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Theorem fislw 19658

Description: The sylow subgroups of a finite group are exactly the groups which have cardinality equal to the maximum power of 𝑃 dividing the group. (Contributed by Mario Carneiro, 16-Jan-2015.)

**Hypothesis**
Ref	Expression
fislw.1	⊢ 𝑋 = (Base‘𝐺)

**Assertion**
Ref	Expression
fislw	⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))))

Proof of Theorem fislw Dummy variables 𝑘 𝑛 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → 𝐻 ∈ (𝑃 pSyl 𝐺))
2		slwsubg 19643	. . . 4 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
3	1, 2	syl 17	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → 𝐻 ∈ (SubGrp‘𝐺))
4		fislw.1	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
5		simpl2 1191	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → 𝑋 ∈ Fin)
6	4, 5, 1	slwhash 19657	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
7	3, 6	jca 511	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))))
8		simpl3 1192	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑃 ∈ ℙ)
9		simprl 771	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 ∈ (SubGrp‘𝐺))
10		simpl2 1191	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑋 ∈ Fin)
11	10	adantr 480	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → 𝑋 ∈ Fin)
12		simprl 771	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → 𝑘 ∈ (SubGrp‘𝐺))
13	4	subgss 19158	. . . . . . . . 9 ⊢ (𝑘 ∈ (SubGrp‘𝐺) → 𝑘 ⊆ 𝑋)
14	12, 13	syl 17	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → 𝑘 ⊆ 𝑋)
15	11, 14	ssfid 9299	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → 𝑘 ∈ Fin)
16		simprrl 781	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → 𝐻 ⊆ 𝑘)
17		ssdomg 9039	. . . . . . . . 9 ⊢ (𝑘 ∈ Fin → (𝐻 ⊆ 𝑘 → 𝐻 ≼ 𝑘))
18	15, 16, 17	sylc 65	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → 𝐻 ≼ 𝑘)
19		simprrr 782	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → 𝑃 pGrp (𝐺 ↾_s 𝑘))
20		eqid 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ↾_s 𝑘) = (𝐺 ↾_s 𝑘)
21	20	subggrp 19160	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (SubGrp‘𝐺) → (𝐺 ↾_s 𝑘) ∈ Grp)
22	12, 21	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (𝐺 ↾_s 𝑘) ∈ Grp)
23	20	subgbas 19161	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (SubGrp‘𝐺) → 𝑘 = (Base‘(𝐺 ↾_s 𝑘)))
24	12, 23	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → 𝑘 = (Base‘(𝐺 ↾_s 𝑘)))
25	24, 15	eqeltrrd 2840	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (Base‘(𝐺 ↾_s 𝑘)) ∈ Fin)
26		eqid 2735	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(𝐺 ↾_s 𝑘)) = (Base‘(𝐺 ↾_s 𝑘))
27	26	pgpfi 19638	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ↾_s 𝑘) ∈ Grp ∧ (Base‘(𝐺 ↾_s 𝑘)) ∈ Fin) → (𝑃 pGrp (𝐺 ↾_s 𝑘) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ₀ (♯‘(Base‘(𝐺 ↾_s 𝑘))) = (𝑃↑𝑛))))
28	22, 25, 27	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (𝑃 pGrp (𝐺 ↾_s 𝑘) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ₀ (♯‘(Base‘(𝐺 ↾_s 𝑘))) = (𝑃↑𝑛))))
29	19, 28	mpbid 232	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ₀ (♯‘(Base‘(𝐺 ↾_s 𝑘))) = (𝑃↑𝑛)))
30	29	simpld 494	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → 𝑃 ∈ ℙ)
31		prmnn 16708	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
32	30, 31	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → 𝑃 ∈ ℕ)
33	32	nnred 12279	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → 𝑃 ∈ ℝ)
34	32	nnge1d 12312	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → 1 ≤ 𝑃)
35		eqid 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
36	35	subg0cl 19165	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ 𝑘)
37	12, 36	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (0_g‘𝐺) ∈ 𝑘)
38	37	ne0d 4348	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → 𝑘 ≠ ∅)
39		hashnncl 14402	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ Fin → ((♯‘𝑘) ∈ ℕ ↔ 𝑘 ≠ ∅))
40	15, 39	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → ((♯‘𝑘) ∈ ℕ ↔ 𝑘 ≠ ∅))
41	38, 40	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (♯‘𝑘) ∈ ℕ)
42	30, 41	pccld 16884	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (𝑃 pCnt (♯‘𝑘)) ∈ ℕ₀)
43	42	nn0zd 12637	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (𝑃 pCnt (♯‘𝑘)) ∈ ℤ)
44		simpl1 1190	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐺 ∈ Grp)
45	4	grpbn0 18997	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
46	44, 45	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑋 ≠ ∅)
47		hashnncl 14402	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
48	10, 47	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
49	46, 48	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝑋) ∈ ℕ)
50	8, 49	pccld 16884	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ₀)
51	50	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ₀)
52	51	nn0zd 12637	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (𝑃 pCnt (♯‘𝑋)) ∈ ℤ)
53		oveq1 7438	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑃 → (𝑝 pCnt (♯‘𝑘)) = (𝑃 pCnt (♯‘𝑘)))
54		oveq1 7438	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑃 → (𝑝 pCnt (♯‘𝑋)) = (𝑃 pCnt (♯‘𝑋)))
55	53, 54	breq12d 5161	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑃 → ((𝑝 pCnt (♯‘𝑘)) ≤ (𝑝 pCnt (♯‘𝑋)) ↔ (𝑃 pCnt (♯‘𝑘)) ≤ (𝑃 pCnt (♯‘𝑋))))
56	4	lagsubg 19226	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (♯‘𝑘) ∥ (♯‘𝑋))
57	12, 11, 56	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (♯‘𝑘) ∥ (♯‘𝑋))
58	41	nnzd 12638	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (♯‘𝑘) ∈ ℤ)
59	49	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (♯‘𝑋) ∈ ℕ)
60	59	nnzd 12638	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (♯‘𝑋) ∈ ℤ)
61		pc2dvds 16913	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝑘) ∈ ℤ ∧ (♯‘𝑋) ∈ ℤ) → ((♯‘𝑘) ∥ (♯‘𝑋) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (♯‘𝑘)) ≤ (𝑝 pCnt (♯‘𝑋))))
62	58, 60, 61	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → ((♯‘𝑘) ∥ (♯‘𝑋) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (♯‘𝑘)) ≤ (𝑝 pCnt (♯‘𝑋))))
63	57, 62	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → ∀𝑝 ∈ ℙ (𝑝 pCnt (♯‘𝑘)) ≤ (𝑝 pCnt (♯‘𝑋)))
64	55, 63, 30	rspcdva 3623	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (𝑃 pCnt (♯‘𝑘)) ≤ (𝑃 pCnt (♯‘𝑋)))
65		eluz2 12882	. . . . . . . . . . . 12 ⊢ ((𝑃 pCnt (♯‘𝑋)) ∈ (ℤ_≥‘(𝑃 pCnt (♯‘𝑘))) ↔ ((𝑃 pCnt (♯‘𝑘)) ∈ ℤ ∧ (𝑃 pCnt (♯‘𝑋)) ∈ ℤ ∧ (𝑃 pCnt (♯‘𝑘)) ≤ (𝑃 pCnt (♯‘𝑋))))
66	43, 52, 64, 65	syl3anbrc 1342	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (𝑃 pCnt (♯‘𝑋)) ∈ (ℤ_≥‘(𝑃 pCnt (♯‘𝑘))))
67	33, 34, 66	leexp2ad 14290	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (𝑃↑(𝑃 pCnt (♯‘𝑘))) ≤ (𝑃↑(𝑃 pCnt (♯‘𝑋))))
68	29	simprd 495	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → ∃𝑛 ∈ ℕ₀ (♯‘(Base‘(𝐺 ↾_s 𝑘))) = (𝑃↑𝑛))
69	24	fveqeq2d 6915	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → ((♯‘𝑘) = (𝑃↑𝑛) ↔ (♯‘(Base‘(𝐺 ↾_s 𝑘))) = (𝑃↑𝑛)))
70	69	rexbidv 3177	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (∃𝑛 ∈ ℕ₀ (♯‘𝑘) = (𝑃↑𝑛) ↔ ∃𝑛 ∈ ℕ₀ (♯‘(Base‘(𝐺 ↾_s 𝑘))) = (𝑃↑𝑛)))
71	68, 70	mpbird 257	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → ∃𝑛 ∈ ℕ₀ (♯‘𝑘) = (𝑃↑𝑛))
72		pcprmpw 16917	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝑘) ∈ ℕ) → (∃𝑛 ∈ ℕ₀ (♯‘𝑘) = (𝑃↑𝑛) ↔ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑘)))))
73	30, 41, 72	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (∃𝑛 ∈ ℕ₀ (♯‘𝑘) = (𝑃↑𝑛) ↔ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑘)))))
74	71, 73	mpbid 232	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑘))))
75		simplrr 778	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
76	67, 74, 75	3brtr4d 5180	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → (♯‘𝑘) ≤ (♯‘𝐻))
77	4	subgss 19158	. . . . . . . . . . . . 13 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋)
78	77	ad2antrl 728	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 ⊆ 𝑋)
79	10, 78	ssfid 9299	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 ∈ Fin)
80	79	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → 𝐻 ∈ Fin)
81		hashdom 14415	. . . . . . . . . 10 ⊢ ((𝑘 ∈ Fin ∧ 𝐻 ∈ Fin) → ((♯‘𝑘) ≤ (♯‘𝐻) ↔ 𝑘 ≼ 𝐻))
82	15, 80, 81	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → ((♯‘𝑘) ≤ (♯‘𝐻) ↔ 𝑘 ≼ 𝐻))
83	76, 82	mpbid 232	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → 𝑘 ≼ 𝐻)
84		sbth 9132	. . . . . . . 8 ⊢ ((𝐻 ≼ 𝑘 ∧ 𝑘 ≼ 𝐻) → 𝐻 ≈ 𝑘)
85	18, 83, 84	syl2anc 584	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → 𝐻 ≈ 𝑘)
86		fisseneq 9291	. . . . . . 7 ⊢ ((𝑘 ∈ Fin ∧ 𝐻 ⊆ 𝑘 ∧ 𝐻 ≈ 𝑘) → 𝐻 = 𝑘)
87	15, 16, 85, 86	syl3anc 1370	. . . . . 6 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)))) → 𝐻 = 𝑘)
88	87	expr 456	. . . . 5 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)) → 𝐻 = 𝑘))
89		eqid 2735	. . . . . . . . . . . . 13 ⊢ (𝐺 ↾_s 𝐻) = (𝐺 ↾_s 𝐻)
90	89	subgbas 19161	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 = (Base‘(𝐺 ↾_s 𝐻)))
91	90	ad2antrl 728	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 = (Base‘(𝐺 ↾_s 𝐻)))
92	91	fveq2d 6911	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝐻) = (♯‘(Base‘(𝐺 ↾_s 𝐻))))
93		simprr 773	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
94	92, 93	eqtr3d 2777	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘(Base‘(𝐺 ↾_s 𝐻))) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
95		oveq2 7439	. . . . . . . . . 10 ⊢ (𝑛 = (𝑃 pCnt (♯‘𝑋)) → (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
96	95	rspceeqv 3645	. . . . . . . . 9 ⊢ (((𝑃 pCnt (♯‘𝑋)) ∈ ℕ₀ ∧ (♯‘(Base‘(𝐺 ↾_s 𝐻))) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) → ∃𝑛 ∈ ℕ₀ (♯‘(Base‘(𝐺 ↾_s 𝐻))) = (𝑃↑𝑛))
97	50, 94, 96	syl2anc 584	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → ∃𝑛 ∈ ℕ₀ (♯‘(Base‘(𝐺 ↾_s 𝐻))) = (𝑃↑𝑛))
98	89	subggrp 19160	. . . . . . . . . 10 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → (𝐺 ↾_s 𝐻) ∈ Grp)
99	98	ad2antrl 728	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (𝐺 ↾_s 𝐻) ∈ Grp)
100	91, 79	eqeltrrd 2840	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (Base‘(𝐺 ↾_s 𝐻)) ∈ Fin)
101		eqid 2735	. . . . . . . . . 10 ⊢ (Base‘(𝐺 ↾_s 𝐻)) = (Base‘(𝐺 ↾_s 𝐻))
102	101	pgpfi 19638	. . . . . . . . 9 ⊢ (((𝐺 ↾_s 𝐻) ∈ Grp ∧ (Base‘(𝐺 ↾_s 𝐻)) ∈ Fin) → (𝑃 pGrp (𝐺 ↾_s 𝐻) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ₀ (♯‘(Base‘(𝐺 ↾_s 𝐻))) = (𝑃↑𝑛))))
103	99, 100, 102	syl2anc 584	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (𝑃 pGrp (𝐺 ↾_s 𝐻) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ₀ (♯‘(Base‘(𝐺 ↾_s 𝐻))) = (𝑃↑𝑛))))
104	8, 97, 103	mpbir2and 713	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑃 pGrp (𝐺 ↾_s 𝐻))
105	104	adantr 480	. . . . . 6 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺 ↾_s 𝐻))
106		oveq2 7439	. . . . . . . 8 ⊢ (𝐻 = 𝑘 → (𝐺 ↾_s 𝐻) = (𝐺 ↾_s 𝑘))
107	106	breq2d 5160	. . . . . . 7 ⊢ (𝐻 = 𝑘 → (𝑃 pGrp (𝐺 ↾_s 𝐻) ↔ 𝑃 pGrp (𝐺 ↾_s 𝑘)))
108		eqimss 4054	. . . . . . . 8 ⊢ (𝐻 = 𝑘 → 𝐻 ⊆ 𝑘)
109	108	biantrurd 532	. . . . . . 7 ⊢ (𝐻 = 𝑘 → (𝑃 pGrp (𝐺 ↾_s 𝑘) ↔ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘))))
110	107, 109	bitrd 279	. . . . . 6 ⊢ (𝐻 = 𝑘 → (𝑃 pGrp (𝐺 ↾_s 𝐻) ↔ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘))))
111	105, 110	syl5ibcom 245	. . . . 5 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → (𝐻 = 𝑘 → (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘))))
112	88, 111	impbid 212	. . . 4 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)) ↔ 𝐻 = 𝑘))
113	112	ralrimiva 3144	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)) ↔ 𝐻 = 𝑘))
114		isslw 19641	. . 3 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑘)) ↔ 𝐻 = 𝑘)))
115	8, 9, 113, 114	syl3anbrc 1342	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 ∈ (𝑃 pSyl 𝐺))
116	7, 115	impbida 801	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 ∅c0 4339 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ≈ cen 8981 ≼ cdom 8982 Fincfn 8984 ≤ cle 11294 ℕcn 12264 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12876 ↑cexp 14099 ♯chash 14366 ∥ cdvds 16287 ℙcprime 16705 pCnt cpc 16870 Basecbs 17245 ↾_s cress 17274 0_gc0g 17486 Grpcgrp 18964 SubGrpcsubg 19151 pGrp cpgp 19559 pSyl cslw 19560

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-dvds 16288 df-gcd 16529 df-prm 16706 df-pc 16871 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-eqg 19156 df-ghm 19244 df-ga 19321 df-od 19561 df-pgp 19563 df-slw 19564

This theorem is referenced by: sylow3lem1 19660

W3C validator