Theorem sylow2alem2 18417

Description: Lemma for sylow2a 18418. 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 8549	. . . . 5 ⊢ (𝑌 ∈ Fin ↔ 𝒫 𝑌 ∈ Fin)
3	1, 2	sylib 210	. . . 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 18124	. . . . . 6 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)
8	4, 7	syl 17	. . . . 5 ⊢ (𝜑 → ∼ Er 𝑌)
9	8	qsss 8091	. . . 4 ⊢ (𝜑 → (𝑌 / ∼ ) ⊆ 𝒫 𝑌)
10	3, 9	ssfid 8471	. . 3 ⊢ (𝜑 → (𝑌 / ∼ ) ∈ Fin)
11		diffi 8480	. . 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 18108	. . . . . . 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 18404	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛))))
18	15, 16, 17	syl2anc 579	. . . . 5 ⊢ (𝜑 → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛))))
19	13, 18	mpbid 224	. . . 4 ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛)))
20	19	simpld 490	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℙ)
21		prmz 15794	. . 3 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
22	20, 21	syl 17	. 2 ⊢ (𝜑 → 𝑃 ∈ ℤ)
23		eldifi 3955	. . . . 5 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍) → 𝑧 ∈ (𝑌 / ∼ ))
24	1	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑌 ∈ Fin)
25	9	sselda 3821	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ 𝒫 𝑌)
26	25	elpwid 4391	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ⊆ 𝑌)
27	24, 26	ssfid 8471	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ Fin)
28	23, 27	sylan2 586	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → 𝑧 ∈ Fin)
29		hashcl 13462	. . . 4 ⊢ (𝑧 ∈ Fin → (♯‘𝑧) ∈ ℕ0)
30	28, 29	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → (♯‘𝑧) ∈ ℕ0)
31	30	nn0zd 11832	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → (♯‘𝑧) ∈ ℤ)
32		eldif 3802	. . 3 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ↔ (𝑧 ∈ (𝑌 / ∼ ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍))
33		eqid 2778	. . . . 5 ⊢ (𝑌 / ∼ ) = (𝑌 / ∼ )
34		sseq1 3845	. . . . . . . 8 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ⊆ 𝑍))
35		selpw 4386	. . . . . . . 8 ⊢ (𝑧 ∈ 𝒫 𝑍 ↔ 𝑧 ⊆ 𝑍)
36	34, 35	syl6bbr 281	. . . . . . 7 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ∈ 𝒫 𝑍))
37	36	notbid 310	. . . . . 6 ⊢ ([𝑤] ∼ = 𝑧 → (¬ [𝑤] ∼ ⊆ 𝑍 ↔ ¬ 𝑧 ∈ 𝒫 𝑍))
38		fveq2 6446	. . . . . . 7 ⊢ ([𝑤] ∼ = 𝑧 → (♯‘[𝑤] ∼ ) = (♯‘𝑧))
39	38	breq2d 4898	. . . . . 6 ⊢ ([𝑤] ∼ = 𝑧 → (𝑃 ∥ (♯‘[𝑤] ∼ ) ↔ 𝑃 ∥ (♯‘𝑧)))
40	37, 39	imbi12d 336	. . . . 5 ⊢ ([𝑤] ∼ = 𝑧 → ((¬ [𝑤] ∼ ⊆ 𝑍 → 𝑃 ∥ (♯‘[𝑤] ∼ )) ↔ (¬ 𝑧 ∈ 𝒫 𝑍 → 𝑃 ∥ (♯‘𝑧))))
41	20	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑃 ∈ ℙ)
42	8	adantr 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ∼ Er 𝑌)
43		simpr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ 𝑌)
44	42, 43	erref 8046	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∼ 𝑤)
45		vex 3401	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
46	45, 45	elec 8068	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ [𝑤] ∼ ↔ 𝑤 ∼ 𝑤)
47	44, 46	sylibr 226	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ [𝑤] ∼ )
48	47	ne0d 4150	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → [𝑤] ∼ ≠ ∅)
49	8	ecss 8070	. . . . . . . . . . . . . 14 ⊢ (𝜑 → [𝑤] ∼ ⊆ 𝑌)
50	1, 49	ssfid 8471	. . . . . . . . . . . . 13 ⊢ (𝜑 → [𝑤] ∼ ∈ Fin)
51	50	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → [𝑤] ∼ ∈ Fin)
52		hashnncl 13472	. . . . . . . . . . . 12 ⊢ ([𝑤] ∼ ∈ Fin → ((♯‘[𝑤] ∼ ) ∈ ℕ ↔ [𝑤] ∼ ≠ ∅))
53	51, 52	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((♯‘[𝑤] ∼ ) ∈ ℕ ↔ [𝑤] ∼ ≠ ∅))
54	48, 53	mpbird 249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (♯‘[𝑤] ∼ ) ∈ ℕ)
55		pceq0 15979	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘[𝑤] ∼ ) ∈ ℕ) → ((𝑃 pCnt (♯‘[𝑤] ∼ )) = 0 ↔ ¬ 𝑃 ∥ (♯‘[𝑤] ∼ )))
56	41, 54, 55	syl2anc 579	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃 pCnt (♯‘[𝑤] ∼ )) = 0 ↔ ¬ 𝑃 ∥ (♯‘[𝑤] ∼ )))
57		oveq2 6930	. . . . . . . . . 10 ⊢ ((𝑃 pCnt (♯‘[𝑤] ∼ )) = 0 → (𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))) = (𝑃↑0))
58		hashcl 13462	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ([𝑤] ∼ ∈ Fin → (♯‘[𝑤] ∼ ) ∈ ℕ0)
59	50, 58	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (♯‘[𝑤] ∼ ) ∈ ℕ0)
60	59	nn0zd 11832	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘[𝑤] ∼ ) ∈ ℤ)
61		ssrab2 3908	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ⊆ 𝑋
62		ssfi 8468	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑋 ∈ Fin ∧ {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ⊆ 𝑋) → {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ∈ Fin)
63	16, 61, 62	sylancl 580	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ∈ Fin)
64		hashcl 13462	. . . . . . . . . . . . . . . . . . . . . 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 11832	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) ∈ ℤ)
67		dvdsmul1 15410	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((♯‘[𝑤] ∼ ) ∈ ℤ ∧ (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) ∈ ℤ) → (♯‘[𝑤] ∼ ) ∥ ((♯‘[𝑤] ∼ ) · (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
68	60, 66, 67	syl2anc 579	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (♯‘[𝑤] ∼ ) ∥ ((♯‘[𝑤] ∼ ) · (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
69	68	adantr 474	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (♯‘[𝑤] ∼ ) ∥ ((♯‘[𝑤] ∼ ) · (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
70	4	adantr 474	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ⊕ ∈ (𝐺 GrpAct 𝑌))
71	16	adantr 474	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑋 ∈ Fin)
72		eqid 2778	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} = {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}
73		eqid 2778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺 ~QG {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) = (𝐺 ~QG {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})
74	6, 72, 73, 5	orbsta2 18130	. . . . . . . . . . . . . . . . . . 19 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑤 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘[𝑤] ∼ ) · (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
75	70, 43, 71, 74	syl21anc 828	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (♯‘𝑋) = ((♯‘[𝑤] ∼ ) · (♯‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
76	69, 75	breqtrrd 4914	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (♯‘[𝑤] ∼ ) ∥ (♯‘𝑋))
77	19	simprd 491	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛))
78	77	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛))
79		breq2 4890	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑋) = (𝑃↑𝑛) → ((♯‘[𝑤] ∼ ) ∥ (♯‘𝑋) ↔ (♯‘[𝑤] ∼ ) ∥ (𝑃↑𝑛)))
80	79	biimpcd 241	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘[𝑤] ∼ ) ∥ (♯‘𝑋) → ((♯‘𝑋) = (𝑃↑𝑛) → (♯‘[𝑤] ∼ ) ∥ (𝑃↑𝑛)))
81	80	reximdv 3197	. . . . . . . . . . . . . . . . 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 15990	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘[𝑤] ∼ ) ∈ ℕ) → (∃𝑛 ∈ ℕ0 (♯‘[𝑤] ∼ ) ∥ (𝑃↑𝑛) ↔ (♯‘[𝑤] ∼ ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ )))))
84	41, 54, 83	syl2anc 579	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (∃𝑛 ∈ ℕ0 (♯‘[𝑤] ∼ ) ∥ (𝑃↑𝑛) ↔ (♯‘[𝑤] ∼ ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ )))))
85	82, 84	mpbid 224	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (♯‘[𝑤] ∼ ) = (𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))))
86	85	eqcomd 2784	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))) = (♯‘[𝑤] ∼ ))
87	22	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑃 ∈ ℤ)
88	87	zcnd 11835	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑃 ∈ ℂ)
89	88	exp0d 13321	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑃↑0) = 1)
90		hash1 13506	. . . . . . . . . . . . . . 15 ⊢ (♯‘1o) = 1
91	89, 90	syl6eqr 2832	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑃↑0) = (♯‘1o))
92	86, 91	eqeq12d 2793	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))) = (𝑃↑0) ↔ (♯‘[𝑤] ∼ ) = (♯‘1o)))
93		df1o2 7856	. . . . . . . . . . . . . . 15 ⊢ 1o = {∅}
94		snfi 8326	. . . . . . . . . . . . . . 15 ⊢ {∅} ∈ Fin
95	93, 94	eqeltri 2855	. . . . . . . . . . . . . 14 ⊢ 1o ∈ Fin
96		hashen 13452	. . . . . . . . . . . . . 14 ⊢ (([𝑤] ∼ ∈ Fin ∧ 1o ∈ Fin) → ((♯‘[𝑤] ∼ ) = (♯‘1o) ↔ [𝑤] ∼ ≈ 1o))
97	51, 95, 96	sylancl 580	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((♯‘[𝑤] ∼ ) = (♯‘1o) ↔ [𝑤] ∼ ≈ 1o))
98	92, 97	bitrd 271	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))) = (𝑃↑0) ↔ [𝑤] ∼ ≈ 1o))
99		en1b 8309	. . . . . . . . . . . 12 ⊢ ([𝑤] ∼ ≈ 1o ↔ [𝑤] ∼ = {∪ [𝑤] ∼ })
100	98, 99	syl6bb 279	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃↑(𝑃 pCnt (♯‘[𝑤] ∼ ))) = (𝑃↑0) ↔ [𝑤] ∼ = {∪ [𝑤] ∼ }))
101	43	adantr 474	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 ∈ 𝑌)
102	4	ad2antrr 716	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → ⊕ ∈ (𝐺 GrpAct 𝑌))
103	6	gaf 18111	. . . . . . . . . . . . . . . . . . . 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 761	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → ℎ ∈ 𝑋)
106	104, 105, 101	fovrnd 7083	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) ∈ 𝑌)
107		eqid 2778	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ ⊕ 𝑤) = (ℎ ⊕ 𝑤)
108		oveq1 6929	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = ℎ → (𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤))
109	108	eqeq1d 2780	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = ℎ → ((𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤) ↔ (ℎ ⊕ 𝑤) = (ℎ ⊕ 𝑤)))
110	109	rspcev 3511	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ ∈ 𝑋 ∧ (ℎ ⊕ 𝑤) = (ℎ ⊕ 𝑤)) → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤))
111	105, 107, 110	sylancl 580	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤))
112	5	gaorb 18123	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∼ (ℎ ⊕ 𝑤) ↔ (𝑤 ∈ 𝑌 ∧ (ℎ ⊕ 𝑤) ∈ 𝑌 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤)))
113	101, 106, 111, 112	syl3anbrc 1400	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 ∼ (ℎ ⊕ 𝑤))
114		ovex 6954	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ ⊕ 𝑤) ∈ V
115	114, 45	elec 8068	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ ⊕ 𝑤) ∈ [𝑤] ∼ ↔ 𝑤 ∼ (ℎ ⊕ 𝑤))
116	113, 115	sylibr 226	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) ∈ [𝑤] ∼ )
117		simprr 763	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → [𝑤] ∼ = {∪ [𝑤] ∼ })
118	116, 117	eleqtrd 2861	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) ∈ {∪ [𝑤] ∼ })
119	114	elsn 4413	. . . . . . . . . . . . . . 15 ⊢ ((ℎ ⊕ 𝑤) ∈ {∪ [𝑤] ∼ } ↔ (ℎ ⊕ 𝑤) = ∪ [𝑤] ∼ )
120	118, 119	sylib 210	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) = ∪ [𝑤] ∼ )
121	47	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 ∈ [𝑤] ∼ )
122	121, 117	eleqtrd 2861	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 ∈ {∪ [𝑤] ∼ })
123	45	elsn 4413	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ {∪ [𝑤] ∼ } ↔ 𝑤 = ∪ [𝑤] ∼ )
124	122, 123	sylib 210	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 = ∪ [𝑤] ∼ )
125	120, 124	eqtr4d 2817	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) = 𝑤)
126	125	expr 450	. . . . . . . . . . . 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 232	. . . . . . . . . 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 252	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ∼ ) → ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
131		oveq2 6930	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑤 → (ℎ ⊕ 𝑢) = (ℎ ⊕ 𝑤))
132		id 22	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑤 → 𝑢 = 𝑤)
133	131, 132	eqeq12d 2793	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑤 → ((ℎ ⊕ 𝑢) = 𝑢 ↔ (ℎ ⊕ 𝑤) = 𝑤))
134	133	ralbidv 3168	. . . . . . . . . . 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 3576	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑍 ↔ (𝑤 ∈ 𝑌 ∧ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
137	136	baib 531	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑌 → (𝑤 ∈ 𝑍 ↔ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
138	137	adantl 475	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑤 ∈ 𝑍 ↔ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
139	130, 138	sylibrd 251	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ∼ ) → 𝑤 ∈ 𝑍))
140	6, 4, 13, 16, 1, 135, 5	sylow2alem1 18416	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ = {𝑤})
141		simpr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑍)
142	141	snssd 4571	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ⊆ 𝑍)
143	140, 142	eqsstrd 3858	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ⊆ 𝑍)
144	143	ex 403	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ 𝑍 → [𝑤] ∼ ⊆ 𝑍))
145	144	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑤 ∈ 𝑍 → [𝑤] ∼ ⊆ 𝑍))
146	139, 145	syld 47	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (¬ 𝑃 ∥ (♯‘[𝑤] ∼ ) → [𝑤] ∼ ⊆ 𝑍))
147	146	con1d 142	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (¬ [𝑤] ∼ ⊆ 𝑍 → 𝑃 ∥ (♯‘[𝑤] ∼ )))
148	33, 40, 147	ectocld 8097	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (¬ 𝑧 ∈ 𝒫 𝑍 → 𝑃 ∥ (♯‘𝑧)))
149	148	impr 448	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 / ∼ ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍)) → 𝑃 ∥ (♯‘𝑧))
150	32, 149	sylan2b 587	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → 𝑃 ∥ (♯‘𝑧))
151	12, 22, 31, 150	fsumdvds 15437	1 ⊢ (𝜑 → 𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ∀wral 3090  ∃wrex 3091  {crab 3094   ∖ cdif 3789   ⊆ wss 3792  ∅c0 4141  𝒫 cpw 4379  {csn 4398  {cpr 4400  ∪ cuni 4671   class class class wbr 4886  {copab 4948   × cxp 5353  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922  1_oc1o 7836   Er wer 8023  [cec 8024   / cqs 8025   ≈ cen 8238  Fincfn 8241  0cc0 10272  1c1 10273   · cmul 10277  ℕcn 11374  ℕ₀cn0 11642  ℤcz 11728  ↑cexp 13178  ♯chash 13435  Σcsu 14824   ∥ cdvds 15387  ℙcprime 15790   pCnt cpc 15945  Basecbs 16255  Grpcgrp 17809   ~_QG cqg 17974   GrpAct cga 18105   pGrp cpgp 18330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-disj 4855  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-omul 7848  df-er 8026  df-ec 8028  df-qs 8032  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-sup 8636  df-inf 8637  df-oi 8704  df-card 9098  df-acn 9101  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-xnn0 11715  df-z 11729  df-uz 11993  df-q 12096  df-rp 12138  df-fz 12644  df-fzo 12785  df-fl 12912  df-mod 12988  df-seq 13120  df-exp 13179  df-fac 13379  df-bc 13408  df-hash 13436  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-clim 14627  df-sum 14825  df-dvds 15388  df-gcd 15623  df-prm 15791  df-pc 15946  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-0g 16488  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-submnd 17722  df-grp 17812  df-minusg 17813  df-sbg 17814  df-mulg 17928  df-subg 17975  df-eqg 17977  df-ga 18106  df-od 18332  df-pgp 18334

This theorem is referenced by:  sylow2a  18418