Theorem sylow2alem2 19683

Description: Lemma for sylow2a 19684. 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 9274	. . . . 5 ⊢ (𝑌 ∈ Fin ↔ 𝒫 𝑌 ∈ Fin)
3	1, 2	sylib 221	. . . 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 19373	. . . . . 6 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)
8	4, 7	syl 18	. . . . 5 ⊢ (𝜑 → ∼ Er 𝑌)
9	8	qsss 8769	. . . 4 ⊢ (𝜑 → (𝑌 / ∼ ) ⊆ 𝒫 𝑌)
10	3, 9	ssfid 9225	. . 3 ⊢ (𝜑 → (𝑌 / ∼ ) ∈ Fin)
11		diffi 9155	. . 3 ⊢ ((𝑌 / ∼ ) ∈ Fin → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
12	10, 11	syl 18	. 2 ⊢ (𝜑 → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
13		sylow2a.p	. . . . 5 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
14		gagrp 19357	. . . . . . 7 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
15	4, 14	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
16		sylow2a.f	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin)
17	6	pgpfi 19670	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛))))
18	15, 16, 17	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛))))
19	13, 18	mpbid 235	. . . 4 ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛)))
20	19	simpld 499	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℙ)
21		prmz 16728	. . 3 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
22	20, 21	syl 18	. 2 ⊢ (𝜑 → 𝑃 ∈ ℤ)
23		eldifi 4085	. . . . 5 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍) → 𝑧 ∈ (𝑌 / ∼ ))
24	1	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑌 ∈ Fin)
25	9	sselda 3937	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ 𝒫 𝑌)
26	25	elpwid 4571	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ⊆ 𝑌)
27	24, 26	ssfid 9225	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ Fin)
28	23, 27	sylan2 604	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → 𝑧 ∈ Fin)
29		hashcl 14388	. . . 4 ⊢ (𝑧 ∈ Fin → (♯‘𝑧) ∈ ℕ0)
30	28, 29	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → (♯‘𝑧) ∈ ℕ0)
31	30	nn0zd 12611	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → (♯‘𝑧) ∈ ℤ)
32		eldif 3915	. . 3 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ↔ (𝑧 ∈ (𝑌 / ∼ ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍))
33		eqid 2763	. . . . 5 ⊢ (𝑌 / ∼ ) = (𝑌 / ∼ )
34		sseq1 3962	. . . . . . . 8 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ⊆ 𝑍))
35		velpw 4567	. . . . . . . 8 ⊢ (𝑧 ∈ 𝒫 𝑍 ↔ 𝑧 ⊆ 𝑍)
36	34, 35	bitr4di 292	. . . . . . 7 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ∈ 𝒫 𝑍))
37	36	notbid 321	. . . . . 6 ⊢ ([𝑤] ∼ = 𝑧 → (¬ [𝑤] ∼ ⊆ 𝑍 ↔ ¬ 𝑧 ∈ 𝒫 𝑍))
38		fveq2 6881	. . . . . . 7 ⊢ ([𝑤] ∼ = 𝑧 → (♯‘[𝑤] ∼ ) = (♯‘𝑧))
39	38	breq2d 5121	. . . . . 6 ⊢ ([𝑤] ∼ = 𝑧 → (𝑃 ∥ (♯‘[𝑤] ∼ ) ↔ 𝑃 ∥ (♯‘𝑧)))
40	37, 39	imbi12d 347	. . . . 5 ⊢ ([𝑤] ∼ = 𝑧 → ((¬ [𝑤] ∼ ⊆ 𝑍 → 𝑃 ∥ (♯‘[𝑤] ∼ )) ↔ (¬ 𝑧 ∈ 𝒫 𝑍 → 𝑃 ∥ (♯‘𝑧))))
41	20	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑃 ∈ ℙ)
42	8	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ∼ Er 𝑌)
43		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ 𝑌)
44	42, 43	erref 8711	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∼ 𝑤)
45		vex 3459	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
46	45, 45	elec 8737	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ [𝑤] ∼ ↔ 𝑤 ∼ 𝑤)
47	44, 46	sylibr 237	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ [𝑤] ∼ )
48	47	ne0d 4295	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → [𝑤] ∼ ≠ ∅)
49	8	ecss 8742	. . . . . . . . . . . . . 14 ⊢ (𝜑 → [𝑤] ∼ ⊆ 𝑌)
50	1, 49	ssfid 9225	. . . . . . . . . . . . 13 ⊢ (𝜑 → [𝑤] ∼ ∈ Fin)
51	50	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → [𝑤] ∼ ∈ Fin)
52		hashnncl 14398	. . . . . . . . . . . 12 ⊢ ([𝑤] ∼ ∈ Fin → ((♯‘[𝑤] ∼ ) ∈ ℕ ↔ [𝑤] ∼ ≠ ∅))
53	51, 52	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((♯‘[𝑤] ∼ ) ∈ ℕ ↔ [𝑤] ∼ ≠ ∅))
54	48, 53	mpbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (♯‘[𝑤] ∼ ) ∈ ℕ)
55		pceq0 16926	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘[𝑤] ∼ ) ∈ ℕ) → ((𝑃 pCnt (♯‘[𝑤] ∼ )) = 0 ↔ ¬ 𝑃 ∥ (♯‘[𝑤] ∼ )))
56	41, 54, 55	syl2anc 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃 pCnt (♯‘[𝑤] ∼ )) = 0 ↔ ¬ 𝑃 ∥ (♯‘[𝑤] ∼ )))
57		oveq2 7418	. . . . . . . . . 10 ⊢ ((𝑃 pCnt (♯‘[𝑤] ∼ )) = 0 → (𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))) = (𝑃↑0))
58		hashcl 14388	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ([𝑤] ∼ ∈ Fin → (♯‘[𝑤] ∼ ) ∈ ℕ0)
59	50, 58	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (♯‘[𝑤] ∼ ) ∈ ℕ0)
60	59	nn0zd 12611	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘[𝑤] ∼ ) ∈ ℤ)
61		ssrab2 4034	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ⊆ 𝑋
62		ssfi 9153	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑋 ∈ Fin ∧ {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ⊆ 𝑋) → {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ∈ Fin)
63	16, 61, 62	sylancl 597	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ∈ Fin)
64		hashcl 14388	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ∈ Fin → (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) ∈ ℕ0)
65	63, 64	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) ∈ ℕ0)
66	65	nn0zd 12611	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) ∈ ℤ)
67		dvdsmul1 16330	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((♯‘[𝑤] ∼ ) ∈ ℤ ∧ (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) ∈ ℤ) → (♯‘[𝑤] ∼ ) ∥ ((♯‘[𝑤] ∼ ) · (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
68	60, 66, 67	syl2anc 595	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (♯‘[𝑤] ∼ ) ∥ ((♯‘[𝑤] ∼ ) · (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
69	68	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (♯‘[𝑤] ∼ ) ∥ ((♯‘[𝑤] ∼ ) · (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
70	4	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ⊕ ∈ (𝐺 GrpAct 𝑌))
71	16	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑋 ∈ Fin)
72		eqid 2763	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} = {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}
73		eqid 2763	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺 ~QG {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) = (𝐺 ~QG {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})
74	6, 72, 73, 5	orbsta2 19379	. . . . . . . . . . . . . . . . . . 19 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑤 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘[𝑤] ∼ ) · (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
75	70, 43, 71, 74	syl21anc 850	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (♯‘𝑋) = ((♯‘[𝑤] ∼ ) · (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
76	69, 75	breqtrrd 5139	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (♯‘[𝑤] ∼ ) ∥ (♯‘𝑋))
77	19	simprd 500	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛))
78	77	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛))
79		breq2 5113	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑋) = (𝑃↑𝑛) → ((♯‘[𝑤] ∼ ) ∥ (♯‘𝑋) ↔ (♯‘[𝑤] ∼ ) ∥ (𝑃↑𝑛)))
80	79	biimpcd 252	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘[𝑤] ∼ ) ∥ (♯‘𝑋) → ((♯‘𝑋) = (𝑃↑𝑛) → (♯‘[𝑤] ∼ ) ∥ (𝑃↑𝑛)))
81	80	reximdv 3180	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘[𝑤] ∼ ) ∥ (♯‘𝑋) → (∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛) → ∃𝑛 ∈ ℕ0 (♯‘[𝑤] ∼ ) ∥ (𝑃↑𝑛)))
82	76, 78, 81	sylc 66	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ∃𝑛 ∈ ℕ0 (♯‘[𝑤] ∼ ) ∥ (𝑃↑𝑛))
83		pcprmpw2 16937	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘[𝑤] ∼ ) ∈ ℕ) → (∃𝑛 ∈ ℕ0 (♯‘[𝑤] ∼ ) ∥ (𝑃↑𝑛) ↔ (♯‘[𝑤] ∼ ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ )))))
84	41, 54, 83	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (∃𝑛 ∈ ℕ0 (♯‘[𝑤] ∼ ) ∥ (𝑃↑𝑛) ↔ (♯‘[𝑤] ∼ ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ )))))
85	82, 84	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (♯‘[𝑤] ∼ ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))))
86	85	eqcomd 2769	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))) = (♯‘[𝑤] ∼ ))
87	22	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑃 ∈ ℤ)
88	87	zcnd 12696	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑃 ∈ ℂ)
89	88	exp0d 14172	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑃↑0) = 1)
90		hash1 14436	. . . . . . . . . . . . . . 15 ⊢ (♯‘1o) = 1
91	89, 90	eqtr4di 2816	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑃↑0) = (♯‘1o))
92	86, 91	eqeq12d 2779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))) = (𝑃↑0) ↔ (♯‘[𝑤] ∼ ) = (♯‘1o)))
93		df1o2 8456	. . . . . . . . . . . . . . 15 ⊢ 1o = {∅}
94		snfi 9036	. . . . . . . . . . . . . . 15 ⊢ {∅} ∈ Fin
95	93, 94	eqeltri 2859	. . . . . . . . . . . . . 14 ⊢ 1o ∈ Fin
96		hashen 14379	. . . . . . . . . . . . . 14 ⊢ (([𝑤] ∼ ∈ Fin ∧ 1o ∈ Fin) → ((♯‘[𝑤] ∼ ) = (♯‘1o) ↔ [𝑤] ∼ ≈ 1o))
97	51, 95, 96	sylancl 597	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((♯‘[𝑤] ∼ ) = (♯‘1o) ↔ [𝑤] ∼ ≈ 1o))
98	92, 97	bitrd 282	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))) = (𝑃↑0) ↔ [𝑤] ∼ ≈ 1o))
99		en1b 9018	. . . . . . . . . . . 12 ⊢ ([𝑤] ∼ ≈ 1o ↔ [𝑤] ∼ = {∪ [𝑤] ∼ })
100	98, 99	bitrdi 290	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))) = (𝑃↑0) ↔ [𝑤] ∼ = {∪ [𝑤] ∼ }))
101	43	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 ∈ 𝑌)
102	4	ad2antrr 738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → ⊕ ∈ (𝐺 GrpAct 𝑌))
103	6	gaf 19360	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
104	102, 103	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
105		simprl 782	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → ℎ ∈ 𝑋)
106	104, 105, 101	fovcdmd 7582	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) ∈ 𝑌)
107		eqid 2763	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ ⊕ 𝑤) = (ℎ ⊕ 𝑤)
108		oveq1 7417	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = ℎ → (𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤))
109	108	eqeq1d 2765	. . . . . . . . . . . . . . . . . . . 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 597	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤))
112	5	gaorb 19372	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∼ (ℎ ⊕ 𝑤) ↔ (𝑤 ∈ 𝑌 ∧ (ℎ ⊕ 𝑤) ∈ 𝑌 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤)))
113	101, 106, 111, 112	syl3anbrc 1362	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 ∼ (ℎ ⊕ 𝑤))
114		ovex 7443	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ ⊕ 𝑤) ∈ V
115	114, 45	elec 8737	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ ⊕ 𝑤) ∈ [𝑤] ∼ ↔ 𝑤 ∼ (ℎ ⊕ 𝑤))
116	113, 115	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) ∈ [𝑤] ∼ )
117		simprr 784	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → [𝑤] ∼ = {∪ [𝑤] ∼ })
118	116, 117	eleqtrd 2865	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) ∈ {∪ [𝑤] ∼ })
119	114	elsn 4604	. . . . . . . . . . . . . . 15 ⊢ ((ℎ ⊕ 𝑤) ∈ {∪ [𝑤] ∼ } ↔ (ℎ ⊕ 𝑤) = ∪ [𝑤] ∼ )
120	118, 119	sylib 221	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) = ∪ [𝑤] ∼ )
121	47	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 ∈ [𝑤] ∼ )
122	121, 117	eleqtrd 2865	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 ∈ {∪ [𝑤] ∼ })
123	45	elsn 4604	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ {∪ [𝑤] ∼ } ↔ 𝑤 = ∪ [𝑤] ∼ )
124	122, 123	sylib 221	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 = ∪ [𝑤] ∼ )
125	120, 124	eqtr4d 2801	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) = 𝑤)
126	125	expr 461	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ ℎ ∈ 𝑋) → ([𝑤] ∼ = {∪ [𝑤] ∼ } → (ℎ ⊕ 𝑤) = 𝑤))
127	126	ralrimdva 3165	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ([𝑤] ∼ = {∪ [𝑤] ∼ } → ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
128	100, 127	sylbid 243	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))) = (𝑃↑0) → ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
129	57, 128	syl5 35	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃 pCnt (♯‘[𝑤] ∼ )) = 0 → ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
130	56, 129	sylbird 263	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ∼ ) → ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
131		oveq2 7418	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑤 → (ℎ ⊕ 𝑢) = (ℎ ⊕ 𝑤))
132		id 23	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑤 → 𝑢 = 𝑤)
133	131, 132	eqeq12d 2779	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑤 → ((ℎ ⊕ 𝑢) = 𝑢 ↔ (ℎ ⊕ 𝑤) = 𝑤))
134	133	ralbidv 3188	. . . . . . . . . . 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 3654	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑍 ↔ (𝑤 ∈ 𝑌 ∧ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
137	136	baib 544	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑌 → (𝑤 ∈ 𝑍 ↔ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
138	137	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑤 ∈ 𝑍 ↔ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
139	130, 138	sylibrd 262	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ∼ ) → 𝑤 ∈ 𝑍))
140	6, 4, 13, 16, 1, 135, 5	sylow2alem1 19682	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ = {𝑤})
141		simpr 489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑍)
142	141	snssd 4752	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ⊆ 𝑍)
143	140, 142	eqsstrd 3971	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ⊆ 𝑍)
144	143	ex 417	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ 𝑍 → [𝑤] ∼ ⊆ 𝑍))
145	144	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑤 ∈ 𝑍 → [𝑤] ∼ ⊆ 𝑍))
146	139, 145	syld 48	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ∼ ) → [𝑤] ∼ ⊆ 𝑍))
147	146	con1d 146	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (¬ [𝑤] ∼ ⊆ 𝑍 → 𝑃 ∥ (♯‘[𝑤] ∼ )))
148	33, 40, 147	ectocld 8776	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (¬ 𝑧 ∈ 𝒫 𝑍 → 𝑃 ∥ (♯‘𝑧)))
149	148	impr 459	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 / ∼ ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍)) → 𝑃 ∥ (♯‘𝑧))
150	32, 149	sylan2b 605	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → 𝑃 ∥ (♯‘𝑧))
151	12, 22, 31, 150	fsumdvds 16361	1 ⊢ (𝜑 → 𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  {crab 3416   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4562  {csn 4589  {cpr 4591  ∪ cuni 4872   class class class wbr 5109  {copab 5173   × cxp 5659  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  1_oc1o 8442   Er wer 8687  [cec 8688   / cqs 8689   ≈ cen 8936  Fincfn 8939  0cc0 11095  1c1 11096   · cmul 11100  ℕcn 12228  ℕ₀cn0 12499  ℤcz 12586  ↑cexp 14093  ♯chash 14362  Σcsu 15733   ∥ cdvds 16305  ℙcprime 16724   pCnt cpc 16891  Basecbs 17264  Grpcgrp 18995   ~_QG cqg 19183   GrpAct cga 19354   pGrp cpgp 19591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-disj 5077  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-oadd 8453  df-omul 8454  df-er 8690  df-ec 8692  df-qs 8696  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-oi 9468  df-dju 9883  df-card 9921  df-acn 9924  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-n0 12500  df-xnn0 12573  df-z 12587  df-uz 12858  df-q 12968  df-rp 13012  df-fz 13531  df-fzo 13679  df-fl 13821  df-mod 13899  df-seq 14034  df-exp 14094  df-fac 14306  df-bc 14335  df-hash 14363  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-clim 15535  df-sum 15734  df-dvds 16306  df-gcd 16548  df-prm 16725  df-pc 16892  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-0g 17489  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-submnd 18837  df-grp 18998  df-minusg 18999  df-sbg 19000  df-mulg 19129  df-subg 19184  df-eqg 19186  df-ga 19355  df-od 19593  df-pgp 19595

This theorem is referenced by:  sylow2a  19684