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Theorem sylow3lem3 19647

Description: Lemma for sylow3 19651, 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 9357	. . . . . 6 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
3	1, 2	sylib 218	. . . . 5 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
4		slwsubg 19628	. . . . . . 7 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ (SubGrp‘𝐺))
5		sylow3.x	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝐺)
6	5	subgss 19145	. . . . . . . 8 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ 𝑋)
7	4, 6	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ⊆ 𝑋)
8	4, 7	elpwd 4606	. . . . . 6 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ 𝒫 𝑋)
9	8	ssriv 3987	. . . . 5 ⊢ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋
10		ssfi 9213	. . . . 5 ⊢ ((𝒫 𝑋 ∈ Fin ∧ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋) → (𝑃 pSyl 𝐺) ∈ Fin)
11	3, 9, 10	sylancl 586	. . . 4 ⊢ (𝜑 → (𝑃 pSyl 𝐺) ∈ Fin)
12		hashcl 14395	. . . 4 ⊢ ((𝑃 pSyl 𝐺) ∈ Fin → (♯‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
14	13	nn0cnd 12589	. 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 19183	. . . . . . 7 ⊢ (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺))
19		eqid 2737	. . . . . . . 8 ⊢ (𝐺 ~_QG 𝑁) = (𝐺 ~_QG 𝑁)
20	5, 19	eqger 19196	. . . . . . 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 8818	. . . . 5 ⊢ (𝜑 → (𝑋 / (𝐺 ~_QG 𝑁)) ⊆ 𝒫 𝑋)
23	3, 22	ssfid 9301	. . . 4 ⊢ (𝜑 → (𝑋 / (𝐺 ~_QG 𝑁)) ∈ Fin)
24		hashcl 14395	. . . 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 12589	. 2 ⊢ (𝜑 → (♯‘(𝑋 / (𝐺 ~_QG 𝑁))) ∈ ℂ)
27	15, 18	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺))
28		eqid 2737	. . . . . 6 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
29	28	subg0cl 19152	. . . . 5 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ 𝑁)
30		ne0i 4341	. . . . 5 ⊢ ((0_g‘𝐺) ∈ 𝑁 → 𝑁 ≠ ∅)
31	27, 29, 30	3syl 18	. . . 4 ⊢ (𝜑 → 𝑁 ≠ ∅)
32	5	subgss 19145	. . . . . . 7 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ 𝑋)
33	15, 18, 32	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑁 ⊆ 𝑋)
34	1, 33	ssfid 9301	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ Fin)
35		hashnncl 14405	. . . . 5 ⊢ (𝑁 ∈ Fin → ((♯‘𝑁) ∈ ℕ ↔ 𝑁 ≠ ∅))
36	34, 35	syl 17	. . . 4 ⊢ (𝜑 → ((♯‘𝑁) ∈ ℕ ↔ 𝑁 ≠ ∅))
37	31, 36	mpbird 257	. . 3 ⊢ (𝜑 → (♯‘𝑁) ∈ ℕ)
38	37	nncnd 12282	. 2 ⊢ (𝜑 → (♯‘𝑁) ∈ ℂ)
39	37	nnne0d 12316	. 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 19645	. . . 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 2737	. . . . 5 ⊢ (𝐺 ~_QG 𝐻) = (𝐺 ~_QG 𝐻)
47		eqid 2737	. . . . 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 19332	. . . 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 838	. . 3 ⊢ (𝜑 → (♯‘𝑋) = ((♯‘[𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}) · (♯‘𝐻)))
50	5, 19, 27, 1	lagsubg2 19212	. . 3 ⊢ (𝜑 → (♯‘𝑋) = ((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝑁)))
51	47, 5	gaorber 19326	. . . . . . . 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 8793	. . . . . 6 ⊢ (𝜑 → [𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} ⊆ (𝑃 pSyl 𝐺))
54	44	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → 𝐾 ∈ (𝑃 pSyl 𝐺))
55		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → ℎ ∈ (𝑃 pSyl 𝐺))
56	1	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → 𝑋 ∈ Fin)
57	5, 56, 55, 54, 17, 41	sylow2 19644	. . . . . . . . 9 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → ∃𝑢 ∈ 𝑋 ℎ = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)))
58		eqcom 2744	. . . . . . . . . . 11 ⊢ ((𝑢 ⊕ 𝐾) = ℎ ↔ ℎ = (𝑢 ⊕ 𝐾))
59		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → 𝑢 ∈ 𝑋)
60	54	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → 𝐾 ∈ (𝑃 pSyl 𝐺))
61		mptexg 7241	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)) ∈ V)
62		rnexg 7924	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → 𝑦 = 𝐾)
65		simpl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → 𝑥 = 𝑢)
66	65	oveq1d 7446	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → (𝑥 + 𝑧) = (𝑢 + 𝑧))
67	66, 65	oveq12d 7449	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → ((𝑥 + 𝑧) − 𝑥) = ((𝑢 + 𝑧) − 𝑢))
68	64, 67	mpteq12dv 5233	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)))
69	68	rneqd 5949	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)))
70	69, 42	ovmpoga 7587	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → (𝑢 ⊕ 𝐾) = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)))
72	71	eqeq2d 2748	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → (ℎ = (𝑢 ⊕ 𝐾) ↔ ℎ = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢))))
73	58, 72	bitrid 283	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → ((𝑢 ⊕ 𝐾) = ℎ ↔ ℎ = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢))))
74	73	rexbidva 3177	. . . . . . . . 9 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → (∃𝑢 ∈ 𝑋 (𝑢 ⊕ 𝐾) = ℎ ↔ ∃𝑢 ∈ 𝑋 ℎ = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢))))
75	57, 74	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → ∃𝑢 ∈ 𝑋 (𝑢 ⊕ 𝐾) = ℎ)
76	47	gaorb 19325	. . . . . . . 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 1344	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → 𝐾{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}ℎ)
78		elecg 8789	. . . . . . . 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 584	. . . . . . 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 257	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → ℎ ∈ [𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)})
81	53, 80	eqelssd 4005	. . . . 5 ⊢ (𝜑 → [𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} = (𝑃 pSyl 𝐺))
82	81	fveq2d 6910	. . . 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 19646	. . . . 5 ⊢ (𝜑 → 𝐻 = 𝑁)
84	83	fveq2d 6910	. . . 4 ⊢ (𝜑 → (♯‘𝐻) = (♯‘𝑁))
85	82, 84	oveq12d 7449	. . 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 2786	. 2 ⊢ (𝜑 → ((♯‘(𝑃 pSyl 𝐺)) · (♯‘𝑁)) = ((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝑁)))
87	14, 26, 38, 39, 86	mulcan2ad 11899	1 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) = (♯‘(𝑋 / (𝐺 ~_QG 𝑁))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {cpr 4628 class class class wbr 5143 {copab 5205 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 Er wer 8742 [cec 8743 / cqs 8744 Fincfn 8985 · cmul 11160 ℕcn 12266 ℕ₀cn0 12526 ♯chash 14369 ℙcprime 16708 Basecbs 17247 +_gcplusg 17297 0_gc0g 17484 Grpcgrp 18951 -_gcsg 18953 SubGrpcsubg 19138 ~_QG cqg 19140 GrpAct cga 19307 pSyl cslw 19545

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-ec 8747 df-qs 8751 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-dvds 16291 df-gcd 16532 df-prm 16709 df-pc 16875 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-eqg 19143 df-ghm 19231 df-ga 19308 df-od 19546 df-pgp 19548 df-slw 19549

This theorem is referenced by: sylow3lem4 19648

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