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Theorem fislw 19600

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 19585	. . . 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 1194	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → 𝑋 ∈ Fin)
6	4, 5, 1	slwhash 19599	. . 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 1195	. . 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 1194	. . . . . . . . 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 19103	. . . . . . . . 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 9179	. . . . . . 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 8947	. . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ↾_s 𝑘) = (𝐺 ↾_s 𝑘)
21	20	subggrp 19105	. . . . . . . . . . . . . . . . 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 19106	. . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(𝐺 ↾_s 𝑘)) = (Base‘(𝐺 ↾_s 𝑘))
27	26	pgpfi 19580	. . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . 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 16643	. . . . . . . . . . . . 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 12189	. . . . . . . . . . 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 12225	. . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
36	35	subg0cl 19110	. . . . . . . . . . . . . . . . 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 4282	. . . . . . . . . . . . . . 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 14328	. . . . . . . . . . . . . . . 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 16821	. . . . . . . . . . . . 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 12549	. . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐺 ∈ Grp)
45	4	grpbn0 18942	. . . . . . . . . . . . . . . . 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 14328	. . . . . . . . . . . . . . . . 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 16821	. . . . . . . . . . . . . 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 12549	. . . . . . . . . . . 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 7374	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑃 → (𝑝 pCnt (♯‘𝑘)) = (𝑃 pCnt (♯‘𝑘)))
54		oveq1 7374	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑃 → (𝑝 pCnt (♯‘𝑋)) = (𝑃 pCnt (♯‘𝑋)))
55	53, 54	breq12d 5098	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑃 → ((𝑝 pCnt (♯‘𝑘)) ≤ (𝑝 pCnt (♯‘𝑋)) ↔ (𝑃 pCnt (♯‘𝑘)) ≤ (𝑃 pCnt (♯‘𝑋))))
56	4	lagsubg 19170	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (♯‘𝑘) ∥ (♯‘𝑋))
57	12, 11, 56	syl2anc 585	. . . . . . . . . . . . . 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 12550	. . . . . . . . . . . . . . 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 12550	. . . . . . . . . . . . . . 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 16850	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝑘) ∈ ℤ ∧ (♯‘𝑋) ∈ ℤ) → ((♯‘𝑘) ∥ (♯‘𝑋) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (♯‘𝑘)) ≤ (𝑝 pCnt (♯‘𝑋))))
62	58, 60, 61	syl2anc 585	. . . . . . . . . . . . . 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 3565	. . . . . . . . . . . 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 12794	. . . . . . . . . . . 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 1345	. . . . . . . . . . 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 14216	. . . . . . . . . 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 6848	. . . . . . . . . . . . 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 3161	. . . . . . . . . . . 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 16854	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝑘) ∈ ℕ) → (∃𝑛 ∈ ℕ₀ (♯‘𝑘) = (𝑃↑𝑛) ↔ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑘)))))
73	30, 41, 72	syl2anc 585	. . . . . . . . . . 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 5117	. . . . . . . . 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 19103	. . . . . . . . . . . . 13 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋)
78	77	ad2antrl 729	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 ⊆ 𝑋)
79	10, 78	ssfid 9179	. . . . . . . . . . 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 14341	. . . . . . . . . 10 ⊢ ((𝑘 ∈ Fin ∧ 𝐻 ∈ Fin) → ((♯‘𝑘) ≤ (♯‘𝐻) ↔ 𝑘 ≼ 𝐻))
82	15, 80, 81	syl2anc 585	. . . . . . . . 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 9035	. . . . . . . 8 ⊢ ((𝐻 ≼ 𝑘 ∧ 𝑘 ≼ 𝐻) → 𝐻 ≈ 𝑘)
85	18, 83, 84	syl2anc 585	. . . . . . 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 9173	. . . . . . 7 ⊢ ((𝑘 ∈ Fin ∧ 𝐻 ⊆ 𝑘 ∧ 𝐻 ≈ 𝑘) → 𝐻 = 𝑘)
87	15, 16, 85, 86	syl3anc 1374	. . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (𝐺 ↾_s 𝐻) = (𝐺 ↾_s 𝐻)
90	89	subgbas 19106	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 = (Base‘(𝐺 ↾_s 𝐻)))
91	90	ad2antrl 729	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 = (Base‘(𝐺 ↾_s 𝐻)))
92	91	fveq2d 6844	. . . . . . . . . 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 2773	. . . . . . . . 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 7375	. . . . . . . . . 10 ⊢ (𝑛 = (𝑃 pCnt (♯‘𝑋)) → (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
96	95	rspceeqv 3587	. . . . . . . . 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 585	. . . . . . . 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 19105	. . . . . . . . . 10 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → (𝐺 ↾_s 𝐻) ∈ Grp)
99	98	ad2antrl 729	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (𝐺 ↾_s 𝐻) ∈ Grp)
100	91, 79	eqeltrrd 2837	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (Base‘(𝐺 ↾_s 𝐻)) ∈ Fin)
101		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘(𝐺 ↾_s 𝐻)) = (Base‘(𝐺 ↾_s 𝐻))
102	101	pgpfi 19580	. . . . . . . . 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 585	. . . . . . . 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 714	. . . . . . 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 7375	. . . . . . . 8 ⊢ (𝐻 = 𝑘 → (𝐺 ↾_s 𝐻) = (𝐺 ↾_s 𝑘))
107	106	breq2d 5097	. . . . . . 7 ⊢ (𝐻 = 𝑘 → (𝑃 pGrp (𝐺 ↾_s 𝐻) ↔ 𝑃 pGrp (𝐺 ↾_s 𝑘)))
108		eqimss 3980	. . . . . . . 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 3129	. . 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 19583	. . 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 1345	. 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 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 ⊆ wss 3889 ∅c0 4273 class class class wbr 5085 ‘cfv 6498 (class class class)co 7367 ≈ cen 8890 ≼ cdom 8891 Fincfn 8893 ≤ cle 11180 ℕcn 12174 ℕ₀cn0 12437 ℤcz 12524 ℤ_≥cuz 12788 ↑cexp 14023 ♯chash 14292 ∥ cdvds 16221 ℙcprime 16640 pCnt cpc 16807 Basecbs 17179 ↾_s cress 17200 0_gc0g 17402 Grpcgrp 18909 SubGrpcsubg 19096 pGrp cpgp 19501 pSyl cslw 19502

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-disj 5053 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-ec 8645 df-qs 8649 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-xnn0 12511 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-sum 15649 df-dvds 16222 df-gcd 16464 df-prm 16641 df-pc 16808 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-0g 17404 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-eqg 19101 df-ghm 19188 df-ga 19265 df-od 19503 df-pgp 19505 df-slw 19506

This theorem is referenced by: sylow3lem1 19602

W3C validator