sylow3lem6 - Metamath Proof Explorer

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Theorem sylow3lem6 19650

Description: Lemma for sylow3 19651, second part. Using the lemma sylow2a 19637, 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 2737	. . . . 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 19649	. . . . 5 ⊢ (𝜑 → ⊕ ∈ ((𝐺 ↾_s 𝐾) GrpAct (𝑃 pSyl 𝐺)))
11		eqid 2737	. . . . . . 7 ⊢ (𝐺 ↾_s 𝐾) = (𝐺 ↾_s 𝐾)
12	11	slwpgp 19631	. . . . . 6 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp (𝐺 ↾_s 𝐾))
13	8, 12	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾_s 𝐾))
14		slwsubg 19628	. . . . . . . 8 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
15	8, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))
16	11	subgbas 19148	. . . . . . 7 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 = (Base‘(𝐺 ↾_s 𝐾)))
17	15, 16	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 = (Base‘(𝐺 ↾_s 𝐾)))
18	2	subgss 19145	. . . . . . . 8 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋)
19	15, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ⊆ 𝑋)
20	4, 19	ssfid 9301	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Fin)
21	17, 20	eqeltrrd 2842	. . . . 5 ⊢ (𝜑 → (Base‘(𝐺 ↾_s 𝐾)) ∈ Fin)
22		pwfi 9357	. . . . . . 7 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
23	4, 22	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
24		slwsubg 19628	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ (SubGrp‘𝐺))
25	2	subgss 19145	. . . . . . . . 9 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ 𝑋)
26	24, 25	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ⊆ 𝑋)
27	24, 26	elpwd 4606	. . . . . . 7 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ 𝒫 𝑋)
28	27	ssriv 3987	. . . . . 6 ⊢ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋
29		ssfi 9213	. . . . . 6 ⊢ ((𝒫 𝑋 ∈ Fin ∧ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋) → (𝑃 pSyl 𝐺) ∈ Fin)
30	23, 28, 29	sylancl 586	. . . . 5 ⊢ (𝜑 → (𝑃 pSyl 𝐺) ∈ Fin)
31		eqid 2737	. . . . 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 2737	. . . . 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 19637	. . . 4 ⊢ (𝜑 → 𝑃 ∥ ((♯‘(𝑃 pSyl 𝐺)) − (♯‘{𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠})))
34		eqcom 2744	. . . . . . . . . . . . . 14 ⊢ (ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) = 𝑠 ↔ 𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
35	19	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → 𝐾 ⊆ 𝑋)
36	35	sselda 3983	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → 𝑔 ∈ 𝑋)
37	36	biantrurd 532	. . . . . . . . . . . . . 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 283	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → 𝑔 ∈ 𝐾)
40		simplr 769	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → 𝑠 ∈ (𝑃 pSyl 𝐺))
41		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
42		simpl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → 𝑥 = 𝑔)
43	42	oveq1d 7446	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → (𝑥 + 𝑧) = (𝑔 + 𝑧))
44	43, 42	oveq12d 7449	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → ((𝑥 + 𝑧) − 𝑥) = ((𝑔 + 𝑧) − 𝑔))
45	41, 44	mpteq12dv 5233	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
46	45	rneqd 5949	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
47		vex 3484	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑠 ∈ V
48	47	mptex 7243	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) ∈ V
49	48	rnex 7932	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) ∈ V
50	46, 9, 49	ovmpoa 7588	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ 𝐾 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (𝑔 ⊕ 𝑠) = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
51	39, 40, 50	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → (𝑔 ⊕ 𝑠) = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
52	51	eqeq1d 2739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → ((𝑔 ⊕ 𝑠) = 𝑠 ↔ ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) = 𝑠))
53		slwsubg 19628	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (𝑃 pSyl 𝐺) → 𝑠 ∈ (SubGrp‘𝐺))
54	53	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → 𝑠 ∈ (SubGrp‘𝐺))
55		eqid 2737	. . . . . . . . . . . . . . 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 19271	. . . . . . . . . . . . . 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 311	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → ((𝑔 ⊕ 𝑠) = 𝑠 ↔ 𝑔 ∈ 𝑁))
60	59	ralbidva 3176	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (∀𝑔 ∈ 𝐾 (𝑔 ⊕ 𝑠) = 𝑠 ↔ ∀𝑔 ∈ 𝐾 𝑔 ∈ 𝑁))
61		dfss3 3972	. . . . . . . . . . 11 ⊢ (𝐾 ⊆ 𝑁 ↔ ∀𝑔 ∈ 𝐾 𝑔 ∈ 𝑁)
62	60, 61	bitr4di 289	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (∀𝑔 ∈ 𝐾 (𝑔 ⊕ 𝑠) = 𝑠 ↔ 𝐾 ⊆ 𝑁))
63	17	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → 𝐾 = (Base‘(𝐺 ↾_s 𝐾)))
64	63	raleqdv 3326	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (∀𝑔 ∈ 𝐾 (𝑔 ⊕ 𝑠) = 𝑠 ↔ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠))
65		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐺 ↾_s 𝑁)) = (Base‘(𝐺 ↾_s 𝑁))
66	3	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝐺 ∈ Grp)
67	56, 2, 6	nmzsubg 19183	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺))
68	66, 67	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑁 ∈ (SubGrp‘𝐺))
69		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ↾_s 𝑁) = (𝐺 ↾_s 𝑁)
70	69	subgbas 19148	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑋 ∈ Fin)
73	2	subgss 19145	. . . . . . . . . . . . . . . 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 9301	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑁 ∈ Fin)
76	71, 75	eqeltrrd 2842	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → (Base‘(𝐺 ↾_s 𝑁)) ∈ Fin)
77	8	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝐾 ∈ (𝑃 pSyl 𝐺))
78		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝐾 ⊆ 𝑁)
79	69	subgslw 19634	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑁) → 𝐾 ∈ (𝑃 pSyl (𝐺 ↾_s 𝑁)))
80	68, 77, 78, 79	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝐾 ∈ (𝑃 pSyl (𝐺 ↾_s 𝑁)))
81		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ∈ (𝑃 pSyl 𝐺))
82	53	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ∈ (SubGrp‘𝐺))
83	56, 2, 6	ssnmz 19184	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (SubGrp‘𝐺) → 𝑠 ⊆ 𝑁)
84	82, 83	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ⊆ 𝑁)
85	69	subgslw 19634	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑠 ∈ (𝑃 pSyl 𝐺) ∧ 𝑠 ⊆ 𝑁) → 𝑠 ∈ (𝑃 pSyl (𝐺 ↾_s 𝑁)))
86	68, 81, 84, 85	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ∈ (𝑃 pSyl (𝐺 ↾_s 𝑁)))
87	2	fvexi 6920	. . . . . . . . . . . . . . 15 ⊢ 𝑋 ∈ V
88	56, 87	rabex2 5341	. . . . . . . . . . . . . 14 ⊢ 𝑁 ∈ V
89	69, 6	ressplusg 17334	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ V → + = (+_g‘(𝐺 ↾_s 𝑁)))
90	88, 89	ax-mp 5	. . . . . . . . . . . . 13 ⊢ + = (+_g‘(𝐺 ↾_s 𝑁))
91		eqid 2737	. . . . . . . . . . . . 13 ⊢ (-_g‘(𝐺 ↾_s 𝑁)) = (-_g‘(𝐺 ↾_s 𝑁))
92	65, 76, 80, 86, 90, 91	sylow2 19644	. . . . . . . . . . . 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 19186	. . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . 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 19272	. . . . . . . . . . . . . . 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 580	. . . . . . . . . . . . . 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 2749	. . . . . . . . . . . . . 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 247	. . . . . . . . . . . . 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 3155	. . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑠 = 𝐾) → 𝑠 = 𝐾)
103	15	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑠 = 𝐾) → 𝐾 ∈ (SubGrp‘𝐺))
104	102, 103	eqeltrd 2841	. . . . . . . . . . . . 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 4019	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑠 = 𝐾) → 𝐾 ⊆ 𝑁)
107	101, 106	impbida 801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (𝐾 ⊆ 𝑁 ↔ 𝑠 = 𝐾))
108	62, 64, 107	3bitr3d 309	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠 ↔ 𝑠 = 𝐾))
109	108	rabbidva 3443	. . . . . . . 8 ⊢ (𝜑 → {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠} = {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ 𝑠 = 𝐾})
110		rabsn 4721	. . . . . . . . 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 2777	. . . . . . 7 ⊢ (𝜑 → {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠} = {𝐾})
113	112	fveq2d 6910	. . . . . 6 ⊢ (𝜑 → (♯‘{𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠}) = (♯‘{𝐾}))
114		hashsng 14408	. . . . . . 7 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → (♯‘{𝐾}) = 1)
115	8, 114	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘{𝐾}) = 1)
116	113, 115	eqtrd 2777	. . . . 5 ⊢ (𝜑 → (♯‘{𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠}) = 1)
117	116	oveq2d 7447	. . . 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 5169	. . 3 ⊢ (𝜑 → 𝑃 ∥ ((♯‘(𝑃 pSyl 𝐺)) − 1))
119		prmnn 16711	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
120	5, 119	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ)
121		hashcl 14395	. . . . . 6 ⊢ ((𝑃 pSyl 𝐺) ∈ Fin → (♯‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
122	30, 121	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
123	122	nn0zd 12639	. . . 4 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∈ ℤ)
124		1zzd 12648	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
125		moddvds 16301	. . . 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 1373	. . 3 ⊢ (𝜑 → (((♯‘(𝑃 pSyl 𝐺)) mod 𝑃) = (1 mod 𝑃) ↔ 𝑃 ∥ ((♯‘(𝑃 pSyl 𝐺)) − 1)))
127	118, 126	mpbird 257	. 2 ⊢ (𝜑 → ((♯‘(𝑃 pSyl 𝐺)) mod 𝑃) = (1 mod 𝑃))
128		prmuz2 16733	. . 3 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
129		eluz2b2 12963	. . . 4 ⊢ (𝑃 ∈ (ℤ_≥‘2) ↔ (𝑃 ∈ ℕ ∧ 1 < 𝑃))
130		nnre 12273	. . . . 5 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ)
131		1mod 13943	. . . . 5 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
132	130, 131	sylan 580	. . . 4 ⊢ ((𝑃 ∈ ℕ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
133	129, 132	sylbi 217	. . 3 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (1 mod 𝑃) = 1)
134	5, 128, 133	3syl 18	. 2 ⊢ (𝜑 → (1 mod 𝑃) = 1)
135	127, 134	eqtrd 2777	1 ⊢ (𝜑 → ((♯‘(𝑃 pSyl 𝐺)) mod 𝑃) = 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 {csn 4626 {cpr 4628 class class class wbr 5143 {copab 5205 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 Fincfn 8985 ℝcr 11154 1c1 11156 < clt 11295 − cmin 11492 ℕcn 12266 2c2 12321 ℕ₀cn0 12526 ℤcz 12613 ℤ_≥cuz 12878 mod cmo 13909 ♯chash 14369 ∥ cdvds 16290 ℙcprime 16708 Basecbs 17247 ↾_s cress 17274 +_gcplusg 17297 Grpcgrp 18951 -_gcsg 18953 SubGrpcsubg 19138 NrmSGrpcnsg 19139 pGrp cpgp 19544 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-nsg 19142 df-eqg 19143 df-ghm 19231 df-ga 19308 df-od 19546 df-pgp 19548 df-slw 19549

This theorem is referenced by: sylow3 19651

W3C validator