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Theorem sylow3lem3 19604

Description: Lemma for sylow3 19608, 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 9229	. . . . . 6 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
3	1, 2	sylib 218	. . . . 5 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
4		slwsubg 19585	. . . . . . 7 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ (SubGrp‘𝐺))
5		sylow3.x	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝐺)
6	5	subgss 19103	. . . . . . . 8 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ 𝑋)
7	4, 6	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ⊆ 𝑋)
8	4, 7	elpwd 4547	. . . . . 6 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ 𝒫 𝑋)
9	8	ssriv 3925	. . . . 5 ⊢ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋
10		ssfi 9107	. . . . 5 ⊢ ((𝒫 𝑋 ∈ Fin ∧ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋) → (𝑃 pSyl 𝐺) ∈ Fin)
11	3, 9, 10	sylancl 587	. . . 4 ⊢ (𝜑 → (𝑃 pSyl 𝐺) ∈ Fin)
12		hashcl 14318	. . . 4 ⊢ ((𝑃 pSyl 𝐺) ∈ Fin → (♯‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
14	13	nn0cnd 12500	. 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 19140	. . . . . . 7 ⊢ (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺))
19		eqid 2736	. . . . . . . 8 ⊢ (𝐺 ~_QG 𝑁) = (𝐺 ~_QG 𝑁)
20	5, 19	eqger 19153	. . . . . . 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 8722	. . . . 5 ⊢ (𝜑 → (𝑋 / (𝐺 ~_QG 𝑁)) ⊆ 𝒫 𝑋)
23	3, 22	ssfid 9179	. . . 4 ⊢ (𝜑 → (𝑋 / (𝐺 ~_QG 𝑁)) ∈ Fin)
24		hashcl 14318	. . . 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 12500	. 2 ⊢ (𝜑 → (♯‘(𝑋 / (𝐺 ~_QG 𝑁))) ∈ ℂ)
27	15, 18	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺))
28		eqid 2736	. . . . . 6 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
29	28	subg0cl 19110	. . . . 5 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ 𝑁)
30		ne0i 4281	. . . . 5 ⊢ ((0_g‘𝐺) ∈ 𝑁 → 𝑁 ≠ ∅)
31	27, 29, 30	3syl 18	. . . 4 ⊢ (𝜑 → 𝑁 ≠ ∅)
32	5	subgss 19103	. . . . . . 7 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ 𝑋)
33	15, 18, 32	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑁 ⊆ 𝑋)
34	1, 33	ssfid 9179	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ Fin)
35		hashnncl 14328	. . . . 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 12190	. 2 ⊢ (𝜑 → (♯‘𝑁) ∈ ℂ)
39	37	nnne0d 12227	. 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 19602	. . . 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 2736	. . . . 5 ⊢ (𝐺 ~_QG 𝐻) = (𝐺 ~_QG 𝐻)
47		eqid 2736	. . . . 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 19289	. . . 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 19169	. . 3 ⊢ (𝜑 → (♯‘𝑋) = ((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝑁)))
51	47, 5	gaorber 19283	. . . . . . . 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 8695	. . . . . 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 19601	. . . . . . . . 9 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → ∃𝑢 ∈ 𝑋 ℎ = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)))
58		eqcom 2743	. . . . . . . . . . 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 7176	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)) ∈ V)
62		rnexg 7853	. . . . . . . . . . . . . 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 7382	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → (𝑥 + 𝑧) = (𝑢 + 𝑧))
67	66, 65	oveq12d 7385	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → ((𝑥 + 𝑧) − 𝑥) = ((𝑢 + 𝑧) − 𝑢))
68	64, 67	mpteq12dv 5172	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)))
69	68	rneqd 5893	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)))
70	69, 42	ovmpoga 7521	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → (𝑢 ⊕ 𝐾) = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)))
72	71	eqeq2d 2747	. . . . . . . . . . 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 3159	. . . . . . . . 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 19282	. . . . . . . 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 1345	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → 𝐾{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}ℎ)
78		elecg 8688	. . . . . . . 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 585	. . . . . . 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 3943	. . . . 5 ⊢ (𝜑 → [𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} = (𝑃 pSyl 𝐺))
82	81	fveq2d 6844	. . . 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 19603	. . . . 5 ⊢ (𝜑 → 𝐻 = 𝑁)
84	83	fveq2d 6844	. . . 4 ⊢ (𝜑 → (♯‘𝐻) = (♯‘𝑁))
85	82, 84	oveq12d 7385	. . 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 2780	. 2 ⊢ (𝜑 → ((♯‘(𝑃 pSyl 𝐺)) · (♯‘𝑁)) = ((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝑁)))
87	14, 26, 38, 39, 86	mulcan2ad 11786	1 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) = (♯‘(𝑋 / (𝐺 ~_QG 𝑁))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 {crab 3389 Vcvv 3429 ⊆ wss 3889 ∅c0 4273 𝒫 cpw 4541 {cpr 4569 class class class wbr 5085 {copab 5147 ↦ cmpt 5166 ran crn 5632 ‘cfv 6498 (class class class)co 7367 ∈ cmpo 7369 Er wer 8640 [cec 8641 / cqs 8642 Fincfn 8893 · cmul 11043 ℕcn 12174 ℕ₀cn0 12437 ♯chash 14292 ℙcprime 16640 Basecbs 17179 +_gcplusg 17220 0_gc0g 17402 Grpcgrp 18909 -_gcsg 18911 SubGrpcsubg 19096 ~_QG cqg 19098 GrpAct cga 19264 pSyl cslw 19502

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-disj 5053 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-ec 8645 df-qs 8649 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-xnn0 12511 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-sum 15649 df-dvds 16222 df-gcd 16464 df-prm 16641 df-pc 16808 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-0g 17404 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-eqg 19101 df-ghm 19188 df-ga 19265 df-od 19503 df-pgp 19505 df-slw 19506

This theorem is referenced by: sylow3lem4 19605

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