sylow3lem6 - Metamath Proof Explorer

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Theorem sylow3lem6 19598

Description: Lemma for sylow3 19599, second part. Using the lemma sylow2a 19585, show that the number of sylow subgroups is equivalent mod 𝑃 to the number of fixed points under the group action. But 𝐾 is the unique element of the set of Sylow subgroups that is fixed under the group action, so there is exactly one fixed point and so ((♯‘(𝑃 pSyl 𝐺)) mod 𝑃) = 1. (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)
sylow3lem5.a	⊢ + = (+_g‘𝐺)
sylow3lem5.d	⊢ − = (-_g‘𝐺)
sylow3lem5.k	⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺))
sylow3lem5.m	⊢ ⊕ = (𝑥 ∈ 𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)))
sylow3lem6.n	⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)}

**Assertion**
Ref	Expression
sylow3lem6	⊢ (𝜑 → ((♯‘(𝑃 pSyl 𝐺)) mod 𝑃) = 1)

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

Proof of Theorem sylow3lem6 Dummy variables 𝑤 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2739	. . . . 5 ⊢ (Base‘(𝐺 ↾_s 𝐾)) = (Base‘(𝐺 ↾_s 𝐾))
2		sylow3.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝐺)
3		sylow3.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
4		sylow3.xf	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin)
5		sylow3.p	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℙ)
6		sylow3lem5.a	. . . . . 6 ⊢ + = (+_g‘𝐺)
7		sylow3lem5.d	. . . . . 6 ⊢ − = (-_g‘𝐺)
8		sylow3lem5.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺))
9		sylow3lem5.m	. . . . . 6 ⊢ ⊕ = (𝑥 ∈ 𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)))
10	2, 3, 4, 5, 6, 7, 8, 9	sylow3lem5 19597	. . . . 5 ⊢ (𝜑 → ⊕ ∈ ((𝐺 ↾_s 𝐾) GrpAct (𝑃 pSyl 𝐺)))
11		eqid 2739	. . . . . . 7 ⊢ (𝐺 ↾_s 𝐾) = (𝐺 ↾_s 𝐾)
12	11	slwpgp 19579	. . . . . 6 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp (𝐺 ↾_s 𝐾))
13	8, 12	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾_s 𝐾))
14		slwsubg 19576	. . . . . . . 8 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
15	8, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))
16	11	subgbas 19097	. . . . . . 7 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 = (Base‘(𝐺 ↾_s 𝐾)))
17	15, 16	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 = (Base‘(𝐺 ↾_s 𝐾)))
18	2	subgss 19094	. . . . . . . 8 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋)
19	15, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ⊆ 𝑋)
20	4, 19	ssfid 9169	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Fin)
21	17, 20	eqeltrrd 2840	. . . . 5 ⊢ (𝜑 → (Base‘(𝐺 ↾_s 𝐾)) ∈ Fin)
22		pwfi 9219	. . . . . . 7 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
23	4, 22	sylib 219	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
24		slwsubg 19576	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ (SubGrp‘𝐺))
25	2	subgss 19094	. . . . . . . . 9 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ 𝑋)
26	24, 25	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ⊆ 𝑋)
27	24, 26	elpwd 4535	. . . . . . 7 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ 𝒫 𝑋)
28	27	ssriv 3919	. . . . . 6 ⊢ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋
29		ssfi 9097	. . . . . 6 ⊢ ((𝒫 𝑋 ∈ Fin ∧ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋) → (𝑃 pSyl 𝐺) ∈ Fin)
30	23, 28, 29	sylancl 592	. . . . 5 ⊢ (𝜑 → (𝑃 pSyl 𝐺) ∈ Fin)
31		eqid 2739	. . . . 5 ⊢ {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠} = {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠}
32		eqid 2739	. . . . 5 ⊢ {⟨𝑧, 𝑤⟩ ∣ ({𝑧, 𝑤} ⊆ (𝑃 pSyl 𝐺) ∧ ∃ℎ ∈ (Base‘(𝐺 ↾_s 𝐾))(ℎ ⊕ 𝑧) = 𝑤)} = {⟨𝑧, 𝑤⟩ ∣ ({𝑧, 𝑤} ⊆ (𝑃 pSyl 𝐺) ∧ ∃ℎ ∈ (Base‘(𝐺 ↾_s 𝐾))(ℎ ⊕ 𝑧) = 𝑤)}
33	1, 10, 13, 21, 30, 31, 32	sylow2a 19585	. . . 4 ⊢ (𝜑 → 𝑃 ∥ ((♯‘(𝑃 pSyl 𝐺)) − (♯‘{𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠})))
34		eqcom 2746	. . . . . . . . . . . . . 14 ⊢ (ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) = 𝑠 ↔ 𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
35	19	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → 𝐾 ⊆ 𝑋)
36	35	sselda 3915	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → 𝑔 ∈ 𝑋)
37	36	biantrurd 537	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → (𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) ↔ (𝑔 ∈ 𝑋 ∧ 𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))))
38	34, 37	bitrid 284	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → (ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) = 𝑠 ↔ (𝑔 ∈ 𝑋 ∧ 𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))))
39		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → 𝑔 ∈ 𝐾)
40		simplr 774	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → 𝑠 ∈ (𝑃 pSyl 𝐺))
41		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
42		simpl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → 𝑥 = 𝑔)
43	42	oveq1d 7371	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → (𝑥 + 𝑧) = (𝑔 + 𝑧))
44	43, 42	oveq12d 7374	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → ((𝑥 + 𝑧) − 𝑥) = ((𝑔 + 𝑧) − 𝑔))
45	41, 44	mpteq12dv 5159	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
46	45	rneqd 5880	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
47		vex 3435	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑠 ∈ V
48	47	mptex 7167	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) ∈ V
49	48	rnex 7850	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) ∈ V
50	46, 9, 49	ovmpoa 7511	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ 𝐾 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (𝑔 ⊕ 𝑠) = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
51	39, 40, 50	syl2anc 590	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → (𝑔 ⊕ 𝑠) = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
52	51	eqeq1d 2741	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → ((𝑔 ⊕ 𝑠) = 𝑠 ↔ ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) = 𝑠))
53		slwsubg 19576	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (𝑃 pSyl 𝐺) → 𝑠 ∈ (SubGrp‘𝐺))
54	53	ad2antlr 733	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → 𝑠 ∈ (SubGrp‘𝐺))
55		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) = (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔))
56		sylow3lem6.n	. . . . . . . . . . . . . . 15 ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)}
57	2, 6, 7, 55, 56	conjnmzb 19219	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (SubGrp‘𝐺) → (𝑔 ∈ 𝑁 ↔ (𝑔 ∈ 𝑋 ∧ 𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))))
58	54, 57	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → (𝑔 ∈ 𝑁 ↔ (𝑔 ∈ 𝑋 ∧ 𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))))
59	38, 52, 58	3bitr4d 312	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → ((𝑔 ⊕ 𝑠) = 𝑠 ↔ 𝑔 ∈ 𝑁))
60	59	ralbidva 3160	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (∀𝑔 ∈ 𝐾 (𝑔 ⊕ 𝑠) = 𝑠 ↔ ∀𝑔 ∈ 𝐾 𝑔 ∈ 𝑁))
61		dfss3 3904	. . . . . . . . . . 11 ⊢ (𝐾 ⊆ 𝑁 ↔ ∀𝑔 ∈ 𝐾 𝑔 ∈ 𝑁)
62	60, 61	bitr4di 290	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (∀𝑔 ∈ 𝐾 (𝑔 ⊕ 𝑠) = 𝑠 ↔ 𝐾 ⊆ 𝑁))
63	17	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → 𝐾 = (Base‘(𝐺 ↾_s 𝐾)))
64	63	raleqdv 3297	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (∀𝑔 ∈ 𝐾 (𝑔 ⊕ 𝑠) = 𝑠 ↔ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠))
65		eqid 2739	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐺 ↾_s 𝑁)) = (Base‘(𝐺 ↾_s 𝑁))
66	3	ad2antrr 732	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝐺 ∈ Grp)
67	56, 2, 6	nmzsubg 19131	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺))
68	66, 67	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑁 ∈ (SubGrp‘𝐺))
69		eqid 2739	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ↾_s 𝑁) = (𝐺 ↾_s 𝑁)
70	69	subgbas 19097	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘(𝐺 ↾_s 𝑁)))
71	68, 70	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑁 = (Base‘(𝐺 ↾_s 𝑁)))
72	4	ad2antrr 732	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑋 ∈ Fin)
73	2	subgss 19094	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ 𝑋)
74	68, 73	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑁 ⊆ 𝑋)
75	72, 74	ssfid 9169	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑁 ∈ Fin)
76	71, 75	eqeltrrd 2840	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → (Base‘(𝐺 ↾_s 𝑁)) ∈ Fin)
77	8	ad2antrr 732	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝐾 ∈ (𝑃 pSyl 𝐺))
78		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝐾 ⊆ 𝑁)
79	69	subgslw 19582	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑁) → 𝐾 ∈ (𝑃 pSyl (𝐺 ↾_s 𝑁)))
80	68, 77, 78, 79	syl3anc 1379	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝐾 ∈ (𝑃 pSyl (𝐺 ↾_s 𝑁)))
81		simplr 774	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ∈ (𝑃 pSyl 𝐺))
82	53	ad2antlr 733	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ∈ (SubGrp‘𝐺))
83	56, 2, 6	ssnmz 19132	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (SubGrp‘𝐺) → 𝑠 ⊆ 𝑁)
84	82, 83	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ⊆ 𝑁)
85	69	subgslw 19582	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑠 ∈ (𝑃 pSyl 𝐺) ∧ 𝑠 ⊆ 𝑁) → 𝑠 ∈ (𝑃 pSyl (𝐺 ↾_s 𝑁)))
86	68, 81, 84, 85	syl3anc 1379	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ∈ (𝑃 pSyl (𝐺 ↾_s 𝑁)))
87	2	fvexi 6841	. . . . . . . . . . . . . . 15 ⊢ 𝑋 ∈ V
88	56, 87	rabex2 5269	. . . . . . . . . . . . . 14 ⊢ 𝑁 ∈ V
89	69, 6	ressplusg 17245	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ V → + = (+_g‘(𝐺 ↾_s 𝑁)))
90	88, 89	ax-mp 5	. . . . . . . . . . . . 13 ⊢ + = (+_g‘(𝐺 ↾_s 𝑁))
91		eqid 2739	. . . . . . . . . . . . 13 ⊢ (-_g‘(𝐺 ↾_s 𝑁)) = (-_g‘(𝐺 ↾_s 𝑁))
92	65, 76, 80, 86, 90, 91	sylow2 19592	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → ∃𝑔 ∈ (Base‘(𝐺 ↾_s 𝑁))𝐾 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧)(-_g‘(𝐺 ↾_s 𝑁))𝑔)))
93	56, 2, 6, 69	nmznsg 19134	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (SubGrp‘𝐺) → 𝑠 ∈ (NrmSGrp‘(𝐺 ↾_s 𝑁)))
94	82, 93	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ∈ (NrmSGrp‘(𝐺 ↾_s 𝑁)))
95		eqid 2739	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧)(-_g‘(𝐺 ↾_s 𝑁))𝑔)) = (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧)(-_g‘(𝐺 ↾_s 𝑁))𝑔))
96	65, 90, 91, 95	conjnsg 19220	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ (NrmSGrp‘(𝐺 ↾_s 𝑁)) ∧ 𝑔 ∈ (Base‘(𝐺 ↾_s 𝑁))) → 𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧)(-_g‘(𝐺 ↾_s 𝑁))𝑔)))
97	94, 96	sylan 586	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) ∧ 𝑔 ∈ (Base‘(𝐺 ↾_s 𝑁))) → 𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧)(-_g‘(𝐺 ↾_s 𝑁))𝑔)))
98		eqeq2 2751	. . . . . . . . . . . . . 14 ⊢ (𝐾 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧)(-_g‘(𝐺 ↾_s 𝑁))𝑔)) → (𝑠 = 𝐾 ↔ 𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧)(-_g‘(𝐺 ↾_s 𝑁))𝑔))))
99	97, 98	syl5ibrcom 248	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) ∧ 𝑔 ∈ (Base‘(𝐺 ↾_s 𝑁))) → (𝐾 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧)(-_g‘(𝐺 ↾_s 𝑁))𝑔)) → 𝑠 = 𝐾))
100	99	rexlimdva 3140	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → (∃𝑔 ∈ (Base‘(𝐺 ↾_s 𝑁))𝐾 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧)(-_g‘(𝐺 ↾_s 𝑁))𝑔)) → 𝑠 = 𝐾))
101	92, 100	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 = 𝐾)
102		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑠 = 𝐾) → 𝑠 = 𝐾)
103	15	ad2antrr 732	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑠 = 𝐾) → 𝐾 ∈ (SubGrp‘𝐺))
104	102, 103	eqeltrd 2839	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑠 = 𝐾) → 𝑠 ∈ (SubGrp‘𝐺))
105	104, 83	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑠 = 𝐾) → 𝑠 ⊆ 𝑁)
106	102, 105	eqsstrrd 3950	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑠 = 𝐾) → 𝐾 ⊆ 𝑁)
107	101, 106	impbida 806	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (𝐾 ⊆ 𝑁 ↔ 𝑠 = 𝐾))
108	62, 64, 107	3bitr3d 310	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠 ↔ 𝑠 = 𝐾))
109	108	rabbidva 3397	. . . . . . . 8 ⊢ (𝜑 → {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠} = {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ 𝑠 = 𝐾})
110		rabsn 4653	. . . . . . . . 9 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ 𝑠 = 𝐾} = {𝐾})
111	8, 110	syl 17	. . . . . . . 8 ⊢ (𝜑 → {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ 𝑠 = 𝐾} = {𝐾})
112	109, 111	eqtrd 2774	. . . . . . 7 ⊢ (𝜑 → {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠} = {𝐾})
113	112	fveq2d 6831	. . . . . 6 ⊢ (𝜑 → (♯‘{𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠}) = (♯‘{𝐾}))
114		hashsng 14322	. . . . . . 7 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → (♯‘{𝐾}) = 1)
115	8, 114	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘{𝐾}) = 1)
116	113, 115	eqtrd 2774	. . . . 5 ⊢ (𝜑 → (♯‘{𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠}) = 1)
117	116	oveq2d 7372	. . . 4 ⊢ (𝜑 → ((♯‘(𝑃 pSyl 𝐺)) − (♯‘{𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠})) = ((♯‘(𝑃 pSyl 𝐺)) − 1))
118	33, 117	breqtrd 5098	. . 3 ⊢ (𝜑 → 𝑃 ∥ ((♯‘(𝑃 pSyl 𝐺)) − 1))
119		prmnn 16634	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
120	5, 119	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ)
121		hashcl 14309	. . . . . 6 ⊢ ((𝑃 pSyl 𝐺) ∈ Fin → (♯‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
122	30, 121	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
123	122	nn0zd 12540	. . . 4 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∈ ℤ)
124		1zzd 12549	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
125		moddvds 16223	. . . 4 ⊢ ((𝑃 ∈ ℕ ∧ (♯‘(𝑃 pSyl 𝐺)) ∈ ℤ ∧ 1 ∈ ℤ) → (((♯‘(𝑃 pSyl 𝐺)) mod 𝑃) = (1 mod 𝑃) ↔ 𝑃 ∥ ((♯‘(𝑃 pSyl 𝐺)) − 1)))
126	120, 123, 124, 125	syl3anc 1379	. . 3 ⊢ (𝜑 → (((♯‘(𝑃 pSyl 𝐺)) mod 𝑃) = (1 mod 𝑃) ↔ 𝑃 ∥ ((♯‘(𝑃 pSyl 𝐺)) − 1)))
127	118, 126	mpbird 258	. 2 ⊢ (𝜑 → ((♯‘(𝑃 pSyl 𝐺)) mod 𝑃) = (1 mod 𝑃))
128		prmuz2 16656	. . 3 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
129		eluz2b2 12862	. . . 4 ⊢ (𝑃 ∈ (ℤ_≥‘2) ↔ (𝑃 ∈ ℕ ∧ 1 < 𝑃))
130		nnre 12172	. . . . 5 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ)
131		1mod 13853	. . . . 5 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
132	130, 131	sylan 586	. . . 4 ⊢ ((𝑃 ∈ ℕ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
133	129, 132	sylbi 218	. . 3 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (1 mod 𝑃) = 1)
134	5, 128, 133	3syl 18	. 2 ⊢ (𝜑 → (1 mod 𝑃) = 1)
135	127, 134	eqtrd 2774	1 ⊢ (𝜑 → ((♯‘(𝑃 pSyl 𝐺)) mod 𝑃) = 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∀wral 3053 ∃wrex 3063 {crab 3391 Vcvv 3431 ⊆ wss 3883 𝒫 cpw 4529 {csn 4555 {cpr 4557 class class class wbr 5072 {copab 5134 ↦ cmpt 5153 ran crn 5619 ‘cfv 6485 (class class class)co 7356 ∈ cmpo 7358 Fincfn 8883 ℝcr 11028 1c1 11030 < clt 11170 − cmin 11368 ℕcn 12165 2c2 12227 ℕ₀cn0 12428 ℤcz 12515 ℤ_≥cuz 12779 mod cmo 13819 ♯chash 14283 ∥ cdvds 16212 ℙcprime 16631 Basecbs 17170 ↾_s cress 17191 +_gcplusg 17211 Grpcgrp 18900 -_gcsg 18902 SubGrpcsubg 19087 NrmSGrpcnsg 19088 pGrp cpgp 19492 pSyl cslw 19493

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-disj 5040 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8633 df-ec 8635 df-qs 8639 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9816 df-card 9854 df-acn 9857 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-xnn0 12502 df-z 12516 df-uz 12780 df-q 12890 df-rp 12934 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-sum 15640 df-dvds 16213 df-gcd 16455 df-prm 16632 df-pc 16799 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-0g 17395 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-nsg 19091 df-eqg 19092 df-ghm 19179 df-ga 19256 df-od 19494 df-pgp 19496 df-slw 19497

This theorem is referenced by: sylow3 19599

W3C validator