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Theorem sylow2alem2 18033

Description: Lemma for sylow2a 18034. 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 8261	. . . . 5 ⊢ (𝑌 ∈ Fin ↔ 𝒫 𝑌 ∈ Fin)
3	1, 2	sylib 208	. . . 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 17741	. . . . . 6 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)
8	4, 7	syl 17	. . . . 5 ⊢ (𝜑 → ∼ Er 𝑌)
9	8	qsss 7808	. . . 4 ⊢ (𝜑 → (𝑌 / ∼ ) ⊆ 𝒫 𝑌)
10		ssfi 8180	. . . 4 ⊢ ((𝒫 𝑌 ∈ Fin ∧ (𝑌 / ∼ ) ⊆ 𝒫 𝑌) → (𝑌 / ∼ ) ∈ Fin)
11	3, 9, 10	syl2anc 693	. . 3 ⊢ (𝜑 → (𝑌 / ∼ ) ∈ Fin)
12		diffi 8192	. . 3 ⊢ ((𝑌 / ∼ ) ∈ Fin → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
13	11, 12	syl 17	. 2 ⊢ (𝜑 → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
14		sylow2a.p	. . . . 5 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
15		gagrp 17725	. . . . . . 7 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
16	4, 15	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
17		sylow2a.f	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin)
18	6	pgpfi 18020	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ₀ (#‘𝑋) = (𝑃↑𝑛))))
19	16, 17, 18	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ₀ (#‘𝑋) = (𝑃↑𝑛))))
20	14, 19	mpbid 222	. . . 4 ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ₀ (#‘𝑋) = (𝑃↑𝑛)))
21	20	simpld 475	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℙ)
22		prmz 15389	. . 3 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
23	21, 22	syl 17	. 2 ⊢ (𝜑 → 𝑃 ∈ ℤ)
24		eldifi 3732	. . . . 5 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍) → 𝑧 ∈ (𝑌 / ∼ ))
25	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑌 ∈ Fin)
26	9	sselda 3603	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ 𝒫 𝑌)
27	26	elpwid 4170	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ⊆ 𝑌)
28		ssfi 8180	. . . . . 6 ⊢ ((𝑌 ∈ Fin ∧ 𝑧 ⊆ 𝑌) → 𝑧 ∈ Fin)
29	25, 27, 28	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ Fin)
30	24, 29	sylan2 491	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → 𝑧 ∈ Fin)
31		hashcl 13147	. . . 4 ⊢ (𝑧 ∈ Fin → (#‘𝑧) ∈ ℕ₀)
32	30, 31	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → (#‘𝑧) ∈ ℕ₀)
33	32	nn0zd 11480	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → (#‘𝑧) ∈ ℤ)
34		eldif 3584	. . 3 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ↔ (𝑧 ∈ (𝑌 / ∼ ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍))
35		eqid 2622	. . . . 5 ⊢ (𝑌 / ∼ ) = (𝑌 / ∼ )
36		sseq1 3626	. . . . . . . 8 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ⊆ 𝑍))
37		selpw 4165	. . . . . . . 8 ⊢ (𝑧 ∈ 𝒫 𝑍 ↔ 𝑧 ⊆ 𝑍)
38	36, 37	syl6bbr 278	. . . . . . 7 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ∈ 𝒫 𝑍))
39	38	notbid 308	. . . . . 6 ⊢ ([𝑤] ∼ = 𝑧 → (¬ [𝑤] ∼ ⊆ 𝑍 ↔ ¬ 𝑧 ∈ 𝒫 𝑍))
40		fveq2 6191	. . . . . . 7 ⊢ ([𝑤] ∼ = 𝑧 → (#‘[𝑤] ∼ ) = (#‘𝑧))
41	40	breq2d 4665	. . . . . 6 ⊢ ([𝑤] ∼ = 𝑧 → (𝑃 ∥ (#‘[𝑤] ∼ ) ↔ 𝑃 ∥ (#‘𝑧)))
42	39, 41	imbi12d 334	. . . . 5 ⊢ ([𝑤] ∼ = 𝑧 → ((¬ [𝑤] ∼ ⊆ 𝑍 → 𝑃 ∥ (#‘[𝑤] ∼ )) ↔ (¬ 𝑧 ∈ 𝒫 𝑍 → 𝑃 ∥ (#‘𝑧))))
43	21	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑃 ∈ ℙ)
44	8	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ∼ Er 𝑌)
45		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ 𝑌)
46	44, 45	erref 7762	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∼ 𝑤)
47		vex 3203	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
48	47, 47	elec 7786	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ [𝑤] ∼ ↔ 𝑤 ∼ 𝑤)
49	46, 48	sylibr 224	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ [𝑤] ∼ )
50		ne0i 3921	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ [𝑤] ∼ → [𝑤] ∼ ≠ ∅)
51	49, 50	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → [𝑤] ∼ ≠ ∅)
52	8	ecss 7788	. . . . . . . . . . . . . 14 ⊢ (𝜑 → [𝑤] ∼ ⊆ 𝑌)
53		ssfi 8180	. . . . . . . . . . . . . 14 ⊢ ((𝑌 ∈ Fin ∧ [𝑤] ∼ ⊆ 𝑌) → [𝑤] ∼ ∈ Fin)
54	1, 52, 53	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → [𝑤] ∼ ∈ Fin)
55	54	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → [𝑤] ∼ ∈ Fin)
56		hashnncl 13157	. . . . . . . . . . . 12 ⊢ ([𝑤] ∼ ∈ Fin → ((#‘[𝑤] ∼ ) ∈ ℕ ↔ [𝑤] ∼ ≠ ∅))
57	55, 56	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((#‘[𝑤] ∼ ) ∈ ℕ ↔ [𝑤] ∼ ≠ ∅))
58	51, 57	mpbird 247	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (#‘[𝑤] ∼ ) ∈ ℕ)
59		pceq0 15575	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ (#‘[𝑤] ∼ ) ∈ ℕ) → ((𝑃 pCnt (#‘[𝑤] ∼ )) = 0 ↔ ¬ 𝑃 ∥ (#‘[𝑤] ∼ )))
60	43, 58, 59	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃 pCnt (#‘[𝑤] ∼ )) = 0 ↔ ¬ 𝑃 ∥ (#‘[𝑤] ∼ )))
61		oveq2 6658	. . . . . . . . . 10 ⊢ ((𝑃 pCnt (#‘[𝑤] ∼ )) = 0 → (𝑃↑(𝑃 pCnt (#‘[𝑤] ∼ ))) = (𝑃↑0))
62		hashcl 13147	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ([𝑤] ∼ ∈ Fin → (#‘[𝑤] ∼ ) ∈ ℕ₀)
63	54, 62	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (#‘[𝑤] ∼ ) ∈ ℕ₀)
64	63	nn0zd 11480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (#‘[𝑤] ∼ ) ∈ ℤ)
65		ssrab2 3687	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ⊆ 𝑋
66		ssfi 8180	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑋 ∈ Fin ∧ {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ⊆ 𝑋) → {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ∈ Fin)
67	17, 65, 66	sylancl 694	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ∈ Fin)
68		hashcl 13147	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} ∈ Fin → (#‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) ∈ ℕ₀)
69	67, 68	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (#‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) ∈ ℕ₀)
70	69	nn0zd 11480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (#‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) ∈ ℤ)
71		dvdsmul1 15003	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((#‘[𝑤] ∼ ) ∈ ℤ ∧ (#‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) ∈ ℤ) → (#‘[𝑤] ∼ ) ∥ ((#‘[𝑤] ∼ ) · (#‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
72	64, 70, 71	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (#‘[𝑤] ∼ ) ∥ ((#‘[𝑤] ∼ ) · (#‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
73	72	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (#‘[𝑤] ∼ ) ∥ ((#‘[𝑤] ∼ ) · (#‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
74	4	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ⊕ ∈ (𝐺 GrpAct 𝑌))
75	17	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑋 ∈ Fin)
76		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤} = {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}
77		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺 ~_QG {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤}) = (𝐺 ~_QG {𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})
78	6, 76, 77, 5	orbsta2 17747	. . . . . . . . . . . . . . . . . . 19 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑤 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (#‘𝑋) = ((#‘[𝑤] ∼ ) · (#‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
79	74, 45, 75, 78	syl21anc 1325	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (#‘𝑋) = ((#‘[𝑤] ∼ ) · (#‘{𝑣 ∈ 𝑋 ∣ (𝑣 ⊕ 𝑤) = 𝑤})))
80	73, 79	breqtrrd 4681	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (#‘[𝑤] ∼ ) ∥ (#‘𝑋))
81	20	simprd 479	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∃𝑛 ∈ ℕ₀ (#‘𝑋) = (𝑃↑𝑛))
82	81	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ∃𝑛 ∈ ℕ₀ (#‘𝑋) = (𝑃↑𝑛))
83		breq2 4657	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑋) = (𝑃↑𝑛) → ((#‘[𝑤] ∼ ) ∥ (#‘𝑋) ↔ (#‘[𝑤] ∼ ) ∥ (𝑃↑𝑛)))
84	83	biimpcd 239	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘[𝑤] ∼ ) ∥ (#‘𝑋) → ((#‘𝑋) = (𝑃↑𝑛) → (#‘[𝑤] ∼ ) ∥ (𝑃↑𝑛)))
85	84	reximdv 3016	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘[𝑤] ∼ ) ∥ (#‘𝑋) → (∃𝑛 ∈ ℕ₀ (#‘𝑋) = (𝑃↑𝑛) → ∃𝑛 ∈ ℕ₀ (#‘[𝑤] ∼ ) ∥ (𝑃↑𝑛)))
86	80, 82, 85	sylc 65	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ∃𝑛 ∈ ℕ₀ (#‘[𝑤] ∼ ) ∥ (𝑃↑𝑛))
87		pcprmpw2 15586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ ℙ ∧ (#‘[𝑤] ∼ ) ∈ ℕ) → (∃𝑛 ∈ ℕ₀ (#‘[𝑤] ∼ ) ∥ (𝑃↑𝑛) ↔ (#‘[𝑤] ∼ ) = (𝑃↑(𝑃 pCnt (#‘[𝑤] ∼ )))))
88	43, 58, 87	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (∃𝑛 ∈ ℕ₀ (#‘[𝑤] ∼ ) ∥ (𝑃↑𝑛) ↔ (#‘[𝑤] ∼ ) = (𝑃↑(𝑃 pCnt (#‘[𝑤] ∼ )))))
89	86, 88	mpbid 222	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (#‘[𝑤] ∼ ) = (𝑃↑(𝑃 pCnt (#‘[𝑤] ∼ ))))
90	89	eqcomd 2628	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑃↑(𝑃 pCnt (#‘[𝑤] ∼ ))) = (#‘[𝑤] ∼ ))
91	23	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑃 ∈ ℤ)
92	91	zcnd 11483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑃 ∈ ℂ)
93	92	exp0d 13002	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑃↑0) = 1)
94		hash1 13192	. . . . . . . . . . . . . . 15 ⊢ (#‘1_𝑜) = 1
95	93, 94	syl6eqr 2674	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑃↑0) = (#‘1_𝑜))
96	90, 95	eqeq12d 2637	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃↑(𝑃 pCnt (#‘[𝑤] ∼ ))) = (𝑃↑0) ↔ (#‘[𝑤] ∼ ) = (#‘1_𝑜)))
97		df1o2 7572	. . . . . . . . . . . . . . 15 ⊢ 1_𝑜 = {∅}
98		snfi 8038	. . . . . . . . . . . . . . 15 ⊢ {∅} ∈ Fin
99	97, 98	eqeltri 2697	. . . . . . . . . . . . . 14 ⊢ 1_𝑜 ∈ Fin
100		hashen 13135	. . . . . . . . . . . . . 14 ⊢ (([𝑤] ∼ ∈ Fin ∧ 1_𝑜 ∈ Fin) → ((#‘[𝑤] ∼ ) = (#‘1_𝑜) ↔ [𝑤] ∼ ≈ 1_𝑜))
101	55, 99, 100	sylancl 694	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((#‘[𝑤] ∼ ) = (#‘1_𝑜) ↔ [𝑤] ∼ ≈ 1_𝑜))
102	96, 101	bitrd 268	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃↑(𝑃 pCnt (#‘[𝑤] ∼ ))) = (𝑃↑0) ↔ [𝑤] ∼ ≈ 1_𝑜))
103		en1b 8024	. . . . . . . . . . . 12 ⊢ ([𝑤] ∼ ≈ 1_𝑜 ↔ [𝑤] ∼ = {∪ [𝑤] ∼ })
104	102, 103	syl6bb 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃↑(𝑃 pCnt (#‘[𝑤] ∼ ))) = (𝑃↑0) ↔ [𝑤] ∼ = {∪ [𝑤] ∼ }))
105	45	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 ∈ 𝑌)
106	4	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → ⊕ ∈ (𝐺 GrpAct 𝑌))
107	6	gaf 17728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
108	106, 107	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
109		simprl 794	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → ℎ ∈ 𝑋)
110	108, 109, 105	fovrnd 6806	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) ∈ 𝑌)
111		eqid 2622	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ ⊕ 𝑤) = (ℎ ⊕ 𝑤)
112		oveq1 6657	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = ℎ → (𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤))
113	112	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = ℎ → ((𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤) ↔ (ℎ ⊕ 𝑤) = (ℎ ⊕ 𝑤)))
114	113	rspcev 3309	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ ∈ 𝑋 ∧ (ℎ ⊕ 𝑤) = (ℎ ⊕ 𝑤)) → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤))
115	109, 111, 114	sylancl 694	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤))
116	5	gaorb 17740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∼ (ℎ ⊕ 𝑤) ↔ (𝑤 ∈ 𝑌 ∧ (ℎ ⊕ 𝑤) ∈ 𝑌 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑤) = (ℎ ⊕ 𝑤)))
117	105, 110, 115, 116	syl3anbrc 1246	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 ∼ (ℎ ⊕ 𝑤))
118		ovex 6678	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ ⊕ 𝑤) ∈ V
119	118, 47	elec 7786	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ ⊕ 𝑤) ∈ [𝑤] ∼ ↔ 𝑤 ∼ (ℎ ⊕ 𝑤))
120	117, 119	sylibr 224	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) ∈ [𝑤] ∼ )
121		simprr 796	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → [𝑤] ∼ = {∪ [𝑤] ∼ })
122	120, 121	eleqtrd 2703	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) ∈ {∪ [𝑤] ∼ })
123	118	elsn 4192	. . . . . . . . . . . . . . 15 ⊢ ((ℎ ⊕ 𝑤) ∈ {∪ [𝑤] ∼ } ↔ (ℎ ⊕ 𝑤) = ∪ [𝑤] ∼ )
124	122, 123	sylib 208	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) = ∪ [𝑤] ∼ )
125	49	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 ∈ [𝑤] ∼ )
126	125, 121	eleqtrd 2703	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 ∈ {∪ [𝑤] ∼ })
127	47	elsn 4192	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ {∪ [𝑤] ∼ } ↔ 𝑤 = ∪ [𝑤] ∼ )
128	126, 127	sylib 208	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → 𝑤 = ∪ [𝑤] ∼ )
129	124, 128	eqtr4d 2659	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ (ℎ ∈ 𝑋 ∧ [𝑤] ∼ = {∪ [𝑤] ∼ })) → (ℎ ⊕ 𝑤) = 𝑤)
130	129	expr 643	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑌) ∧ ℎ ∈ 𝑋) → ([𝑤] ∼ = {∪ [𝑤] ∼ } → (ℎ ⊕ 𝑤) = 𝑤))
131	130	ralrimdva 2969	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ([𝑤] ∼ = {∪ [𝑤] ∼ } → ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
132	104, 131	sylbid 230	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃↑(𝑃 pCnt (#‘[𝑤] ∼ ))) = (𝑃↑0) → ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
133	61, 132	syl5 34	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ((𝑃 pCnt (#‘[𝑤] ∼ )) = 0 → ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
134	60, 133	sylbird 250	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (¬ 𝑃 ∥ (#‘[𝑤] ∼ ) → ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
135		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑤 → (ℎ ⊕ 𝑢) = (ℎ ⊕ 𝑤))
136		id 22	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑤 → 𝑢 = 𝑤)
137	135, 136	eqeq12d 2637	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑤 → ((ℎ ⊕ 𝑢) = 𝑢 ↔ (ℎ ⊕ 𝑤) = 𝑤))
138	137	ralbidv 2986	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑤 → (∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢 ↔ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
139		sylow2a.z	. . . . . . . . . . 11 ⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢}
140	138, 139	elrab2 3366	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑍 ↔ (𝑤 ∈ 𝑌 ∧ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
141	140	baib 944	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑌 → (𝑤 ∈ 𝑍 ↔ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
142	141	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑤 ∈ 𝑍 ↔ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑤) = 𝑤))
143	134, 142	sylibrd 249	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (¬ 𝑃 ∥ (#‘[𝑤] ∼ ) → 𝑤 ∈ 𝑍))
144	6, 4, 14, 17, 1, 139, 5	sylow2alem1 18032	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ = {𝑤})
145		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑍)
146	145	snssd 4340	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ⊆ 𝑍)
147	144, 146	eqsstrd 3639	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ⊆ 𝑍)
148	147	ex 450	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ 𝑍 → [𝑤] ∼ ⊆ 𝑍))
149	148	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑤 ∈ 𝑍 → [𝑤] ∼ ⊆ 𝑍))
150	143, 149	syld 47	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (¬ 𝑃 ∥ (#‘[𝑤] ∼ ) → [𝑤] ∼ ⊆ 𝑍))
151	150	con1d 139	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (¬ [𝑤] ∼ ⊆ 𝑍 → 𝑃 ∥ (#‘[𝑤] ∼ )))
152	35, 42, 151	ectocld 7814	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (¬ 𝑧 ∈ 𝒫 𝑍 → 𝑃 ∥ (#‘𝑧)))
153	152	impr 649	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 / ∼ ) ∧ ¬ 𝑧 ∈ 𝒫 𝑍)) → 𝑃 ∥ (#‘𝑧))
154	34, 153	sylan2b 492	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → 𝑃 ∥ (#‘𝑧))
155	13, 23, 33, 154	fsumdvds 15030	1 ⊢ (𝜑 → 𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 ∖ cdif 3571 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 {csn 4177 {cpr 4179 ∪ cuni 4436 class class class wbr 4653 {copab 4712 × cxp 5112 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1_𝑜c1o 7553 Er wer 7739 [cec 7740 / cqs 7741 ≈ cen 7952 Fincfn 7955 0cc0 9936 1c1 9937 · cmul 9941 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ↑cexp 12860 #chash 13117 Σcsu 14416 ∥ cdvds 14983 ℙcprime 15385 pCnt cpc 15541 Basecbs 15857 Grpcgrp 17422 ~_QG cqg 17590 GrpAct cga 17722 pGrp cpgp 17946

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-eqg 17593 df-ga 17723 df-od 17948 df-pgp 17950

This theorem is referenced by: sylow2a 18034

W3C validator