Theorem sylow1lem5 17840

Description: Lemma for sylow1 17841. 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 17837	. . 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 17565	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑆) ∧ 𝐵 ∈ 𝑆) → 𝐻 ∈ (SubGrp‘𝐺))
14	10, 11, 13	syl2anc 691	. 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 17839	. . 3 ⊢ (𝜑 → (#‘𝐻) ≤ (𝑃↑𝑁))
17		sylow1lem5.l	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt (#‘[𝐵] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
18	15, 1	gaorber 17564	. . . . . . . . . . . . . . . 16 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑆) → ∼ Er 𝑆)
19	10, 18	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∼ Er 𝑆)
20		erdm 7639	. . . . . . . . . . . . . . 15 ⊢ ( ∼ Er 𝑆 → dom ∼ = 𝑆)
21	19, 20	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom ∼ = 𝑆)
22	11, 21	eleqtrrd 2691	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ dom ∼ )
23		ecdmn0 7676	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ dom ∼ ↔ [𝐵] ∼ ≠ ∅)
24	22, 23	sylib 207	. . . . . . . . . . . 12 ⊢ (𝜑 → [𝐵] ∼ ≠ ∅)
25		pwfi 8144	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
26	3, 25	sylib 207	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
27		ssrab2 3650	. . . . . . . . . . . . . . . 16 ⊢ {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃↑𝑁)} ⊆ 𝒫 𝑋
28	8, 27	eqsstri 3598	. . . . . . . . . . . . . . 15 ⊢ 𝑆 ⊆ 𝒫 𝑋
29		ssfi 8065	. . . . . . . . . . . . . . 15 ⊢ ((𝒫 𝑋 ∈ Fin ∧ 𝑆 ⊆ 𝒫 𝑋) → 𝑆 ∈ Fin)
30	26, 28, 29	sylancl 693	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ Fin)
31	19	ecss 7675	. . . . . . . . . . . . . 14 ⊢ (𝜑 → [𝐵] ∼ ⊆ 𝑆)
32		ssfi 8065	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ Fin ∧ [𝐵] ∼ ⊆ 𝑆) → [𝐵] ∼ ∈ Fin)
33	30, 31, 32	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (𝜑 → [𝐵] ∼ ∈ Fin)
34		hashnncl 13018	. . . . . . . . . . . . 13 ⊢ ([𝐵] ∼ ∈ Fin → ((#‘[𝐵] ∼ ) ∈ ℕ ↔ [𝐵] ∼ ≠ ∅))
35	33, 34	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((#‘[𝐵] ∼ ) ∈ ℕ ↔ [𝐵] ∼ ≠ ∅))
36	24, 35	mpbird 246	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘[𝐵] ∼ ) ∈ ℕ)
37	4, 36	pccld 15393	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (#‘[𝐵] ∼ )) ∈ ℕ0)
38	37	nn0red 11229	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 pCnt (#‘[𝐵] ∼ )) ∈ ℝ)
39	5	nn0red 11229	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
40	1	grpbn0 17274	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
41	2, 40	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ≠ ∅)
42		hashnncl 13018	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ Fin → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
43	3, 42	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
44	41, 43	mpbird 246	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝑋) ∈ ℕ)
45	4, 44	pccld 15393	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℕ0)
46	45	nn0red 11229	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℝ)
47		leaddsub 10383	. . . . . . . . 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 1318	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 pCnt (#‘[𝐵] ∼ )) + 𝑁) ≤ (𝑃 pCnt (#‘𝑋)) ↔ (𝑃 pCnt (#‘[𝐵] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
49	17, 48	mpbird 246	. . . . . . 7 ⊢ (𝜑 → ((𝑃 pCnt (#‘[𝐵] ∼ )) + 𝑁) ≤ (𝑃 pCnt (#‘𝑋)))
50		eqid 2610	. . . . . . . . . . 11 ⊢ (𝐺 ~QG 𝐻) = (𝐺 ~QG 𝐻)
51	1, 12, 50, 15	orbsta2 17570	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑆) ∧ 𝐵 ∈ 𝑆) ∧ 𝑋 ∈ Fin) → (#‘𝑋) = ((#‘[𝐵] ∼ ) · (#‘𝐻)))
52	10, 11, 3, 51	syl21anc 1317	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝑋) = ((#‘[𝐵] ∼ ) · (#‘𝐻)))
53	52	oveq2d 6565	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑋)) = (𝑃 pCnt ((#‘[𝐵] ∼ ) · (#‘𝐻))))
54	36	nnzd 11357	. . . . . . . . 9 ⊢ (𝜑 → (#‘[𝐵] ∼ ) ∈ ℤ)
55	36	nnne0d 10942	. . . . . . . . 9 ⊢ (𝜑 → (#‘[𝐵] ∼ ) ≠ 0)
56		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (0g‘𝐺) = (0g‘𝐺)
57	56	subg0cl 17425	. . . . . . . . . . . . 13 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝐻)
58	14, 57	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝐻)
59		ne0i 3880	. . . . . . . . . . . 12 ⊢ ((0g‘𝐺) ∈ 𝐻 → 𝐻 ≠ ∅)
60	58, 59	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ≠ ∅)
61		ssrab2 3650	. . . . . . . . . . . . . 14 ⊢ {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐵) = 𝐵} ⊆ 𝑋
62	12, 61	eqsstri 3598	. . . . . . . . . . . . 13 ⊢ 𝐻 ⊆ 𝑋
63		ssfi 8065	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ Fin ∧ 𝐻 ⊆ 𝑋) → 𝐻 ∈ Fin)
64	3, 62, 63	sylancl 693	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ Fin)
65		hashnncl 13018	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ Fin → ((#‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
66	64, 65	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((#‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
67	60, 66	mpbird 246	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝐻) ∈ ℕ)
68	67	nnzd 11357	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝐻) ∈ ℤ)
69	67	nnne0d 10942	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝐻) ≠ 0)
70		pcmul 15394	. . . . . . . . 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 1327	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt ((#‘[𝐵] ∼ ) · (#‘𝐻))) = ((𝑃 pCnt (#‘[𝐵] ∼ )) + (𝑃 pCnt (#‘𝐻))))
72	53, 71	eqtrd 2644	. . . . . . 7 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑋)) = ((𝑃 pCnt (#‘[𝐵] ∼ )) + (𝑃 pCnt (#‘𝐻))))
73	49, 72	breqtrd 4609	. . . . . 6 ⊢ (𝜑 → ((𝑃 pCnt (#‘[𝐵] ∼ )) + 𝑁) ≤ ((𝑃 pCnt (#‘[𝐵] ∼ )) + (𝑃 pCnt (#‘𝐻))))
74	4, 67	pccld 15393	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt (#‘𝐻)) ∈ ℕ0)
75	74	nn0red 11229	. . . . . . 7 ⊢ (𝜑 → (𝑃 pCnt (#‘𝐻)) ∈ ℝ)
76	39, 75, 38	leadd2d 10501	. . . . . 6 ⊢ (𝜑 → (𝑁 ≤ (𝑃 pCnt (#‘𝐻)) ↔ ((𝑃 pCnt (#‘[𝐵] ∼ )) + 𝑁) ≤ ((𝑃 pCnt (#‘[𝐵] ∼ )) + (𝑃 pCnt (#‘𝐻)))))
77	73, 76	mpbird 246	. . . . 5 ⊢ (𝜑 → 𝑁 ≤ (𝑃 pCnt (#‘𝐻)))
78		pcdvdsb 15411	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (#‘𝐻) ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝑁 ≤ (𝑃 pCnt (#‘𝐻)) ↔ (𝑃↑𝑁) ∥ (#‘𝐻)))
79	4, 68, 5, 78	syl3anc 1318	. . . . 5 ⊢ (𝜑 → (𝑁 ≤ (𝑃 pCnt (#‘𝐻)) ↔ (𝑃↑𝑁) ∥ (#‘𝐻)))
80	77, 79	mpbid 221	. . . 4 ⊢ (𝜑 → (𝑃↑𝑁) ∥ (#‘𝐻))
81		prmnn 15226	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
82	4, 81	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ)
83	82, 5	nnexpcld 12892	. . . . . 6 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℕ)
84	83	nnzd 11357	. . . . 5 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℤ)
85		dvdsle 14870	. . . . 5 ⊢ (((𝑃↑𝑁) ∈ ℤ ∧ (#‘𝐻) ∈ ℕ) → ((𝑃↑𝑁) ∥ (#‘𝐻) → (𝑃↑𝑁) ≤ (#‘𝐻)))
86	84, 67, 85	syl2anc 691	. . . 4 ⊢ (𝜑 → ((𝑃↑𝑁) ∥ (#‘𝐻) → (𝑃↑𝑁) ≤ (#‘𝐻)))
87	80, 86	mpd 15	. . 3 ⊢ (𝜑 → (𝑃↑𝑁) ≤ (#‘𝐻))
88		hashcl 13009	. . . . . 6 ⊢ (𝐻 ∈ Fin → (#‘𝐻) ∈ ℕ0)
89	64, 88	syl 17	. . . . 5 ⊢ (𝜑 → (#‘𝐻) ∈ ℕ0)
90	89	nn0red 11229	. . . 4 ⊢ (𝜑 → (#‘𝐻) ∈ ℝ)
91	83	nnred 10912	. . . 4 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℝ)
92	90, 91	letri3d 10058	. . 3 ⊢ (𝜑 → ((#‘𝐻) = (𝑃↑𝑁) ↔ ((#‘𝐻) ≤ (𝑃↑𝑁) ∧ (𝑃↑𝑁) ≤ (#‘𝐻))))
93	16, 87, 92	mpbir2and 959	. 2 ⊢ (𝜑 → (#‘𝐻) = (𝑃↑𝑁))
94		fveq2 6103	. . . 4 ⊢ (ℎ = 𝐻 → (#‘ℎ) = (#‘𝐻))
95	94	eqeq1d 2612	. . 3 ⊢ (ℎ = 𝐻 → ((#‘ℎ) = (𝑃↑𝑁) ↔ (#‘𝐻) = (𝑃↑𝑁)))
96	95	rspcev 3282	. 2 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑𝑁)) → ∃ℎ ∈ (SubGrp‘𝐺)(#‘ℎ) = (𝑃↑𝑁))
97	14, 93, 96	syl2anc 691	1 ⊢ (𝜑 → ∃ℎ ∈ (SubGrp‘𝐺)(#‘ℎ) = (𝑃↑𝑁))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {cpr 4127   class class class wbr 4583  {copab 4642   ↦ cmpt 4643  dom cdm 5038  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   Er wer 7626  [cec 7627  Fincfn 7841  ℝcr 9814  0cc0 9815   + caddc 9818   · cmul 9820   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℕ₀cn0 11169  ℤcz 11254  ↑cexp 12722  #chash 12979   ∥ cdvds 14821  ℙcprime 15223   pCnt cpc 15379  Basecbs 15695  +_gcplusg 15768  0_gc0g 15923  Grpcgrp 17245  SubGrpcsubg 17411   ~_QG cqg 17413   GrpAct cga 17545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-dvds 14822  df-gcd 15055  df-prm 15224  df-pc 15380  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-subg 17414  df-eqg 17416  df-ga 17546

This theorem is referenced by:  sylow1  17841