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Theorem sylow3lem6 19573

Description: Lemma for sylow3 19574, second part. Using the lemma sylow2a 19560, 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 19572	. . . . 5 ⊢ (𝜑 → ⊕ ∈ ((𝐺 ↾_s 𝐾) GrpAct (𝑃 pSyl 𝐺)))
11		eqid 2737	. . . . . . 7 ⊢ (𝐺 ↾_s 𝐾) = (𝐺 ↾_s 𝐾)
12	11	slwpgp 19554	. . . . . 6 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp (𝐺 ↾_s 𝐾))
13	8, 12	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾_s 𝐾))
14		slwsubg 19551	. . . . . . . 8 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
15	8, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))
16	11	subgbas 19072	. . . . . . 7 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 = (Base‘(𝐺 ↾_s 𝐾)))
17	15, 16	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 = (Base‘(𝐺 ↾_s 𝐾)))
18	2	subgss 19069	. . . . . . . 8 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋)
19	15, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ⊆ 𝑋)
20	4, 19	ssfid 9181	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Fin)
21	17, 20	eqeltrrd 2838	. . . . 5 ⊢ (𝜑 → (Base‘(𝐺 ↾_s 𝐾)) ∈ Fin)
22		pwfi 9231	. . . . . . 7 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
23	4, 22	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
24		slwsubg 19551	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ (SubGrp‘𝐺))
25	2	subgss 19069	. . . . . . . . 9 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ 𝑋)
26	24, 25	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ⊆ 𝑋)
27	24, 26	elpwd 4562	. . . . . . 7 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ 𝒫 𝑋)
28	27	ssriv 3939	. . . . . 6 ⊢ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋
29		ssfi 9109	. . . . . 6 ⊢ ((𝒫 𝑋 ∈ Fin ∧ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋) → (𝑃 pSyl 𝐺) ∈ Fin)
30	23, 28, 29	sylancl 587	. . . . 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 19560	. . . 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 3935	. . . . . . . . . . . . . . 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 7383	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → (𝑥 + 𝑧) = (𝑔 + 𝑧))
44	43, 42	oveq12d 7386	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → ((𝑥 + 𝑧) − 𝑥) = ((𝑔 + 𝑧) − 𝑔))
45	41, 44	mpteq12dv 5187	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
46	45	rneqd 5895	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
47		vex 3446	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑠 ∈ V
48	47	mptex 7179	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) ∈ V
49	48	rnex 7862	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) ∈ V
50	46, 9, 49	ovmpoa 7523	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ 𝐾 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (𝑔 ⊕ 𝑠) = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
51	39, 40, 50	syl2anc 585	. . . . . . . . . . . . . 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 19551	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (𝑃 pSyl 𝐺) → 𝑠 ∈ (SubGrp‘𝐺))
54	53	ad2antlr 728	. . . . . . . . . . . . . 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 19194	. . . . . . . . . . . . . 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 3159	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (∀𝑔 ∈ 𝐾 (𝑔 ⊕ 𝑠) = 𝑠 ↔ ∀𝑔 ∈ 𝐾 𝑔 ∈ 𝑁))
61		dfss3 3924	. . . . . . . . . . 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 3298	. . . . . . . . . 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 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝐺 ∈ Grp)
67	56, 2, 6	nmzsubg 19106	. . . . . . . . . . . . . . . 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 19072	. . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑋 ∈ Fin)
73	2	subgss 19069	. . . . . . . . . . . . . . . 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 9181	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑁 ∈ Fin)
76	71, 75	eqeltrrd 2838	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → (Base‘(𝐺 ↾_s 𝑁)) ∈ Fin)
77	8	ad2antrr 727	. . . . . . . . . . . . . 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 19557	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑁) → 𝐾 ∈ (𝑃 pSyl (𝐺 ↾_s 𝑁)))
80	68, 77, 78, 79	syl3anc 1374	. . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ∈ (SubGrp‘𝐺))
83	56, 2, 6	ssnmz 19107	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (SubGrp‘𝐺) → 𝑠 ⊆ 𝑁)
84	82, 83	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ⊆ 𝑁)
85	69	subgslw 19557	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑠 ∈ (𝑃 pSyl 𝐺) ∧ 𝑠 ⊆ 𝑁) → 𝑠 ∈ (𝑃 pSyl (𝐺 ↾_s 𝑁)))
86	68, 81, 84, 85	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ∈ (𝑃 pSyl (𝐺 ↾_s 𝑁)))
87	2	fvexi 6856	. . . . . . . . . . . . . . 15 ⊢ 𝑋 ∈ V
88	56, 87	rabex2 5288	. . . . . . . . . . . . . 14 ⊢ 𝑁 ∈ V
89	69, 6	ressplusg 17223	. . . . . . . . . . . . . 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 19567	. . . . . . . . . . . 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 19109	. . . . . . . . . . . . . . . 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 19195	. . . . . . . . . . . . . . 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 581	. . . . . . . . . . . . . 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 3139	. . . . . . . . . . . 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 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑠 = 𝐾) → 𝐾 ∈ (SubGrp‘𝐺))
104	102, 103	eqeltrd 2837	. . . . . . . . . . . . 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 3971	. . . . . . . . . . 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 3407	. . . . . . . 8 ⊢ (𝜑 → {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠} = {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ 𝑠 = 𝐾})
110		rabsn 4680	. . . . . . . . 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 2772	. . . . . . 7 ⊢ (𝜑 → {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠} = {𝐾})
113	112	fveq2d 6846	. . . . . 6 ⊢ (𝜑 → (♯‘{𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠}) = (♯‘{𝐾}))
114		hashsng 14304	. . . . . . 7 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → (♯‘{𝐾}) = 1)
115	8, 114	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘{𝐾}) = 1)
116	113, 115	eqtrd 2772	. . . . 5 ⊢ (𝜑 → (♯‘{𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠}) = 1)
117	116	oveq2d 7384	. . . 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 5126	. . 3 ⊢ (𝜑 → 𝑃 ∥ ((♯‘(𝑃 pSyl 𝐺)) − 1))
119		prmnn 16613	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
120	5, 119	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ)
121		hashcl 14291	. . . . . 6 ⊢ ((𝑃 pSyl 𝐺) ∈ Fin → (♯‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
122	30, 121	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
123	122	nn0zd 12525	. . . 4 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∈ ℤ)
124		1zzd 12534	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
125		moddvds 16202	. . . 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 1374	. . 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 16635	. . 3 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
129		eluz2b2 12846	. . . 4 ⊢ (𝑃 ∈ (ℤ_≥‘2) ↔ (𝑃 ∈ ℕ ∧ 1 < 𝑃))
130		nnre 12164	. . . . 5 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ)
131		1mod 13835	. . . . 5 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
132	130, 131	sylan 581	. . . 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 2772	1 ⊢ (𝜑 → ((♯‘(𝑃 pSyl 𝐺)) mod 𝑃) = 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 {crab 3401 Vcvv 3442 ⊆ wss 3903 𝒫 cpw 4556 {csn 4582 {cpr 4584 class class class wbr 5100 {copab 5162 ↦ cmpt 5181 ran crn 5633 ‘cfv 6500 (class class class)co 7368 ∈ cmpo 7370 Fincfn 8895 ℝcr 11037 1c1 11039 < clt 11178 − cmin 11376 ℕcn 12157 2c2 12212 ℕ₀cn0 12413 ℤcz 12500 ℤ_≥cuz 12763 mod cmo 13801 ♯chash 14265 ∥ cdvds 16191 ℙcprime 16610 Basecbs 17148 ↾_s cress 17169 +_gcplusg 17189 Grpcgrp 18875 -_gcsg 18877 SubGrpcsubg 19062 NrmSGrpcnsg 19063 pGrp cpgp 19467 pSyl cslw 19468

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 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-disj 5068 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-omul 8412 df-er 8645 df-ec 8647 df-qs 8651 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-oi 9427 df-dju 9825 df-card 9863 df-acn 9866 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-xnn0 12487 df-z 12501 df-uz 12764 df-q 12874 df-rp 12918 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-fac 14209 df-bc 14238 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-sum 15622 df-dvds 16192 df-gcd 16434 df-prm 16611 df-pc 16777 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-0g 17373 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19010 df-subg 19065 df-nsg 19066 df-eqg 19067 df-ghm 19154 df-ga 19231 df-od 19469 df-pgp 19471 df-slw 19472

This theorem is referenced by: sylow3 19574

W3C validator