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Theorem sylow3lem6 19021

Description: Lemma for sylow3 19022, second part. Using the lemma sylow2a 19008, 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 19020	. . . . 5 ⊢ (𝜑 → ⊕ ∈ ((𝐺 ↾_s 𝐾) GrpAct (𝑃 pSyl 𝐺)))
11		eqid 2737	. . . . . . 7 ⊢ (𝐺 ↾_s 𝐾) = (𝐺 ↾_s 𝐾)
12	11	slwpgp 19002	. . . . . 6 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp (𝐺 ↾_s 𝐾))
13	8, 12	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾_s 𝐾))
14		slwsubg 18999	. . . . . . . 8 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
15	8, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))
16	11	subgbas 18547	. . . . . . 7 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 = (Base‘(𝐺 ↾_s 𝐾)))
17	15, 16	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 = (Base‘(𝐺 ↾_s 𝐾)))
18	2	subgss 18544	. . . . . . . 8 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋)
19	15, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ⊆ 𝑋)
20	4, 19	ssfid 8898	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Fin)
21	17, 20	eqeltrrd 2839	. . . . 5 ⊢ (𝜑 → (Base‘(𝐺 ↾_s 𝐾)) ∈ Fin)
22		pwfi 8856	. . . . . . 7 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
23	4, 22	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
24		slwsubg 18999	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ (SubGrp‘𝐺))
25	2	subgss 18544	. . . . . . . . 9 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ 𝑋)
26	24, 25	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ⊆ 𝑋)
27	24, 26	elpwd 4521	. . . . . . 7 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ 𝒫 𝑋)
28	27	ssriv 3905	. . . . . 6 ⊢ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋
29		ssfi 8851	. . . . . 6 ⊢ ((𝒫 𝑋 ∈ Fin ∧ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋) → (𝑃 pSyl 𝐺) ∈ Fin)
30	23, 28, 29	sylancl 589	. . . . 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 19008	. . . 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 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → 𝐾 ⊆ 𝑋)
36	35	sselda 3901	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → 𝑔 ∈ 𝑋)
37	36	biantrurd 536	. . . . . . . . . . . . . 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	syl5bb 286	. . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
42		simpl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → 𝑥 = 𝑔)
43	42	oveq1d 7228	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → (𝑥 + 𝑧) = (𝑔 + 𝑧))
44	43, 42	oveq12d 7231	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → ((𝑥 + 𝑧) − 𝑥) = ((𝑔 + 𝑧) − 𝑔))
45	41, 44	mpteq12dv 5140	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
46	45	rneqd 5807	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
47		vex 3412	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑠 ∈ V
48	47	mptex 7039	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) ∈ V
49	48	rnex 7690	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) ∈ V
50	46, 9, 49	ovmpoa 7364	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ 𝐾 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (𝑔 ⊕ 𝑠) = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
51	39, 40, 50	syl2anc 587	. . . . . . . . . . . . . 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 18999	. . . . . . . . . . . . . . 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 18657	. . . . . . . . . . . . . 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 314	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → ((𝑔 ⊕ 𝑠) = 𝑠 ↔ 𝑔 ∈ 𝑁))
60	59	ralbidva 3117	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (∀𝑔 ∈ 𝐾 (𝑔 ⊕ 𝑠) = 𝑠 ↔ ∀𝑔 ∈ 𝐾 𝑔 ∈ 𝑁))
61		dfss3 3888	. . . . . . . . . . 11 ⊢ (𝐾 ⊆ 𝑁 ↔ ∀𝑔 ∈ 𝐾 𝑔 ∈ 𝑁)
62	60, 61	bitr4di 292	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (∀𝑔 ∈ 𝐾 (𝑔 ⊕ 𝑠) = 𝑠 ↔ 𝐾 ⊆ 𝑁))
63	17	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → 𝐾 = (Base‘(𝐺 ↾_s 𝐾)))
64	63	raleqdv 3325	. . . . . . . . . 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 18581	. . . . . . . . . . . . . . . 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 18547	. . . . . . . . . . . . . . 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 18544	. . . . . . . . . . . . . . . 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 8898	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑁 ∈ Fin)
76	71, 75	eqeltrrd 2839	. . . . . . . . . . . . 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 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝐾 ⊆ 𝑁)
79	69	subgslw 19005	. . . . . . . . . . . . . 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 18582	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (SubGrp‘𝐺) → 𝑠 ⊆ 𝑁)
84	82, 83	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ⊆ 𝑁)
85	69	subgslw 19005	. . . . . . . . . . . . . 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 6731	. . . . . . . . . . . . . . 15 ⊢ 𝑋 ∈ V
88	56, 87	rabex2 5227	. . . . . . . . . . . . . 14 ⊢ 𝑁 ∈ V
89	69, 6	ressplusg 16834	. . . . . . . . . . . . . 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 19015	. . . . . . . . . . . 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 18584	. . . . . . . . . . . . . . . 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 18658	. . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . 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 250	. . . . . . . . . . . . 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 3203	. . . . . . . . . . . 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 488	. . . . . . . . . . . 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 2838	. . . . . . . . . . . . 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 3940	. . . . . . . . . . 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 312	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠 ↔ 𝑠 = 𝐾))
109	108	rabbidva 3388	. . . . . . . 8 ⊢ (𝜑 → {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠} = {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ 𝑠 = 𝐾})
110		rabsn 4637	. . . . . . . . 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 6721	. . . . . 6 ⊢ (𝜑 → (♯‘{𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠}) = (♯‘{𝐾}))
114		hashsng 13936	. . . . . . 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 7229	. . . 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 5079	. . 3 ⊢ (𝜑 → 𝑃 ∥ ((♯‘(𝑃 pSyl 𝐺)) − 1))
119		prmnn 16231	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
120	5, 119	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ)
121		hashcl 13923	. . . . . 6 ⊢ ((𝑃 pSyl 𝐺) ∈ Fin → (♯‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
122	30, 121	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
123	122	nn0zd 12280	. . . 4 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∈ ℤ)
124		1zzd 12208	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
125		moddvds 15826	. . . 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 260	. 2 ⊢ (𝜑 → ((♯‘(𝑃 pSyl 𝐺)) mod 𝑃) = (1 mod 𝑃))
128		prmuz2 16253	. . 3 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
129		eluz2b2 12517	. . . 4 ⊢ (𝑃 ∈ (ℤ_≥‘2) ↔ (𝑃 ∈ ℕ ∧ 1 < 𝑃))
130		nnre 11837	. . . . 5 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ)
131		1mod 13476	. . . . 5 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
132	130, 131	sylan 583	. . . 4 ⊢ ((𝑃 ∈ ℕ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
133	129, 132	sylbi 220	. . 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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ∃wrex 3062 {crab 3065 Vcvv 3408 ⊆ wss 3866 𝒫 cpw 4513 {csn 4541 {cpr 4543 class class class wbr 5053 {copab 5115 ↦ cmpt 5135 ran crn 5552 ‘cfv 6380 (class class class)co 7213 ∈ cmpo 7215 Fincfn 8626 ℝcr 10728 1c1 10730 < clt 10867 − cmin 11062 ℕcn 11830 2c2 11885 ℕ₀cn0 12090 ℤcz 12176 ℤ_≥cuz 12438 mod cmo 13442 ♯chash 13896 ∥ cdvds 15815 ℙcprime 16228 Basecbs 16760 ↾_s cress 16784 +_gcplusg 16802 Grpcgrp 18365 -_gcsg 18367 SubGrpcsubg 18537 NrmSGrpcnsg 18538 pGrp cpgp 18918 pSyl cslw 18919

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-disj 5019 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-oadd 8206 df-omul 8207 df-er 8391 df-ec 8393 df-qs 8397 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-inf 9059 df-oi 9126 df-dju 9517 df-card 9555 df-acn 9558 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-xnn0 12163 df-z 12177 df-uz 12439 df-q 12545 df-rp 12587 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-seq 13575 df-exp 13636 df-fac 13840 df-bc 13869 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-sum 15250 df-dvds 15816 df-gcd 16054 df-prm 16229 df-pc 16390 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-mulg 18489 df-subg 18540 df-nsg 18541 df-eqg 18542 df-ghm 18620 df-ga 18684 df-od 18920 df-pgp 18922 df-slw 18923

This theorem is referenced by: sylow3 19022

W3C validator