sylow3lem6 - Metamath Proof Explorer

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Theorem sylow3lem6 17870

Description: Lemma for sylow3 17871, second part. Using the lemma sylow2a 17857, 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 2610	. . . . 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 17869	. . . . 5 ⊢ (𝜑 → ⊕ ∈ ((𝐺 ↾_s 𝐾) GrpAct (𝑃 pSyl 𝐺)))
11		eqid 2610	. . . . . . 7 ⊢ (𝐺 ↾_s 𝐾) = (𝐺 ↾_s 𝐾)
12	11	slwpgp 17851	. . . . . 6 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp (𝐺 ↾_s 𝐾))
13	8, 12	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾_s 𝐾))
14		slwsubg 17848	. . . . . . . 8 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
15	8, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))
16	11	subgbas 17421	. . . . . . 7 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 = (Base‘(𝐺 ↾_s 𝐾)))
17	15, 16	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 = (Base‘(𝐺 ↾_s 𝐾)))
18	2	subgss 17418	. . . . . . . 8 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋)
19	15, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ⊆ 𝑋)
20		ssfi 8065	. . . . . . 7 ⊢ ((𝑋 ∈ Fin ∧ 𝐾 ⊆ 𝑋) → 𝐾 ∈ Fin)
21	4, 19, 20	syl2anc 691	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Fin)
22	17, 21	eqeltrrd 2689	. . . . 5 ⊢ (𝜑 → (Base‘(𝐺 ↾_s 𝐾)) ∈ Fin)
23		pwfi 8144	. . . . . . 7 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
24	4, 23	sylib 207	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
25		slwsubg 17848	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ (SubGrp‘𝐺))
26	2	subgss 17418	. . . . . . . . 9 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ 𝑋)
27	25, 26	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ⊆ 𝑋)
28		selpw 4115	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝑋 ↔ 𝑥 ⊆ 𝑋)
29	27, 28	sylibr 223	. . . . . . 7 ⊢ (𝑥 ∈ (𝑃 pSyl 𝐺) → 𝑥 ∈ 𝒫 𝑋)
30	29	ssriv 3572	. . . . . 6 ⊢ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋
31		ssfi 8065	. . . . . 6 ⊢ ((𝒫 𝑋 ∈ Fin ∧ (𝑃 pSyl 𝐺) ⊆ 𝒫 𝑋) → (𝑃 pSyl 𝐺) ∈ Fin)
32	24, 30, 31	sylancl 693	. . . . 5 ⊢ (𝜑 → (𝑃 pSyl 𝐺) ∈ Fin)
33		eqid 2610	. . . . 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}
34		eqid 2610	. . . . 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)}
35	1, 10, 13, 22, 32, 33, 34	sylow2a 17857	. . . 4 ⊢ (𝜑 → 𝑃 ∥ ((#‘(𝑃 pSyl 𝐺)) − (#‘{𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠})))
36		eqcom 2617	. . . . . . . . . . . . . 14 ⊢ (ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) = 𝑠 ↔ 𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
37	19	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → 𝐾 ⊆ 𝑋)
38	37	sselda 3568	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → 𝑔 ∈ 𝑋)
39	38	biantrurd 528	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → (𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) ↔ (𝑔 ∈ 𝑋 ∧ 𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))))
40	36, 39	syl5bb 271	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → (ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) = 𝑠 ↔ (𝑔 ∈ 𝑋 ∧ 𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))))
41		simpr 476	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → 𝑔 ∈ 𝐾)
42		simplr 788	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → 𝑠 ∈ (𝑃 pSyl 𝐺))
43		simpr 476	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
44		simpl 472	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → 𝑥 = 𝑔)
45	44	oveq1d 6564	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → (𝑥 + 𝑧) = (𝑔 + 𝑧))
46	45, 44	oveq12d 6567	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → ((𝑥 + 𝑧) − 𝑥) = ((𝑔 + 𝑧) − 𝑔))
47	43, 46	mpteq12dv 4663	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
48	47	rneqd 5274	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑠) → ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)) = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
49		vex 3176	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑠 ∈ V
50	49	mptex 6390	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) ∈ V
51	50	rnex 6992	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) ∈ V
52	48, 9, 51	ovmpt2a 6689	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ 𝐾 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (𝑔 ⊕ 𝑠) = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
53	41, 42, 52	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → (𝑔 ⊕ 𝑠) = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))
54	53	eqeq1d 2612	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → ((𝑔 ⊕ 𝑠) = 𝑠 ↔ ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) = 𝑠))
55		slwsubg 17848	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (𝑃 pSyl 𝐺) → 𝑠 ∈ (SubGrp‘𝐺))
56	55	ad2antlr 759	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → 𝑠 ∈ (SubGrp‘𝐺))
57		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)) = (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔))
58		sylow3lem6.n	. . . . . . . . . . . . . . 15 ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)}
59	2, 6, 7, 57, 58	conjnmzb 17518	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (SubGrp‘𝐺) → (𝑔 ∈ 𝑁 ↔ (𝑔 ∈ 𝑋 ∧ 𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))))
60	56, 59	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → (𝑔 ∈ 𝑁 ↔ (𝑔 ∈ 𝑋 ∧ 𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧) − 𝑔)))))
61	40, 54, 60	3bitr4d 299	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑔 ∈ 𝐾) → ((𝑔 ⊕ 𝑠) = 𝑠 ↔ 𝑔 ∈ 𝑁))
62	61	ralbidva 2968	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (∀𝑔 ∈ 𝐾 (𝑔 ⊕ 𝑠) = 𝑠 ↔ ∀𝑔 ∈ 𝐾 𝑔 ∈ 𝑁))
63		dfss3 3558	. . . . . . . . . . 11 ⊢ (𝐾 ⊆ 𝑁 ↔ ∀𝑔 ∈ 𝐾 𝑔 ∈ 𝑁)
64	62, 63	syl6bbr 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (∀𝑔 ∈ 𝐾 (𝑔 ⊕ 𝑠) = 𝑠 ↔ 𝐾 ⊆ 𝑁))
65	17	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → 𝐾 = (Base‘(𝐺 ↾_s 𝐾)))
66	65	raleqdv 3121	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (∀𝑔 ∈ 𝐾 (𝑔 ⊕ 𝑠) = 𝑠 ↔ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠))
67		eqid 2610	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐺 ↾_s 𝑁)) = (Base‘(𝐺 ↾_s 𝑁))
68	3	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝐺 ∈ Grp)
69	58, 2, 6	nmzsubg 17458	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺))
70	68, 69	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑁 ∈ (SubGrp‘𝐺))
71		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ↾_s 𝑁) = (𝐺 ↾_s 𝑁)
72	71	subgbas 17421	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘(𝐺 ↾_s 𝑁)))
73	70, 72	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑁 = (Base‘(𝐺 ↾_s 𝑁)))
74	4	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑋 ∈ Fin)
75	2	subgss 17418	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ 𝑋)
76	70, 75	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑁 ⊆ 𝑋)
77		ssfi 8065	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ Fin ∧ 𝑁 ⊆ 𝑋) → 𝑁 ∈ Fin)
78	74, 76, 77	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑁 ∈ Fin)
79	73, 78	eqeltrrd 2689	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → (Base‘(𝐺 ↾_s 𝑁)) ∈ Fin)
80	8	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝐾 ∈ (𝑃 pSyl 𝐺))
81		simpr 476	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝐾 ⊆ 𝑁)
82	71	subgslw 17854	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑁) → 𝐾 ∈ (𝑃 pSyl (𝐺 ↾_s 𝑁)))
83	70, 80, 81, 82	syl3anc 1318	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝐾 ∈ (𝑃 pSyl (𝐺 ↾_s 𝑁)))
84		simplr 788	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ∈ (𝑃 pSyl 𝐺))
85	55	ad2antlr 759	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ∈ (SubGrp‘𝐺))
86	58, 2, 6	ssnmz 17459	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (SubGrp‘𝐺) → 𝑠 ⊆ 𝑁)
87	85, 86	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ⊆ 𝑁)
88	71	subgslw 17854	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑠 ∈ (𝑃 pSyl 𝐺) ∧ 𝑠 ⊆ 𝑁) → 𝑠 ∈ (𝑃 pSyl (𝐺 ↾_s 𝑁)))
89	70, 84, 87, 88	syl3anc 1318	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ∈ (𝑃 pSyl (𝐺 ↾_s 𝑁)))
90		fvex 6113	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐺) ∈ V
91	2, 90	eqeltri 2684	. . . . . . . . . . . . . . 15 ⊢ 𝑋 ∈ V
92	58, 91	rabex2 4742	. . . . . . . . . . . . . 14 ⊢ 𝑁 ∈ V
93	71, 6	ressplusg 15818	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ V → + = (+_g‘(𝐺 ↾_s 𝑁)))
94	92, 93	ax-mp 5	. . . . . . . . . . . . 13 ⊢ + = (+_g‘(𝐺 ↾_s 𝑁))
95		eqid 2610	. . . . . . . . . . . . 13 ⊢ (-_g‘(𝐺 ↾_s 𝑁)) = (-_g‘(𝐺 ↾_s 𝑁))
96	67, 79, 83, 89, 94, 95	sylow2 17864	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → ∃𝑔 ∈ (Base‘(𝐺 ↾_s 𝑁))𝐾 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧)(-_g‘(𝐺 ↾_s 𝑁))𝑔)))
97	58, 2, 6, 71	nmznsg 17461	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (SubGrp‘𝐺) → 𝑠 ∈ (NrmSGrp‘(𝐺 ↾_s 𝑁)))
98	85, 97	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 ∈ (NrmSGrp‘(𝐺 ↾_s 𝑁)))
99		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧)(-_g‘(𝐺 ↾_s 𝑁))𝑔)) = (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧)(-_g‘(𝐺 ↾_s 𝑁))𝑔))
100	67, 94, 95, 99	conjnsg 17519	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ (NrmSGrp‘(𝐺 ↾_s 𝑁)) ∧ 𝑔 ∈ (Base‘(𝐺 ↾_s 𝑁))) → 𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧)(-_g‘(𝐺 ↾_s 𝑁))𝑔)))
101	98, 100	sylan 487	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) ∧ 𝑔 ∈ (Base‘(𝐺 ↾_s 𝑁))) → 𝑠 = ran (𝑧 ∈ 𝑠 ↦ ((𝑔 + 𝑧)(-_g‘(𝐺 ↾_s 𝑁))𝑔)))
102		eqeq2 2621	. . . . . . . . . . . . . 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))))
103	101, 102	syl5ibrcom 236	. . . . . . . . . . . . 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))
104	103	rexlimdva 3013	. . . . . . . . . . . 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))
105	96, 104	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝐾 ⊆ 𝑁) → 𝑠 = 𝐾)
106		simpr 476	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑠 = 𝐾) → 𝑠 = 𝐾)
107	15	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑠 = 𝐾) → 𝐾 ∈ (SubGrp‘𝐺))
108	106, 107	eqeltrd 2688	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑠 = 𝐾) → 𝑠 ∈ (SubGrp‘𝐺))
109	108, 86	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑠 = 𝐾) → 𝑠 ⊆ 𝑁)
110	106, 109	eqsstr3d 3603	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) ∧ 𝑠 = 𝐾) → 𝐾 ⊆ 𝑁)
111	105, 110	impbida 873	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (𝐾 ⊆ 𝑁 ↔ 𝑠 = 𝐾))
112	64, 66, 111	3bitr3d 297	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝑃 pSyl 𝐺)) → (∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠 ↔ 𝑠 = 𝐾))
113	112	rabbidva 3163	. . . . . . . 8 ⊢ (𝜑 → {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠} = {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ 𝑠 = 𝐾})
114		rabsn 4200	. . . . . . . . 9 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ 𝑠 = 𝐾} = {𝐾})
115	8, 114	syl 17	. . . . . . . 8 ⊢ (𝜑 → {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ 𝑠 = 𝐾} = {𝐾})
116	113, 115	eqtrd 2644	. . . . . . 7 ⊢ (𝜑 → {𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠} = {𝐾})
117	116	fveq2d 6107	. . . . . 6 ⊢ (𝜑 → (#‘{𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠}) = (#‘{𝐾}))
118		hashsng 13020	. . . . . . 7 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → (#‘{𝐾}) = 1)
119	8, 118	syl 17	. . . . . 6 ⊢ (𝜑 → (#‘{𝐾}) = 1)
120	117, 119	eqtrd 2644	. . . . 5 ⊢ (𝜑 → (#‘{𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠}) = 1)
121	120	oveq2d 6565	. . . 4 ⊢ (𝜑 → ((#‘(𝑃 pSyl 𝐺)) − (#‘{𝑠 ∈ (𝑃 pSyl 𝐺) ∣ ∀𝑔 ∈ (Base‘(𝐺 ↾_s 𝐾))(𝑔 ⊕ 𝑠) = 𝑠})) = ((#‘(𝑃 pSyl 𝐺)) − 1))
122	35, 121	breqtrd 4609	. . 3 ⊢ (𝜑 → 𝑃 ∥ ((#‘(𝑃 pSyl 𝐺)) − 1))
123		prmnn 15226	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
124	5, 123	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ)
125		hashcl 13009	. . . . . 6 ⊢ ((𝑃 pSyl 𝐺) ∈ Fin → (#‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
126	32, 125	syl 17	. . . . 5 ⊢ (𝜑 → (#‘(𝑃 pSyl 𝐺)) ∈ ℕ₀)
127	126	nn0zd 11356	. . . 4 ⊢ (𝜑 → (#‘(𝑃 pSyl 𝐺)) ∈ ℤ)
128		1zzd 11285	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
129		moddvds 14829	. . . 4 ⊢ ((𝑃 ∈ ℕ ∧ (#‘(𝑃 pSyl 𝐺)) ∈ ℤ ∧ 1 ∈ ℤ) → (((#‘(𝑃 pSyl 𝐺)) mod 𝑃) = (1 mod 𝑃) ↔ 𝑃 ∥ ((#‘(𝑃 pSyl 𝐺)) − 1)))
130	124, 127, 128, 129	syl3anc 1318	. . 3 ⊢ (𝜑 → (((#‘(𝑃 pSyl 𝐺)) mod 𝑃) = (1 mod 𝑃) ↔ 𝑃 ∥ ((#‘(𝑃 pSyl 𝐺)) − 1)))
131	122, 130	mpbird 246	. 2 ⊢ (𝜑 → ((#‘(𝑃 pSyl 𝐺)) mod 𝑃) = (1 mod 𝑃))
132		prmuz2 15246	. . 3 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
133		eluz2b2 11637	. . . 4 ⊢ (𝑃 ∈ (ℤ_≥‘2) ↔ (𝑃 ∈ ℕ ∧ 1 < 𝑃))
134		nnre 10904	. . . . 5 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ)
135		1mod 12564	. . . . 5 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
136	134, 135	sylan 487	. . . 4 ⊢ ((𝑃 ∈ ℕ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
137	133, 136	sylbi 206	. . 3 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (1 mod 𝑃) = 1)
138	5, 132, 137	3syl 18	. 2 ⊢ (𝜑 → (1 mod 𝑃) = 1)
139	131, 138	eqtrd 2644	1 ⊢ (𝜑 → ((#‘(𝑃 pSyl 𝐺)) mod 𝑃) = 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 {csn 4125 {cpr 4127 class class class wbr 4583 {copab 4642 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 Fincfn 7841 ℝcr 9814 1c1 9816 < clt 9953 − cmin 10145 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 mod cmo 12530 #chash 12979 ∥ cdvds 14821 ℙcprime 15223 Basecbs 15695 ↾_s cress 15696 +_gcplusg 15768 Grpcgrp 17245 -_gcsg 17247 SubGrpcsubg 17411 NrmSGrpcnsg 17412 pGrp cpgp 17769 pSyl cslw 17770

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-dvds 14822 df-gcd 15055 df-prm 15224 df-pc 15380 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-nsg 17415 df-eqg 17416 df-ghm 17481 df-ga 17546 df-od 17771 df-pgp 17773 df-slw 17774

This theorem is referenced by: sylow3 17871

W3C validator