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Theorem sylow3lem4 19500

Description: Lemma for sylow3 19503, first part. The number of Sylow subgroups is a divisor of the size of 𝐺 reduced by the size of a Sylow subgroup of 𝐺. (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)
sylow3lem1.a	⊢ + = (+_g‘𝐺)
sylow3lem1.d	⊢ − = (-_g‘𝐺)
sylow3lem1.m	⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)))
sylow3lem2.k	⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺))
sylow3lem2.h	⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐾) = 𝐾}
sylow3lem2.n	⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)}

**Assertion**
Ref	Expression
sylow3lem4	⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))))

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

**Proof of Theorem sylow3lem4**
Step	Hyp	Ref	Expression
1		sylow3.x	. . 3 ⊢ 𝑋 = (Base‘𝐺)
2		sylow3.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
3		sylow3.xf	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
4		sylow3.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℙ)
5		sylow3lem1.a	. . 3 ⊢ + = (+_g‘𝐺)
6		sylow3lem1.d	. . 3 ⊢ − = (-_g‘𝐺)
7		sylow3lem1.m	. . 3 ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)))
8		sylow3lem2.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺))
9		sylow3lem2.h	. . 3 ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐾) = 𝐾}
10		sylow3lem2.n	. . 3 ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)}
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	sylow3lem3 19499	. 2 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) = (♯‘(𝑋 / (𝐺 ~_QG 𝑁))))
12		slwsubg 19480	. . . . . . . . . 10 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
13	8, 12	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))
14		eqid 2732	. . . . . . . . . . 11 ⊢ (𝐺 ↾_s 𝑁) = (𝐺 ↾_s 𝑁)
15	10, 1, 5, 14	nmznsg 19050	. . . . . . . . . 10 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ∈ (NrmSGrp‘(𝐺 ↾_s 𝑁)))
16		nsgsubg 19040	. . . . . . . . . 10 ⊢ (𝐾 ∈ (NrmSGrp‘(𝐺 ↾_s 𝑁)) → 𝐾 ∈ (SubGrp‘(𝐺 ↾_s 𝑁)))
17	15, 16	syl 17	. . . . . . . . 9 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ∈ (SubGrp‘(𝐺 ↾_s 𝑁)))
18	13, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘(𝐺 ↾_s 𝑁)))
19	10, 1, 5	nmzsubg 19047	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺))
20	2, 19	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺))
21	14	subgbas 19012	. . . . . . . . . 10 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘(𝐺 ↾_s 𝑁)))
22	20, 21	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 = (Base‘(𝐺 ↾_s 𝑁)))
23	1	subgss 19009	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 ⊆ 𝑋)
24	20, 23	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ⊆ 𝑋)
25	3, 24	ssfid 9269	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ Fin)
26	22, 25	eqeltrrd 2834	. . . . . . . 8 ⊢ (𝜑 → (Base‘(𝐺 ↾_s 𝑁)) ∈ Fin)
27		eqid 2732	. . . . . . . . 9 ⊢ (Base‘(𝐺 ↾_s 𝑁)) = (Base‘(𝐺 ↾_s 𝑁))
28	27	lagsubg 19074	. . . . . . . 8 ⊢ ((𝐾 ∈ (SubGrp‘(𝐺 ↾_s 𝑁)) ∧ (Base‘(𝐺 ↾_s 𝑁)) ∈ Fin) → (♯‘𝐾) ∥ (♯‘(Base‘(𝐺 ↾_s 𝑁))))
29	18, 26, 28	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐾) ∥ (♯‘(Base‘(𝐺 ↾_s 𝑁))))
30	22	fveq2d 6895	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑁) = (♯‘(Base‘(𝐺 ↾_s 𝑁))))
31	29, 30	breqtrrd 5176	. . . . . 6 ⊢ (𝜑 → (♯‘𝐾) ∥ (♯‘𝑁))
32		eqid 2732	. . . . . . . . . . . 12 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
33	32	subg0cl 19016	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ 𝐾)
34	13, 33	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0_g‘𝐺) ∈ 𝐾)
35	34	ne0d 4335	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ≠ ∅)
36	1	subgss 19009	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋)
37	13, 36	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ⊆ 𝑋)
38	3, 37	ssfid 9269	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Fin)
39		hashnncl 14328	. . . . . . . . . 10 ⊢ (𝐾 ∈ Fin → ((♯‘𝐾) ∈ ℕ ↔ 𝐾 ≠ ∅))
40	38, 39	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝐾) ∈ ℕ ↔ 𝐾 ≠ ∅))
41	35, 40	mpbird 256	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝐾) ∈ ℕ)
42	41	nnzd 12587	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐾) ∈ ℤ)
43		hashcl 14318	. . . . . . . . 9 ⊢ (𝑁 ∈ Fin → (♯‘𝑁) ∈ ℕ₀)
44	25, 43	syl 17	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑁) ∈ ℕ₀)
45	44	nn0zd 12586	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑁) ∈ ℤ)
46		pwfi 9180	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
47	3, 46	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
48		eqid 2732	. . . . . . . . . . . . 13 ⊢ (𝐺 ~_QG 𝑁) = (𝐺 ~_QG 𝑁)
49	1, 48	eqger 19060	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → (𝐺 ~_QG 𝑁) Er 𝑋)
50	20, 49	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ~_QG 𝑁) Er 𝑋)
51	50	qsss 8774	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 / (𝐺 ~_QG 𝑁)) ⊆ 𝒫 𝑋)
52	47, 51	ssfid 9269	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 / (𝐺 ~_QG 𝑁)) ∈ Fin)
53		hashcl 14318	. . . . . . . . 9 ⊢ ((𝑋 / (𝐺 ~_QG 𝑁)) ∈ Fin → (♯‘(𝑋 / (𝐺 ~_QG 𝑁))) ∈ ℕ₀)
54	52, 53	syl 17	. . . . . . . 8 ⊢ (𝜑 → (♯‘(𝑋 / (𝐺 ~_QG 𝑁))) ∈ ℕ₀)
55	54	nn0zd 12586	. . . . . . 7 ⊢ (𝜑 → (♯‘(𝑋 / (𝐺 ~_QG 𝑁))) ∈ ℤ)
56		dvdscmul 16228	. . . . . . 7 ⊢ (((♯‘𝐾) ∈ ℤ ∧ (♯‘𝑁) ∈ ℤ ∧ (♯‘(𝑋 / (𝐺 ~_QG 𝑁))) ∈ ℤ) → ((♯‘𝐾) ∥ (♯‘𝑁) → ((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝐾)) ∥ ((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝑁))))
57	42, 45, 55, 56	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → ((♯‘𝐾) ∥ (♯‘𝑁) → ((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝐾)) ∥ ((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝑁))))
58	31, 57	mpd 15	. . . . 5 ⊢ (𝜑 → ((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝐾)) ∥ ((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝑁)))
59		hashcl 14318	. . . . . . . . 9 ⊢ (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ₀)
60	3, 59	syl 17	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ₀)
61	60	nn0cnd 12536	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑋) ∈ ℂ)
62	41	nncnd 12230	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐾) ∈ ℂ)
63	41	nnne0d 12264	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐾) ≠ 0)
64	61, 62, 63	divcan1d 11993	. . . . . 6 ⊢ (𝜑 → (((♯‘𝑋) / (♯‘𝐾)) · (♯‘𝐾)) = (♯‘𝑋))
65	1, 48, 20, 3	lagsubg2 19073	. . . . . 6 ⊢ (𝜑 → (♯‘𝑋) = ((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝑁)))
66	64, 65	eqtrd 2772	. . . . 5 ⊢ (𝜑 → (((♯‘𝑋) / (♯‘𝐾)) · (♯‘𝐾)) = ((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝑁)))
67	58, 66	breqtrrd 5176	. . . 4 ⊢ (𝜑 → ((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝐾)) ∥ (((♯‘𝑋) / (♯‘𝐾)) · (♯‘𝐾)))
68	1	lagsubg 19074	. . . . . . 7 ⊢ ((𝐾 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (♯‘𝐾) ∥ (♯‘𝑋))
69	13, 3, 68	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (♯‘𝐾) ∥ (♯‘𝑋))
70	60	nn0zd 12586	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑋) ∈ ℤ)
71		dvdsval2 16202	. . . . . . 7 ⊢ (((♯‘𝐾) ∈ ℤ ∧ (♯‘𝐾) ≠ 0 ∧ (♯‘𝑋) ∈ ℤ) → ((♯‘𝐾) ∥ (♯‘𝑋) ↔ ((♯‘𝑋) / (♯‘𝐾)) ∈ ℤ))
72	42, 63, 70, 71	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → ((♯‘𝐾) ∥ (♯‘𝑋) ↔ ((♯‘𝑋) / (♯‘𝐾)) ∈ ℤ))
73	69, 72	mpbid 231	. . . . 5 ⊢ (𝜑 → ((♯‘𝑋) / (♯‘𝐾)) ∈ ℤ)
74		dvdsmulcr 16231	. . . . 5 ⊢ (((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) ∈ ℤ ∧ ((♯‘𝑋) / (♯‘𝐾)) ∈ ℤ ∧ ((♯‘𝐾) ∈ ℤ ∧ (♯‘𝐾) ≠ 0)) → (((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝐾)) ∥ (((♯‘𝑋) / (♯‘𝐾)) · (♯‘𝐾)) ↔ (♯‘(𝑋 / (𝐺 ~_QG 𝑁))) ∥ ((♯‘𝑋) / (♯‘𝐾))))
75	55, 73, 42, 63, 74	syl112anc 1374	. . . 4 ⊢ (𝜑 → (((♯‘(𝑋 / (𝐺 ~_QG 𝑁))) · (♯‘𝐾)) ∥ (((♯‘𝑋) / (♯‘𝐾)) · (♯‘𝐾)) ↔ (♯‘(𝑋 / (𝐺 ~_QG 𝑁))) ∥ ((♯‘𝑋) / (♯‘𝐾))))
76	67, 75	mpbid 231	. . 3 ⊢ (𝜑 → (♯‘(𝑋 / (𝐺 ~_QG 𝑁))) ∥ ((♯‘𝑋) / (♯‘𝐾)))
77	1, 3, 8	slwhash 19494	. . . 4 ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
78	77	oveq2d 7427	. . 3 ⊢ (𝜑 → ((♯‘𝑋) / (♯‘𝐾)) = ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))))
79	76, 78	breqtrd 5174	. 2 ⊢ (𝜑 → (♯‘(𝑋 / (𝐺 ~_QG 𝑁))) ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))))
80	11, 79	eqbrtrd 5170	1 ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 {crab 3432 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 class class class wbr 5148 ↦ cmpt 5231 ran crn 5677 ‘cfv 6543 (class class class)co 7411 ∈ cmpo 7413 Er wer 8702 / cqs 8704 Fincfn 8941 0cc0 11112 · cmul 11117 / cdiv 11873 ℕcn 12214 ℕ₀cn0 12474 ℤcz 12560 ↑cexp 14029 ♯chash 14292 ∥ cdvds 16199 ℙcprime 16610 pCnt cpc 16771 Basecbs 17146 ↾_s cress 17175 +_gcplusg 17199 0_gc0g 17387 Grpcgrp 18821 -_gcsg 18823 SubGrpcsubg 19002 NrmSGrpcnsg 19003 ~_QG cqg 19004 pSyl cslw 19397

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-omul 8473 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-n0 12475 df-xnn0 12547 df-z 12561 df-uz 12825 df-q 12935 df-rp 12977 df-fz 13487 df-fzo 13630 df-fl 13759 df-mod 13837 df-seq 13969 df-exp 14030 df-fac 14236 df-bc 14265 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-sum 15635 df-dvds 16200 df-gcd 16438 df-prm 16611 df-pc 16772 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-0g 17389 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-grp 18824 df-minusg 18825 df-sbg 18826 df-mulg 18953 df-subg 19005 df-nsg 19006 df-eqg 19007 df-ghm 19092 df-ga 19156 df-od 19398 df-pgp 19400 df-slw 19401

This theorem is referenced by: sylow3 19503

W3C validator