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Theorem sylow3lem3 18515

Description: Lemma for sylow3 18519, first part. The number of Sylow subgroups is the same as the index (number of cosets) of the normalizer of the Sylow subgroup 𝐾. (Contributed by Mario Carneiro, 19-Jan-2015.)

**Hypotheses**
Ref	Expression
sylow3.x	⊢ 𝑋 = (Base‘𝐺)
sylow3.g	⊢ (𝜑 → 𝐺 ∈ Grp)
sylow3.xf	⊢ (𝜑 → 𝑋 ∈ Fin)
sylow3.p	⊢ (𝜑 → 𝑃 ∈ ℙ)
sylow3lem1.a	⊢ + = (+_g‘𝐺)
sylow3lem1.d	⊢ − = (-_g‘𝐺)
sylow3lem1.m	⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)))
sylow3lem2.k	⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺))
sylow3lem2.h	⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐾) = 𝐾}
sylow3lem2.n	⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)}

**Assertion**
Ref	Expression
sylow3lem3	⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) = (♯‘(𝑋 / (𝐺 ~_QG 𝑁))))

Distinct variable groups: 𝑥,𝑢,𝑦,𝑧, − 𝑢, ⊕ ,𝑥,𝑦,𝑧 𝑥,𝐻,𝑦 𝑢,𝐾,𝑥,𝑦,𝑧 𝑢,𝑁,𝑧 𝑢,𝑋,𝑥,𝑦,𝑧 𝑢,𝐺,𝑥,𝑦,𝑧 𝜑,𝑢,𝑥,𝑦,𝑧 𝑢, + ,𝑥,𝑦,𝑧 𝑢,𝑃,𝑥,𝑦,𝑧 Allowed substitution hints: 𝐻(𝑧,𝑢) 𝑁(𝑥,𝑦)

Proof of Theorem sylow3lem3 Dummy variables 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sylow3.xf	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin)
2		pwfi 8614	. . . . . 6 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
3	1, 2	sylib 210	. . . . 5 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
4		slwsubg 18496	. . . . . . 7 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ (SubGrp‘𝐺))
5		sylow3.x	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝐺)
6	5	subgss 18064	. . . . . . . 8 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ 𝑋)
7	4, 6	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ⊆ 𝑋)
8	4, 7	elpwd 4431	. . . . . 6 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ 𝒫 𝑋)
9	8	ssriv 3863	. . . . 5 ⊢ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋
10		ssfi 8533	. . . . 5 ⊢ ((𝒫 𝑋 ∈ Fin ∧ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋) → (𝑃 pSyl 𝐺) ∈ Fin)
11	3, 9, 10	sylancl 577	. . . 4 ⊢ (𝜑 → (𝑃 pSyl 𝐺) ∈ Fin)
12		hashcl 13532	. . . 4 ⊢ ((𝑃 pSyl 𝐺) ∈ Fin → (♯‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
14	13	nn0cnd 11769	. 2 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∈ ℂ)
15		sylow3.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Grp)
16		sylow3lem2.n	. . . . . . . 8 ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)}
17		sylow3lem1.a	. . . . . . . 8 ⊢ + = (+_g‘𝐺)
18	16, 5, 17	nmzsubg 18104	. . . . . . 7 ⊢ (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺))
19		eqid 2779	. . . . . . . 8 ⊢ (𝐺 ~_QG 𝑁) = (𝐺 ~_QG 𝑁)
20	5, 19	eqger 18113	. . . . . . 7 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → (𝐺 ~_QG 𝑁) Er 𝑋)
21	15, 18, 20	3syl 18	. . . . . 6 ⊢ (𝜑 → (𝐺 ~_QG 𝑁) Er 𝑋)
22	21	qsss 8158	. . . . 5 ⊢ (𝜑 → (𝑋 / (𝐺 ~_QG 𝑁)) ⊆ 𝒫 𝑋)
23	3, 22	ssfid 8536	. . . 4 ⊢ (𝜑 → (𝑋 / (𝐺 ~_QG 𝑁)) ∈ Fin)
24		hashcl 13532	. . . 4 ⊢ ((𝑋 / (𝐺 ~_QG 𝑁)) ∈ Fin → (♯‘(𝑋 / (𝐺 ~_QG 𝑁))) ∈ ℕ₀)
25	23, 24	syl 17	. . 3 ⊢ (𝜑 → (♯‘(𝑋 / (𝐺 ~_QG 𝑁))) ∈ ℕ₀)
26	25	nn0cnd 11769	. 2 ⊢ (𝜑 → (♯‘(𝑋 / (𝐺 ~_QG 𝑁))) ∈ ℂ)
27	15, 18	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺))
28		eqid 2779	. . . . . 6 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
29	28	subg0cl 18071	. . . . 5 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ 𝑁)
30		ne0i 4187	. . . . 5 ⊢ ((0_g‘𝐺) ∈ 𝑁 → 𝑁 ≠ ∅)
31	27, 29, 30	3syl 18	. . . 4 ⊢ (𝜑 → 𝑁 ≠ ∅)
32	5	subgss 18064	. . . . . . 7 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ 𝑋)
33	15, 18, 32	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑁 ⊆ 𝑋)
34	1, 33	ssfid 8536	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ Fin)
35		hashnncl 13542	. . . . 5 ⊢ (𝑁 ∈ Fin → ((♯‘𝑁) ∈ ℕ ↔ 𝑁 ≠ ∅))
36	34, 35	syl 17	. . . 4 ⊢ (𝜑 → ((♯‘𝑁) ∈ ℕ ↔ 𝑁 ≠ ∅))
37	31, 36	mpbird 249	. . 3 ⊢ (𝜑 → (♯‘𝑁) ∈ ℕ)
38	37	nncnd 11457	. 2 ⊢ (𝜑 → (♯‘𝑁) ∈ ℂ)
39	37	nnne0d 11490	. 2 ⊢ (𝜑 → (♯‘𝑁) ≠ 0)
40		sylow3.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℙ)
41		sylow3lem1.d	. . . . 5 ⊢ − = (-_g‘𝐺)
42		sylow3lem1.m	. . . . 5 ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)))
43	5, 15, 1, 40, 17, 41, 42	sylow3lem1 18513	. . . 4 ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct (𝑃 pSyl 𝐺)))
44		sylow3lem2.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺))
45		sylow3lem2.h	. . . . 5 ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐾) = 𝐾}
46		eqid 2779	. . . . 5 ⊢ (𝐺 ~_QG 𝐻) = (𝐺 ~_QG 𝐻)
47		eqid 2779	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
48	5, 45, 46, 47	orbsta2 18215	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct (𝑃 pSyl 𝐺)) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘[𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}) · (♯‘𝐻)))
49	43, 44, 1, 48	syl21anc 825	. . 3 ⊢ (𝜑 → (♯‘𝑋) = ((♯‘[𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}) · (♯‘𝐻)))
50	5, 19, 27, 1	lagsubg2 18124	. . 3 ⊢ (𝜑 → (♯‘𝑋) = ((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝑁)))
51	47, 5	gaorber 18209	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct (𝑃 pSyl 𝐺)) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} Er (𝑃 pSyl 𝐺))
52	43, 51	syl 17	. . . . . . 7 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} Er (𝑃 pSyl 𝐺))
53	52	ecss 8135	. . . . . 6 ⊢ (𝜑 → [𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} ⊆ (𝑃 pSyl 𝐺))
54	44	adantr 473	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → 𝐾 ∈ (𝑃 pSyl 𝐺))
55		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → ℎ ∈ (𝑃 pSyl 𝐺))
56	1	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → 𝑋 ∈ Fin)
57	5, 56, 55, 54, 17, 41	sylow2 18512	. . . . . . . . 9 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → ∃𝑢 ∈ 𝑋 ℎ = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)))
58		eqcom 2786	. . . . . . . . . . 11 ⊢ ((𝑢 ⊕ 𝐾) = ℎ ↔ ℎ = (𝑢 ⊕ 𝐾))
59		simpr 477	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → 𝑢 ∈ 𝑋)
60	54	adantr 473	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → 𝐾 ∈ (𝑃 pSyl 𝐺))
61		mptexg 6810	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)) ∈ V)
62		rnexg 7429	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)) ∈ V → ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)) ∈ V)
63	60, 61, 62	3syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)) ∈ V)
64		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → 𝑦 = 𝐾)
65		simpl 475	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → 𝑥 = 𝑢)
66	65	oveq1d 6991	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → (𝑥 + 𝑧) = (𝑢 + 𝑧))
67	66, 65	oveq12d 6994	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → ((𝑥 + 𝑧) − 𝑥) = ((𝑢 + 𝑧) − 𝑢))
68	64, 67	mpteq12dv 5012	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)))
69	68	rneqd 5651	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)))
70	69, 42	ovmpoga 7120	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)) ∈ V) → (𝑢 ⊕ 𝐾) = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)))
71	59, 60, 63, 70	syl3anc 1351	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → (𝑢 ⊕ 𝐾) = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)))
72	71	eqeq2d 2789	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → (ℎ = (𝑢 ⊕ 𝐾) ↔ ℎ = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢))))
73	58, 72	syl5bb 275	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → ((𝑢 ⊕ 𝐾) = ℎ ↔ ℎ = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢))))
74	73	rexbidva 3242	. . . . . . . . 9 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → (∃𝑢 ∈ 𝑋 (𝑢 ⊕ 𝐾) = ℎ ↔ ∃𝑢 ∈ 𝑋 ℎ = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢))))
75	57, 74	mpbird 249	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → ∃𝑢 ∈ 𝑋 (𝑢 ⊕ 𝐾) = ℎ)
76	47	gaorb 18208	. . . . . . . 8 ⊢ (𝐾{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}ℎ ↔ (𝐾 ∈ (𝑃 pSyl 𝐺) ∧ ℎ ∈ (𝑃 pSyl 𝐺) ∧ ∃𝑢 ∈ 𝑋 (𝑢 ⊕ 𝐾) = ℎ))
77	54, 55, 75, 76	syl3anbrc 1323	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → 𝐾{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}ℎ)
78		elecg 8132	. . . . . . . 8 ⊢ ((ℎ ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺)) → (ℎ ∈ [𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} ↔ 𝐾{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}ℎ))
79	55, 54, 78	syl2anc 576	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → (ℎ ∈ [𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} ↔ 𝐾{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}ℎ))
80	77, 79	mpbird 249	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → ℎ ∈ [𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)})
81	53, 80	eqelssd 3879	. . . . 5 ⊢ (𝜑 → [𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} = (𝑃 pSyl 𝐺))
82	81	fveq2d 6503	. . . 4 ⊢ (𝜑 → (♯‘[𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}) = (♯‘(𝑃 pSyl 𝐺)))
83	5, 15, 1, 40, 17, 41, 42, 44, 45, 16	sylow3lem2 18514	. . . . 5 ⊢ (𝜑 → 𝐻 = 𝑁)
84	83	fveq2d 6503	. . . 4 ⊢ (𝜑 → (♯‘𝐻) = (♯‘𝑁))
85	82, 84	oveq12d 6994	. . 3 ⊢ (𝜑 → ((♯‘[𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}) · (♯‘𝐻)) = ((♯‘(𝑃 pSyl 𝐺)) · (♯‘𝑁)))
86	49, 50, 85	3eqtr3rd 2824	. 2 ⊢ (𝜑 → ((♯‘(𝑃 pSyl 𝐺)) · (♯‘𝑁)) = ((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝑁)))
87	14, 26, 38, 39, 86	mulcan2ad 11077	1 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) = (♯‘(𝑋 / (𝐺 ~_QG 𝑁))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ≠ wne 2968 ∀wral 3089 ∃wrex 3090 {crab 3093 Vcvv 3416 ⊆ wss 3830 ∅c0 4179 𝒫 cpw 4422 {cpr 4443 class class class wbr 4929 {copab 4991 ↦ cmpt 5008 ran crn 5408 ‘cfv 6188 (class class class)co 6976 ∈ cmpo 6978 Er wer 8086 [cec 8087 / cqs 8088 Fincfn 8306 · cmul 10340 ℕcn 11439 ℕ₀cn0 11707 ♯chash 13505 ℙcprime 15871 Basecbs 16339 +_gcplusg 16421 0_gc0g 16569 Grpcgrp 17891 -_gcsg 17893 SubGrpcsubg 18057 ~_QG cqg 18059 GrpAct cga 18190 pSyl cslw 18417

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-disj 4898 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-oadd 7909 df-omul 7910 df-er 8089 df-ec 8091 df-qs 8095 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-sup 8701 df-inf 8702 df-oi 8769 df-dju 9124 df-card 9162 df-acn 9165 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-n0 11708 df-xnn0 11780 df-z 11794 df-uz 12059 df-q 12163 df-rp 12205 df-fz 12709 df-fzo 12850 df-fl 12977 df-mod 13053 df-seq 13185 df-exp 13245 df-fac 13449 df-bc 13478 df-hash 13506 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-clim 14706 df-sum 14904 df-dvds 15468 df-gcd 15704 df-prm 15872 df-pc 16030 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-0g 16571 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-submnd 17804 df-grp 17894 df-minusg 17895 df-sbg 17896 df-mulg 18012 df-subg 18060 df-eqg 18062 df-ghm 18127 df-ga 18191 df-od 18418 df-pgp 18420 df-slw 18421

This theorem is referenced by: sylow3lem4 18516

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