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Theorem sylow1lem5 18017

Description: Lemma for sylow1 18018. Using Lagrange's theorem and the orbit-stabilizer theorem, show that there is a subgroup with size exactly 𝑃↑𝑁. (Contributed by Mario Carneiro, 16-Jan-2015.)

**Hypotheses**
Ref	Expression
sylow1.x	⊢ 𝑋 = (Base‘𝐺)
sylow1.g	⊢ (𝜑 → 𝐺 ∈ Grp)
sylow1.f	⊢ (𝜑 → 𝑋 ∈ Fin)
sylow1.p	⊢ (𝜑 → 𝑃 ∈ ℙ)
sylow1.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
sylow1.d	⊢ (𝜑 → (𝑃↑𝑁) ∥ (#‘𝑋))
sylow1lem.a	⊢ + = (+_g‘𝐺)
sylow1lem.s	⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃↑𝑁)}
sylow1lem.m	⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
sylow1lem3.1	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
sylow1lem4.b	⊢ (𝜑 → 𝐵 ∈ 𝑆)
sylow1lem4.h	⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐵) = 𝐵}
sylow1lem5.l	⊢ (𝜑 → (𝑃 pCnt (#‘[𝐵] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))

**Assertion**
Ref	Expression
sylow1lem5	⊢ (𝜑 → ∃ℎ ∈ (SubGrp‘𝐺)(#‘ℎ) = (𝑃↑𝑁))

Distinct variable groups: 𝑔,𝑠,𝑢,𝑥,𝑦,𝑧,𝐵 𝑔,ℎ,𝐻,𝑥,𝑦 𝑆,𝑔,𝑢,𝑥,𝑦,𝑧 𝑔,𝑁 ℎ,𝑠,𝑢,𝑧,𝑁,𝑥,𝑦 𝑔,𝑋,ℎ,𝑠,𝑢,𝑥,𝑦,𝑧 + ,𝑠,𝑢,𝑥,𝑦,𝑧 𝑧, ∼ ⊕ ,𝑔,𝑢,𝑥,𝑦,𝑧 𝑔,𝐺,ℎ,𝑠,𝑢,𝑥,𝑦,𝑧 𝑃,𝑔,ℎ,𝑠,𝑢,𝑥,𝑦,𝑧 𝜑,𝑢,𝑥,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑔,ℎ,𝑠) 𝐵(ℎ) + (𝑔,ℎ) ⊕ (ℎ,𝑠) ∼ (𝑥,𝑦,𝑢,𝑔,ℎ,𝑠) 𝑆(ℎ,𝑠) 𝐻(𝑧,𝑢,𝑠)

**Proof of Theorem sylow1lem5**
Step	Hyp	Ref	Expression
1		sylow1.x	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
2		sylow1.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
3		sylow1.f	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin)
4		sylow1.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℙ)
5		sylow1.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6		sylow1.d	. . . 4 ⊢ (𝜑 → (𝑃↑𝑁) ∥ (#‘𝑋))
7		sylow1lem.a	. . . 4 ⊢ + = (+_g‘𝐺)
8		sylow1lem.s	. . . 4 ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃↑𝑁)}
9		sylow1lem.m	. . . 4 ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	sylow1lem2 18014	. . 3 ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑆))
11		sylow1lem4.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
12		sylow1lem4.h	. . . 4 ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐵) = 𝐵}
13	1, 12	gastacl 17742	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑆) ∧ 𝐵 ∈ 𝑆) → 𝐻 ∈ (SubGrp‘𝐺))
14	10, 11, 13	syl2anc 693	. 2 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))
15		sylow1lem3.1	. . . 4 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 11, 12	sylow1lem4 18016	. . 3 ⊢ (𝜑 → (#‘𝐻) ≤ (𝑃↑𝑁))
17		sylow1lem5.l	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt (#‘[𝐵] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
18	15, 1	gaorber 17741	. . . . . . . . . . . . . . . 16 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑆) → ∼ Er 𝑆)
19	10, 18	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∼ Er 𝑆)
20		erdm 7752	. . . . . . . . . . . . . . 15 ⊢ ( ∼ Er 𝑆 → dom ∼ = 𝑆)
21	19, 20	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom ∼ = 𝑆)
22	11, 21	eleqtrrd 2704	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ dom ∼ )
23		ecdmn0 7789	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ dom ∼ ↔ [𝐵] ∼ ≠ ∅)
24	22, 23	sylib 208	. . . . . . . . . . . 12 ⊢ (𝜑 → [𝐵] ∼ ≠ ∅)
25		pwfi 8261	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
26	3, 25	sylib 208	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
27		ssrab2 3687	. . . . . . . . . . . . . . . 16 ⊢ {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃↑𝑁)} ⊆ 𝒫 𝑋
28	8, 27	eqsstri 3635	. . . . . . . . . . . . . . 15 ⊢ 𝑆 ⊆ 𝒫 𝑋
29		ssfi 8180	. . . . . . . . . . . . . . 15 ⊢ ((𝒫 𝑋 ∈ Fin ∧ 𝑆 ⊆ 𝒫 𝑋) → 𝑆 ∈ Fin)
30	26, 28, 29	sylancl 694	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ Fin)
31	19	ecss 7788	. . . . . . . . . . . . . 14 ⊢ (𝜑 → [𝐵] ∼ ⊆ 𝑆)
32		ssfi 8180	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ Fin ∧ [𝐵] ∼ ⊆ 𝑆) → [𝐵] ∼ ∈ Fin)
33	30, 31, 32	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → [𝐵] ∼ ∈ Fin)
34		hashnncl 13157	. . . . . . . . . . . . 13 ⊢ ([𝐵] ∼ ∈ Fin → ((#‘[𝐵] ∼ ) ∈ ℕ ↔ [𝐵] ∼ ≠ ∅))
35	33, 34	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((#‘[𝐵] ∼ ) ∈ ℕ ↔ [𝐵] ∼ ≠ ∅))
36	24, 35	mpbird 247	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘[𝐵] ∼ ) ∈ ℕ)
37	4, 36	pccld 15555	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (#‘[𝐵] ∼ )) ∈ ℕ₀)
38	37	nn0red 11352	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 pCnt (#‘[𝐵] ∼ )) ∈ ℝ)
39	5	nn0red 11352	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
40	1	grpbn0 17451	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
41	2, 40	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ≠ ∅)
42		hashnncl 13157	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ Fin → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
43	3, 42	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
44	41, 43	mpbird 247	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝑋) ∈ ℕ)
45	4, 44	pccld 15555	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℕ₀)
46	45	nn0red 11352	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℝ)
47		leaddsub 10504	. . . . . . . . 9 ⊢ (((𝑃 pCnt (#‘[𝐵] ∼ )) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝑃 pCnt (#‘𝑋)) ∈ ℝ) → (((𝑃 pCnt (#‘[𝐵] ∼ )) + 𝑁) ≤ (𝑃 pCnt (#‘𝑋)) ↔ (𝑃 pCnt (#‘[𝐵] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
48	38, 39, 46, 47	syl3anc 1326	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 pCnt (#‘[𝐵] ∼ )) + 𝑁) ≤ (𝑃 pCnt (#‘𝑋)) ↔ (𝑃 pCnt (#‘[𝐵] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
49	17, 48	mpbird 247	. . . . . . 7 ⊢ (𝜑 → ((𝑃 pCnt (#‘[𝐵] ∼ )) + 𝑁) ≤ (𝑃 pCnt (#‘𝑋)))
50		eqid 2622	. . . . . . . . . . 11 ⊢ (𝐺 ~_QG 𝐻) = (𝐺 ~_QG 𝐻)
51	1, 12, 50, 15	orbsta2 17747	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑆) ∧ 𝐵 ∈ 𝑆) ∧ 𝑋 ∈ Fin) → (#‘𝑋) = ((#‘[𝐵] ∼ ) · (#‘𝐻)))
52	10, 11, 3, 51	syl21anc 1325	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝑋) = ((#‘[𝐵] ∼ ) · (#‘𝐻)))
53	52	oveq2d 6666	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑋)) = (𝑃 pCnt ((#‘[𝐵] ∼ ) · (#‘𝐻))))
54	36	nnzd 11481	. . . . . . . . 9 ⊢ (𝜑 → (#‘[𝐵] ∼ ) ∈ ℤ)
55	36	nnne0d 11065	. . . . . . . . 9 ⊢ (𝜑 → (#‘[𝐵] ∼ ) ≠ 0)
56		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
57	56	subg0cl 17602	. . . . . . . . . . . . 13 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ 𝐻)
58	14, 57	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0_g‘𝐺) ∈ 𝐻)
59		ne0i 3921	. . . . . . . . . . . 12 ⊢ ((0_g‘𝐺) ∈ 𝐻 → 𝐻 ≠ ∅)
60	58, 59	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ≠ ∅)
61		ssrab2 3687	. . . . . . . . . . . . . 14 ⊢ {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐵) = 𝐵} ⊆ 𝑋
62	12, 61	eqsstri 3635	. . . . . . . . . . . . 13 ⊢ 𝐻 ⊆ 𝑋
63		ssfi 8180	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ Fin ∧ 𝐻 ⊆ 𝑋) → 𝐻 ∈ Fin)
64	3, 62, 63	sylancl 694	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ Fin)
65		hashnncl 13157	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ Fin → ((#‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
66	64, 65	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((#‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
67	60, 66	mpbird 247	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝐻) ∈ ℕ)
68	67	nnzd 11481	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝐻) ∈ ℤ)
69	67	nnne0d 11065	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝐻) ≠ 0)
70		pcmul 15556	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ ((#‘[𝐵] ∼ ) ∈ ℤ ∧ (#‘[𝐵] ∼ ) ≠ 0) ∧ ((#‘𝐻) ∈ ℤ ∧ (#‘𝐻) ≠ 0)) → (𝑃 pCnt ((#‘[𝐵] ∼ ) · (#‘𝐻))) = ((𝑃 pCnt (#‘[𝐵] ∼ )) + (𝑃 pCnt (#‘𝐻))))
71	4, 54, 55, 68, 69, 70	syl122anc 1335	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt ((#‘[𝐵] ∼ ) · (#‘𝐻))) = ((𝑃 pCnt (#‘[𝐵] ∼ )) + (𝑃 pCnt (#‘𝐻))))
72	53, 71	eqtrd 2656	. . . . . . 7 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑋)) = ((𝑃 pCnt (#‘[𝐵] ∼ )) + (𝑃 pCnt (#‘𝐻))))
73	49, 72	breqtrd 4679	. . . . . 6 ⊢ (𝜑 → ((𝑃 pCnt (#‘[𝐵] ∼ )) + 𝑁) ≤ ((𝑃 pCnt (#‘[𝐵] ∼ )) + (𝑃 pCnt (#‘𝐻))))
74	4, 67	pccld 15555	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt (#‘𝐻)) ∈ ℕ₀)
75	74	nn0red 11352	. . . . . . 7 ⊢ (𝜑 → (𝑃 pCnt (#‘𝐻)) ∈ ℝ)
76	39, 75, 38	leadd2d 10622	. . . . . 6 ⊢ (𝜑 → (𝑁 ≤ (𝑃 pCnt (#‘𝐻)) ↔ ((𝑃 pCnt (#‘[𝐵] ∼ )) + 𝑁) ≤ ((𝑃 pCnt (#‘[𝐵] ∼ )) + (𝑃 pCnt (#‘𝐻)))))
77	73, 76	mpbird 247	. . . . 5 ⊢ (𝜑 → 𝑁 ≤ (𝑃 pCnt (#‘𝐻)))
78		pcdvdsb 15573	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (#‘𝐻) ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ (𝑃 pCnt (#‘𝐻)) ↔ (𝑃↑𝑁) ∥ (#‘𝐻)))
79	4, 68, 5, 78	syl3anc 1326	. . . . 5 ⊢ (𝜑 → (𝑁 ≤ (𝑃 pCnt (#‘𝐻)) ↔ (𝑃↑𝑁) ∥ (#‘𝐻)))
80	77, 79	mpbid 222	. . . 4 ⊢ (𝜑 → (𝑃↑𝑁) ∥ (#‘𝐻))
81		prmnn 15388	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
82	4, 81	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ)
83	82, 5	nnexpcld 13030	. . . . . 6 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℕ)
84	83	nnzd 11481	. . . . 5 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℤ)
85		dvdsle 15032	. . . . 5 ⊢ (((𝑃↑𝑁) ∈ ℤ ∧ (#‘𝐻) ∈ ℕ) → ((𝑃↑𝑁) ∥ (#‘𝐻) → (𝑃↑𝑁) ≤ (#‘𝐻)))
86	84, 67, 85	syl2anc 693	. . . 4 ⊢ (𝜑 → ((𝑃↑𝑁) ∥ (#‘𝐻) → (𝑃↑𝑁) ≤ (#‘𝐻)))
87	80, 86	mpd 15	. . 3 ⊢ (𝜑 → (𝑃↑𝑁) ≤ (#‘𝐻))
88		hashcl 13147	. . . . . 6 ⊢ (𝐻 ∈ Fin → (#‘𝐻) ∈ ℕ₀)
89	64, 88	syl 17	. . . . 5 ⊢ (𝜑 → (#‘𝐻) ∈ ℕ₀)
90	89	nn0red 11352	. . . 4 ⊢ (𝜑 → (#‘𝐻) ∈ ℝ)
91	83	nnred 11035	. . . 4 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℝ)
92	90, 91	letri3d 10179	. . 3 ⊢ (𝜑 → ((#‘𝐻) = (𝑃↑𝑁) ↔ ((#‘𝐻) ≤ (𝑃↑𝑁) ∧ (𝑃↑𝑁) ≤ (#‘𝐻))))
93	16, 87, 92	mpbir2and 957	. 2 ⊢ (𝜑 → (#‘𝐻) = (𝑃↑𝑁))
94		fveq2 6191	. . . 4 ⊢ (ℎ = 𝐻 → (#‘ℎ) = (#‘𝐻))
95	94	eqeq1d 2624	. . 3 ⊢ (ℎ = 𝐻 → ((#‘ℎ) = (𝑃↑𝑁) ↔ (#‘𝐻) = (𝑃↑𝑁)))
96	95	rspcev 3309	. 2 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑𝑁)) → ∃ℎ ∈ (SubGrp‘𝐺)(#‘ℎ) = (𝑃↑𝑁))
97	14, 93, 96	syl2anc 693	1 ⊢ (𝜑 → ∃ℎ ∈ (SubGrp‘𝐺)(#‘ℎ) = (𝑃↑𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 {crab 2916 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 {cpr 4179 class class class wbr 4653 {copab 4712 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 Er wer 7739 [cec 7740 Fincfn 7955 ℝcr 9935 0cc0 9936 + caddc 9939 · cmul 9941 ≤ cle 10075 − cmin 10266 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ↑cexp 12860 #chash 13117 ∥ cdvds 14983 ℙcprime 15385 pCnt cpc 15541 Basecbs 15857 +_gcplusg 15941 0_gc0g 16100 Grpcgrp 17422 SubGrpcsubg 17588 ~_QG cqg 17590 GrpAct cga 17722

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-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-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-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-grp 17425 df-minusg 17426 df-subg 17591 df-eqg 17593 df-ga 17723

This theorem is referenced by: sylow1 18018

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