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Theorem sylow3lem3 19894

Description: Lemma for sylow3 19898, 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 9348	. . . . . 6 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
3	1, 2	sylib 221	. . . . 5 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
4		slwsubg 19875	. . . . . . 7 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ (SubGrp‘𝐺))
5		sylow3.x	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝐺)
6	5	subgss 19387	. . . . . . . 8 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ 𝑋)
7	4, 6	syl 18	. . . . . . 7 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ⊆ 𝑋)
8	4, 7	elpwd 4566	. . . . . 6 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ 𝒫 𝑋)
9	8	ssriv 3917	. . . . 5 ⊢ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋
10		ssfi 9226	. . . . 5 ⊢ ((𝒫 𝑋 ∈ Fin ∧ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋) → (𝑃 pSyl 𝐺) ∈ Fin)
11	3, 9, 10	sylancl 598	. . . 4 ⊢ (𝜑 → (𝑃 pSyl 𝐺) ∈ Fin)
12		hashcl 14541	. . . 4 ⊢ ((𝑃 pSyl 𝐺) ∈ Fin → (♯‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
13	11, 12	syl 18	. . 3 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
14	13	nn0cnd 12709	. 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 19425	. . . . . . 7 ⊢ (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺))
19		eqid 2763	. . . . . . . 8 ⊢ (𝐺 ~_QG 𝑁) = (𝐺 ~_QG 𝑁)
20	5, 19	eqger 19440	. . . . . . 7 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → (𝐺 ~_QG 𝑁) Er 𝑋)
21	15, 18, 20	3syl 19	. . . . . 6 ⊢ (𝜑 → (𝐺 ~_QG 𝑁) Er 𝑋)
22	21	qsss 8834	. . . . 5 ⊢ (𝜑 → (𝑋 / (𝐺 ~_QG 𝑁)) ⊆ 𝒫 𝑋)
23	3, 22	ssfid 9298	. . . 4 ⊢ (𝜑 → (𝑋 / (𝐺 ~_QG 𝑁)) ∈ Fin)
24		hashcl 14541	. . . 4 ⊢ ((𝑋 / (𝐺 ~_QG 𝑁)) ∈ Fin → (♯‘(𝑋 / (𝐺 ~_QG 𝑁))) ∈ ℕ₀)
25	23, 24	syl 18	. . 3 ⊢ (𝜑 → (♯‘(𝑋 / (𝐺 ~_QG 𝑁))) ∈ ℕ₀)
26	25	nn0cnd 12709	. 2 ⊢ (𝜑 → (♯‘(𝑋 / (𝐺 ~_QG 𝑁))) ∈ ℂ)
27	15, 18	syl 18	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺))
28		eqid 2763	. . . . . 6 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
29	28	subg0cl 19394	. . . . 5 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ 𝑁)
30		ne0i 4290	. . . . 5 ⊢ ((0_g‘𝐺) ∈ 𝑁 → 𝑁 ≠ ∅)
31	27, 29, 30	3syl 19	. . . 4 ⊢ (𝜑 → 𝑁 ≠ ∅)
32	5	subgss 19387	. . . . . . 7 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ 𝑋)
33	15, 18, 32	3syl 19	. . . . . 6 ⊢ (𝜑 → 𝑁 ⊆ 𝑋)
34	1, 33	ssfid 9298	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ Fin)
35		hashnncl 14551	. . . . 5 ⊢ (𝑁 ∈ Fin → ((♯‘𝑁) ∈ ℕ ↔ 𝑁 ≠ ∅))
36	34, 35	syl 18	. . . 4 ⊢ (𝜑 → ((♯‘𝑁) ∈ ℕ ↔ 𝑁 ≠ ∅))
37	31, 36	mpbird 260	. . 3 ⊢ (𝜑 → (♯‘𝑁) ∈ ℕ)
38	37	nncnd 12391	. 2 ⊢ (𝜑 → (♯‘𝑁) ∈ ℂ)
39	37	nnne0d 12428	. 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 19892	. . . 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 2763	. . . . 5 ⊢ (𝐺 ~_QG 𝐻) = (𝐺 ~_QG 𝐻)
47		eqid 2763	. . . . 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 19579	. . . 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 851	. . 3 ⊢ (𝜑 → (♯‘𝑋) = ((♯‘[𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}) · (♯‘𝐻)))
50	5, 19, 27, 1	lagsubg2 19459	. . 3 ⊢ (𝜑 → (♯‘𝑋) = ((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝑁)))
51	47, 5	gaorber 19573	. . . . . . . 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 18	. . . . . . 7 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} Er (𝑃 pSyl 𝐺))
53	52	ecss 8807	. . . . . 6 ⊢ (𝜑 → [𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} ⊆ (𝑃 pSyl 𝐺))
54	44	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → 𝐾 ∈ (𝑃 pSyl 𝐺))
55		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → ℎ ∈ (𝑃 pSyl 𝐺))
56	1	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → 𝑋 ∈ Fin)
57	5, 56, 55, 54, 17, 41	sylow2 19891	. . . . . . . . 9 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → ∃𝑢 ∈ 𝑋 ℎ = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)))
58		eqcom 2770	. . . . . . . . . . 11 ⊢ ((𝑢 ⊕ 𝐾) = ℎ ↔ ℎ = (𝑢 ⊕ 𝐾))
59		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → 𝑢 ∈ 𝑋)
60	54	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → 𝐾 ∈ (𝑃 pSyl 𝐺))
61		mptexg 7258	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)) ∈ V)
62		rnexg 7948	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)) ∈ V → ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)) ∈ V)
63	60, 61, 62	3syl 19	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)) ∈ V)
64		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → 𝑦 = 𝐾)
65		simpl 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → 𝑥 = 𝑢)
66	65	oveq1d 7466	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → (𝑥 + 𝑧) = (𝑢 + 𝑧))
67	66, 65	oveq12d 7469	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → ((𝑥 + 𝑧) − 𝑥) = ((𝑢 + 𝑧) − 𝑢))
68	64, 67	mpteq12dv 5198	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)))
69	68	rneqd 5932	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝐾) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)))
70	69, 42	ovmpoga 7606	. . . . . . . . . . . . 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 1399	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → (𝑢 ⊕ 𝐾) = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢)))
72	71	eqeq2d 2774	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → (ℎ = (𝑢 ⊕ 𝐾) ↔ ℎ = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢))))
73	58, 72	bitrid 286	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) ∧ 𝑢 ∈ 𝑋) → ((𝑢 ⊕ 𝐾) = ℎ ↔ ℎ = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢))))
74	73	rexbidva 3188	. . . . . . . . 9 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → (∃𝑢 ∈ 𝑋 (𝑢 ⊕ 𝐾) = ℎ ↔ ∃𝑢 ∈ 𝑋 ℎ = ran (𝑧 ∈ 𝐾 ↦ ((𝑢 + 𝑧) − 𝑢))))
75	57, 74	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → ∃𝑢 ∈ 𝑋 (𝑢 ⊕ 𝐾) = ℎ)
76	47	gaorb 19572	. . . . . . . 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 1362	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → 𝐾{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}ℎ)
78		elecg 8800	. . . . . . . 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 596	. . . . . . 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 260	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ (𝑃 pSyl 𝐺)) → ℎ ∈ [𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)})
81	53, 80	eqelssd 3934	. . . . 5 ⊢ (𝜑 → [𝐾]{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑃 pSyl 𝐺) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} = (𝑃 pSyl 𝐺))
82	81	fveq2d 6920	. . . 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 19893	. . . . 5 ⊢ (𝜑 → 𝐻 = 𝑁)
84	83	fveq2d 6920	. . . 4 ⊢ (𝜑 → (♯‘𝐻) = (♯‘𝑁))
85	82, 84	oveq12d 7469	. . 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 2807	. 2 ⊢ (𝜑 → ((♯‘(𝑃 pSyl 𝐺)) · (♯‘𝑁)) = ((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝑁)))
87	14, 26, 38, 39, 86	mulcan2ad 11992	1 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) = (♯‘(𝑋 / (𝐺 ~_QG 𝑁))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ∃wrex 3090 {crab 3416 Vcvv 3454 ⊆ wss 3900 ∅c0 4281 𝒫 cpw 4560 {cpr 4590 class class class wbr 5108 {copab 5173 ↦ cmpt 5192 ran crn 5659 ‘cfv 6566 (class class class)co 7451 ∈ cmpo 7453 Er wer 8752 [cec 8753 / cqs 8754 Fincfn 9011 · cmul 11244 ℕcn 12375 ℕ₀cn0 12646 ♯chash 14515 ℙcprime 16890 Basecbs 17431 +_gcplusg 17472 0_gc0g 17660 Grpcgrp 19194 -_gcsg 19196 SubGrpcsubg 19380 ~_QG cqg 19382 GrpAct cga 19554 pSyl cslw 19792

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-disj 5076 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-oadd 8518 df-omul 8519 df-er 8755 df-ec 8757 df-qs 8761 df-map 8887 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-sup 9471 df-inf 9472 df-oi 9541 df-dju 10019 df-card 10057 df-acn 10060 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-n0 12647 df-xnn0 12720 df-z 12734 df-uz 13006 df-q 13116 df-rp 13161 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-fac 14459 df-bc 14488 df-hash 14516 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-clim 15699 df-sum 15898 df-dvds 16467 df-gcd 16709 df-prm 16891 df-pc 17059 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-0g 17662 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-grp 19197 df-minusg 19198 df-sbg 19199 df-mulg 19328 df-subg 19383 df-eqg 19385 df-ghm 19478 df-ga 19555 df-od 19793 df-pgp 19795 df-slw 19796

This theorem is used by: sylow3lem4 19895

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