 pGrp 𝐺)
sylow2a.f	⊢ (𝜑 → 𝑋 ∈ Fin)
sylow2a.y	⊢ (𝜑 → 𝑌 ∈ Fin)
sylow2a.z	⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢}
sylow2a.r	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}

Assertion

Ref	Expression
sylow2alem2	⊢ (𝜑 → 𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧))

Distinct variable groups:   𝑧,ℎ, ∼   𝑔,ℎ,𝑢,𝑥,𝑦   𝑔,𝐺,𝑥,𝑦   𝑧,𝑃   ⊕ ,𝑔,ℎ,𝑢,𝑥,𝑦   𝑔,𝑋,ℎ,𝑢,𝑥,𝑦   𝑧,𝑍   𝜑,ℎ,𝑧   𝑧,𝑔,𝑌,ℎ,𝑢,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑢,𝑔)   𝑃(𝑥,𝑦,𝑢,𝑔,ℎ)   ⊕ (𝑧)   ∼ (𝑥,𝑦,𝑢,𝑔)   𝐺(𝑧,𝑢,ℎ)   𝑋(𝑧)   𝑍(𝑥,𝑦,𝑢,𝑔,ℎ)

Proof of Theorem sylow2alem2

Dummy variables 𝑘 𝑛 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sylow2a.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ Fin)
2		pwfi 9122	. . . . 5 ⊢ (𝑌 ∈ Fin ↔ 𝒫 𝑌 ∈ Fin)
3	1, 2	sylib 217	. . . 4 ⊢ (𝜑 → 𝒫 𝑌 ∈ Fin)
4		sylow2a.m	. . . . . 6 ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑌))
5		sylow2a.r	. . . . . . 7 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
6		sylow2a.x	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐺)
7	5, 6	gaorber 19088	. . . . . 6 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)
8	4, 7	syl 17	. . . . 5 ⊢ (𝜑 → ∼ Er 𝑌)
9	8	qsss 8717	. . . 4 ⊢ (𝜑 → (𝑌 / ∼ ) ⊆ 𝒫 𝑌)
10	3, 9	ssfid 9211	. . 3 ⊢ (𝜑 → (𝑌 / ∼ ) ∈ Fin)
11		diffi 9123	. . 3 ⊢ ((𝑌 / ∼ ) ∈ Fin → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
12	10, 11	syl 17	. 2 ⊢ (𝜑 → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
13		sylow2a.p	. . . . 5 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
14		gagrp 19072	. . . . . . 7 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
15	4, 14	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
16		sylow2a.f	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin)
17	6	pgpfi 19387	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛))))
18	15, 16, 17	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛))))
19	13, 18	mpbid 231	. . . 4 ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛)))
20	19	simpld 495	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℙ)
21		prmz 16551	. . 3 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
22	20, 21	syl 17	. 2 ⊢ (𝜑 → 𝑃 ∈ ℤ)
23		eldifi 4086	. . . . 5 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍) → 𝑧 ∈ (𝑌 / ∼ ))
24	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑌 ∈ Fin)
25	9	sselda 3944	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ 𝒫 𝑌)
26	25	elpwid 4569	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ⊆ 𝑌)
27	24, 26	ssfid 9211	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ Fin)
28	23, 27	sylan2 593	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → 𝑧 ∈ Fin)
29		hashcl 14256	. . . 4 ⊢ (𝑧 ∈ Fin → (♯‘𝑧) ∈ ℕ0)
30	28, 29	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → (♯‘𝑧) ∈ ℕ0)
31	30	nn0zd 12525	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ ((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] ∼ ⊆ 𝑍 ↔ 𝑧 ⊆ 𝑍))
35		velpw 4565	. . . . . . . 8 ⊢ (𝑧 ∈ 𝒫 𝑍 ↔ 𝑧 ⊆ 𝑍)
36	34, 35	bitr4di 288	. . . . . . 7 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ∈ 𝒫 𝑍))
37	36	notbid 317	. . . . . 6 ⊢ ([𝑤] ∼ = 𝑧 → (¬ [𝑤] ∼ ⊆ 𝑍 ↔ ¬ 𝑧 ∈ 𝒫 𝑍))
38		fveq2 6842	. . . . . . 7 ⊢ ([𝑤] ∼ = 𝑧 → (♯‘[𝑤] ∼ ) = (♯‘𝑧))
39	38	breq2d 5117	. . . . . 6 ⊢ ([𝑤] ∼ = 𝑧 → (𝑃 ∥ (♯‘[𝑤] ∼ ) ↔ 𝑃 ∥ (♯‘𝑧)))
40	37, 39	imbi12d 344	. . . . 5 ⊢ ([𝑤] ∼ = 𝑧 → ((¬ [𝑤] ∼ ⊆ 𝑍 → 𝑃 ∥ (♯‘[𝑤] ∼ )) ↔ (¬ 𝑧 ∈ 𝒫 𝑍 → 𝑃 ∥ (♯‘𝑧))))
41	20	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑃 ∈ ℙ)
42	8	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ∼ Er 𝑌)
43		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ 𝑌)
44	42, 43	erref 8668	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∼ 𝑤)
45		vex 3449	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
46	45, 45	elec 8692	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ [𝑤] ∼ ↔ 𝑤 ∼ 𝑤)
47	44, 46	sylibr 233	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ [𝑤] ∼ )
48	47	ne0d 4295	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → [𝑤] ∼ ≠ ∅)
49	8	ecss 8694	. . . . . . . . . . . . . 14 ⊢ (𝜑 → [𝑤] ∼ ⊆ 𝑌)
50	1, 49	ssfid 9211	. . . . . . . . . . . . 13 ⊢ (𝜑 → [𝑤] ∼ ∈ Fin)
51	50	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → [𝑤] ∼ ∈ Fin)
52		hashnncl 14266	. . . . . . . . . . . 12 ⊢ ([𝑤] ∼ ∈ Fin → ((♯‘[𝑤] ∼ ) ∈ ℕ ↔ [𝑤] ∼ ≠ ∅))
53	51, 52	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((♯‘[𝑤] ∼ ) ∈ ℕ ↔ [𝑤] ∼ ≠ ∅))
54	48, 53	mpbird 256	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (♯‘[𝑤] ∼ ) ∈ ℕ)
55		pceq0 16743	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘[𝑤] ∼ ) ∈ ℕ) → ((𝑃 pCnt (♯‘[𝑤] ∼ )) = 0 ↔ ¬ 𝑃 ∥ (♯‘[𝑤] ∼ )))
56	41, 54, 55	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((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)
59	50, 58	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (♯‘[𝑤] ∼ ) ∈ ℕ0)
60	59	nn0zd 12525	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘[𝑤] ∼ ) ∈ ℤ)
61		ssrab2 4037	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ⊆ 𝑋
62		ssfi 9117	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑋 ∈ Fin ∧ {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ⊆ 𝑋) → {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ∈ Fin)
63	16, 61, 62	sylancl 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ∈ Fin)
64		hashcl 14256	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ∈ Fin → (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) ∈ ℕ0)
65	63, 64	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) ∈ ℕ0)
66	65	nn0zd 12525	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) ∈ ℤ)
67		dvdsmul1 16160	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((♯‘[𝑤] ∼ ) ∈ ℤ ∧ (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) ∈ ℤ) → (♯‘[𝑤] ∼ ) ∥ ((♯‘[𝑤] ∼ ) · (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
68	60, 66, 67	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (♯‘[𝑤] ∼ ) ∥ ((♯‘[𝑤] ∼ ) · (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
69	68	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (♯‘[𝑤] ∼ ) ∥ ((♯‘[𝑤] ∼ ) · (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
70	4	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ⊕ ∈ (𝐺 GrpAct 𝑌))
71	16	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑋 ∈ Fin)
72		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} = {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}
73		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺 ~QG {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) = (𝐺 ~QG {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})
74	6, 72, 73, 5	orbsta2 19094	. . . . . . . . . . . . . . . . . . 19 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑤 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘[𝑤] ∼ ) · (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
75	70, 43, 71, 74	syl21anc 836	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (♯‘𝑋) = ((♯‘[𝑤] ∼ ) · (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
76	69, 75	breqtrrd 5133	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (♯‘[𝑤] ∼ ) ∥ (♯‘𝑋))
77	19	simprd 496	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛))
78	77	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛))
79		breq2 5109	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑋) = (𝑃↑𝑛) → ((♯‘[𝑤] ∼ ) ∥ (♯‘𝑋) ↔ (♯‘[𝑤] ∼ ) ∥ (𝑃↑𝑛)))
80	79	biimpcd 248	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘[𝑤] ∼ ) ∥ (♯‘𝑋) → ((♯‘𝑋) = (𝑃↑𝑛) → (♯‘[𝑤] ∼ ) ∥ (𝑃↑𝑛)))
81	80	reximdv 3167	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘[𝑤] ∼ ) ∥ (♯‘𝑋) → (∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛) → ∃𝑛 ∈ ℕ0 (♯‘[𝑤] ∼ ) ∥ (𝑃↑𝑛)))
82	76, 78, 81	sylc 65	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ∃𝑛 ∈ ℕ0 (♯‘[𝑤] ∼ ) ∥ (𝑃↑𝑛))
83		pcprmpw2 16754	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘[𝑤] ∼ ) ∈ ℕ) → (∃𝑛 ∈ ℕ0 (♯‘[𝑤] ∼ ) ∥ (𝑃↑𝑛) ↔ (♯‘[𝑤] ∼ ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ )))))
84	41, 54, 83	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (∃𝑛 ∈ ℕ0 (♯‘[𝑤] ∼ ) ∥ (𝑃↑𝑛) ↔ (♯‘[𝑤] ∼ ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ )))))
85	82, 84	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (♯‘[𝑤] ∼ ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))))
86	85	eqcomd 2742	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))) = (♯‘[𝑤] ∼ ))
87	22	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑃 ∈ ℤ)
88	87	zcnd 12608	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑃 ∈ ℂ)
89	88	exp0d 14045	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑃↑0) = 1)
90		hash1 14304	. . . . . . . . . . . . . . 15 ⊢ (♯‘1o) = 1
91	89, 90	eqtr4di 2794	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑃↑0) = (♯‘1o))
92	86, 91	eqeq12d 2752	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))) = (𝑃↑0) ↔ (♯‘[𝑤] ∼ ) = (♯‘1o)))
93		df1o2 8419	. . . . . . . . . . . . . . 15 ⊢ 1o = {∅}
94		snfi 8988	. . . . . . . . . . . . . . 15 ⊢ {∅} ∈ Fin
95	93, 94	eqeltri 2834	. . . . . . . . . . . . . 14 ⊢ 1o ∈ Fin
96		hashen 14247	. . . . . . . . . . . . . 14 ⊢ (([𝑤] ∼ ∈ Fin ∧ 1o ∈ Fin) → ((♯‘[𝑤] ∼ ) = (♯‘1o) ↔ [𝑤] ∼ ≈ 1o))
97	51, 95, 96	sylancl 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((♯‘[𝑤] ∼ ) = (♯‘1o) ↔ [𝑤] ∼ ≈ 1o))
98	92, 97	bitrd 278	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))) = (𝑃↑0) ↔ [𝑤] ∼ ≈ 1o))
99		en1b 8967	. . . . . . . . . . . 12 ⊢ ([𝑤] ∼ ≈ 1o ↔ [𝑤] ∼ = {∪ [𝑤] ∼ })
100	98, 99	bitrdi 286	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))) = (𝑃↑0) ↔ [𝑤] ∼ = {∪ [𝑤] ∼ }))
101	43	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 ∈ 𝑌)
102	4	ad2antrr 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → ⊕ ∈ (𝐺 GrpAct 𝑌))
103	6	gaf 19075	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
104	102, 103	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
105		simprl 769	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → ℎ ∈ 𝑋)
106	104, 105, 101	fovcdmd 7526	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) ∈ 𝑌)
107		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ ⊕ 𝑤) = (ℎ ⊕ 𝑤)
108		oveq1 7364	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = ℎ → (𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤))
109	108	eqeq1d 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = ℎ → ((𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤) ↔ (ℎ ⊕ 𝑤) = (ℎ ⊕ 𝑤)))
110	109	rspcev 3581	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ ∈ 𝑋 ∧ (ℎ ⊕ 𝑤) = (ℎ ⊕ 𝑤)) → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤))
111	105, 107, 110	sylancl 586	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤))
112	5	gaorb 19087	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∼ (ℎ ⊕ 𝑤) ↔ (𝑤 ∈ 𝑌 ∧ (ℎ ⊕ 𝑤) ∈ 𝑌 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤)))
113	101, 106, 111, 112	syl3anbrc 1343	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 ∼ (ℎ ⊕ 𝑤))
114		ovex 7390	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ ⊕ 𝑤) ∈ V
115	114, 45	elec 8692	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ ⊕ 𝑤) ∈ [𝑤] ∼ ↔ 𝑤 ∼ (ℎ ⊕ 𝑤))
116	113, 115	sylibr 233	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) ∈ [𝑤] ∼ )
117		simprr 771	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → [𝑤] ∼ = {∪ [𝑤] ∼ })
118	116, 117	eleqtrd 2840	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) ∈ {∪ [𝑤] ∼ })
119	114	elsn 4601	. . . . . . . . . . . . . . 15 ⊢ ((ℎ ⊕ 𝑤) ∈ {∪ [𝑤] ∼ } ↔ (ℎ ⊕ 𝑤) = ∪ [𝑤] ∼ )
120	118, 119	sylib 217	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) = ∪ [𝑤] ∼ )
121	47	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 ∈ [𝑤] ∼ )
122	121, 117	eleqtrd 2840	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 ∈ {∪ [𝑤] ∼ })
123	45	elsn 4601	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ {∪ [𝑤] ∼ } ↔ 𝑤 = ∪ [𝑤] ∼ )
124	122, 123	sylib 217	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 = ∪ [𝑤] ∼ )
125	120, 124	eqtr4d 2779	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) = 𝑤)
126	125	expr 457	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ ℎ ∈ 𝑋) → ([𝑤] ∼ = {∪ [𝑤] ∼ } → (ℎ ⊕ 𝑤) = 𝑤))
127	126	ralrimdva 3151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ([𝑤] ∼ = {∪ [𝑤] ∼ } → ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
128	100, 127	sylbid 239	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))) = (𝑃↑0) → ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
129	57, 128	syl5 34	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃 pCnt (♯‘[𝑤] ∼ )) = 0 → ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
130	56, 129	sylbird 259	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ∼ ) → ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
131		oveq2 7365	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑤 → (ℎ ⊕ 𝑢) = (ℎ ⊕ 𝑤))
132		id 22	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑤 → 𝑢 = 𝑤)
133	131, 132	eqeq12d 2752	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑤 → ((ℎ ⊕ 𝑢) = 𝑢 ↔ (ℎ ⊕ 𝑤) = 𝑤))
134	133	ralbidv 3174	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑤 → (∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢 ↔ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
135		sylow2a.z	. . . . . . . . . . 11 ⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢}
136	134, 135	elrab2 3648	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑍 ↔ (𝑤 ∈ 𝑌 ∧ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
137	136	baib 536	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑌 → (𝑤 ∈ 𝑍 ↔ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
138	137	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑤 ∈ 𝑍 ↔ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
139	130, 138	sylibrd 258	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ∼ ) → 𝑤 ∈ 𝑍))
140	6, 4, 13, 16, 1, 135, 5	sylow2alem1 19399	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ = {𝑤})
141		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑍)
142	141	snssd 4769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ⊆ 𝑍)
143	140, 142	eqsstrd 3982	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ⊆ 𝑍)
144	143	ex 413	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ 𝑍 → [𝑤] ∼ ⊆ 𝑍))
145	144	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑤 ∈ 𝑍 → [𝑤] ∼ ⊆ 𝑍))
146	139, 145	syld 47	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ∼ ) → [𝑤] ∼ ⊆ 𝑍))
147	146	con1d 145	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (¬ [𝑤] ∼ ⊆ 𝑍 → 𝑃 ∥ (♯‘[𝑤] ∼ )))
148	33, 40, 147	ectocld 8723	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (¬ 𝑧 ∈ 𝒫 𝑍 → 𝑃 ∥ (♯‘𝑧)))
149	148	impr 455	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 / ∼ ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍)) → 𝑃 ∥ (♯‘𝑧))
150	32, 149	sylan2b 594	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → 𝑃 ∥ (♯‘𝑧))
151	12, 22, 31, 150	fsumdvds 16190	1 ⊢ (𝜑 → 𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧))

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  1_oc1o 8405   Er wer 8645  [cec 8646   / cqs 8647   ≈ cen 8880  Fincfn 8883  0cc0 11051  1c1 11052   · cmul 11056  ℕcn 12153  ℕ₀cn0 12413  ℤcz 12499  ↑cexp 13967  ♯chash 14230  Σcsu 15570   ∥ cdvds 16136  ℙcprime 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