Theorem fislw 17863

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 476	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → 𝐻 ∈ (𝑃 pSyl 𝐺))
2		slwsubg 17848	. . . 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 1058	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → 𝑋 ∈ Fin)
6	4, 5, 1	slwhash 17862	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
7	3, 6	jca 553	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋)))))
8		simpl3 1059	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝑃 ∈ ℙ)
9		simprl 790	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐻 ∈ (SubGrp‘𝐺))
10		simpl2 1058	. . . . . . . . 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 790	. . . . . . . . 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 17418	. . . . . . . . 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		ssfi 8065	. . . . . . . 8 ⊢ ((𝑋 ∈ Fin ∧ 𝑘 ⊆ 𝑋) → 𝑘 ∈ Fin)
16	11, 14, 15	syl2anc 691	. . . . . . 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)
17		simprrl 800	. . . . . . 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)
18		ssdomg 7887	. . . . . . . . 9 ⊢ (𝑘 ∈ Fin → (𝐻 ⊆ 𝑘 → 𝐻 ≼ 𝑘))
19	16, 17, 18	sylc 63	. . . . . . . 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)
20		simprrr 801	. . . . . . . . . . . . . . 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))
21		eqid 2610	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ↾s 𝑘) = (𝐺 ↾s 𝑘)
22	21	subggrp 17420	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑘) ∈ Grp)
23	12, 22	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)
24	21	subgbas 17421	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (SubGrp‘𝐺) → 𝑘 = (Base‘(𝐺 ↾s 𝑘)))
25	12, 24	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)))
26	25, 16	eqeltrrd 2689	. . . . . . . . . . . . . . . 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)
27		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(𝐺 ↾s 𝑘)) = (Base‘(𝐺 ↾s 𝑘))
28	27	pgpfi 17843	. . . . . . . . . . . . . . . 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))))
29	23, 26, 28	syl2anc 691	. . . . . . . . . . . . . . 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))))
30	20, 29	mpbid 221	. . . . . . . . . . . . . 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)))
31	30	simpld 474	. . . . . . . . . . . . 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)
32		prmnn 15226	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
33	31, 32	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)
34	33	nnred 10912	. . . . . . . . . . 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)
35	33	nnge1d 10940	. . . . . . . . . . 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)
36		eqid 2610	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘𝐺) = (0g‘𝐺)
37	36	subg0cl 17425	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑘)
38	12, 37	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)))) → (0g‘𝐺) ∈ 𝑘)
39		ne0i 3880	. . . . . . . . . . . . . . . 16 ⊢ ((0g‘𝐺) ∈ 𝑘 → 𝑘 ≠ ∅)
40	38, 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 ≠ ∅)
41		hashnncl 13018	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ Fin → ((#‘𝑘) ∈ ℕ ↔ 𝑘 ≠ ∅))
42	16, 41	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 ≠ ∅))
43	40, 42	mpbird 246	. . . . . . . . . . . . . 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)
44	31, 43	pccld 15393	. . . . . . . . . . . . 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)
45	44	nn0zd 11356	. . . . . . . . . . . 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)
46		simpl1 1057	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐺 ∈ Grp)
47	4	grpbn0 17274	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
48	46, 47	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝑋 ≠ ∅)
49		hashnncl 13018	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ Fin → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
50	10, 49	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
51	48, 50	mpbird 246	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (#‘𝑋) ∈ ℕ)
52	8, 51	pccld 15393	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (𝑃 pCnt (#‘𝑋)) ∈ ℕ0)
53	52	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)
54	53	nn0zd 11356	. . . . . . . . . . . 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)
55	4	lagsubg 17479	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (#‘𝑘) ∥ (#‘𝑋))
56	12, 11, 55	syl2anc 691	. . . . . . . . . . . . . 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))
57	43	nnzd 11357	. . . . . . . . . . . . . . 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)
58	51	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)
59	58	nnzd 11357	. . . . . . . . . . . . . . 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)
60		pc2dvds 15421	. . . . . . . . . . . . . . 15 ⊢ (((#‘𝑘) ∈ ℤ ∧ (#‘𝑋) ∈ ℤ) → ((#‘𝑘) ∥ (#‘𝑋) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (#‘𝑘)) ≤ (𝑝 pCnt (#‘𝑋))))
61	57, 59, 60	syl2anc 691	. . . . . . . . . . . . . 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))))
62	56, 61	mpbid 221	. . . . . . . . . . . . 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)))
63		oveq1 6556	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑃 → (𝑝 pCnt (#‘𝑘)) = (𝑃 pCnt (#‘𝑘)))
64		oveq1 6556	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑃 → (𝑝 pCnt (#‘𝑋)) = (𝑃 pCnt (#‘𝑋)))
65	63, 64	breq12d 4596	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑃 → ((𝑝 pCnt (#‘𝑘)) ≤ (𝑝 pCnt (#‘𝑋)) ↔ (𝑃 pCnt (#‘𝑘)) ≤ (𝑃 pCnt (#‘𝑋))))
66	65	rspcv 3278	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → (∀𝑝 ∈ ℙ (𝑝 pCnt (#‘𝑘)) ≤ (𝑝 pCnt (#‘𝑋)) → (𝑃 pCnt (#‘𝑘)) ≤ (𝑃 pCnt (#‘𝑋))))
67	31, 62, 66	sylc 63	. . . . . . . . . . . 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)))
68		eluz2 11569	. . . . . . . . . . . 12 ⊢ ((𝑃 pCnt (#‘𝑋)) ∈ (ℤ≥‘(𝑃 pCnt (#‘𝑘))) ↔ ((𝑃 pCnt (#‘𝑘)) ∈ ℤ ∧ (𝑃 pCnt (#‘𝑋)) ∈ ℤ ∧ (𝑃 pCnt (#‘𝑘)) ≤ (𝑃 pCnt (#‘𝑋))))
69	45, 54, 67, 68	syl3anbrc 1239	. . . . . . . . . . 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))))
70	34, 35, 69	leexp2ad 12903	. . . . . . . . . 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))))
71	30	simprd 478	. . . . . . . . . . . 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))
72	25	fveq2d 6107	. . . . . . . . . . . . . 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) = (#‘(Base‘(𝐺 ↾s 𝑘))))
73	72	eqeq1d 2612	. . . . . . . . . . . . 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)))
74	73	rexbidv 3034	. . . . . . . . . . . 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)))
75	71, 74	mpbird 246	. . . . . . . . . . 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))
76		pcprmpw 15425	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ (#‘𝑘) ∈ ℕ) → (∃𝑛 ∈ ℕ0 (#‘𝑘) = (𝑃↑𝑛) ↔ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑘)))))
77	31, 43, 76	syl2anc 691	. . . . . . . . . . 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)))))
78	75, 77	mpbid 221	. . . . . . . . . 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))))
79		simplrr 797	. . . . . . . . . 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))))
80	70, 78, 79	3brtr4d 4615	. . . . . . . . 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))
81	4	subgss 17418	. . . . . . . . . . . . 13 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋)
82	81	ad2antrl 760	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐻 ⊆ 𝑋)
83		ssfi 8065	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Fin ∧ 𝐻 ⊆ 𝑋) → 𝐻 ∈ Fin)
84	10, 82, 83	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐻 ∈ Fin)
85	84	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)
86		hashdom 13029	. . . . . . . . . 10 ⊢ ((𝑘 ∈ Fin ∧ 𝐻 ∈ Fin) → ((#‘𝑘) ≤ (#‘𝐻) ↔ 𝑘 ≼ 𝐻))
87	16, 85, 86	syl2anc 691	. . . . . . . . 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))
88	80, 87	mpbid 221	. . . . . . . 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)
89		sbth 7965	. . . . . . . 8 ⊢ ((𝐻 ≼ 𝑘 ∧ 𝑘 ≼ 𝐻) → 𝐻 ≈ 𝑘)
90	19, 88, 89	syl2anc 691	. . . . . . 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)
91		fisseneq 8056	. . . . . . 7 ⊢ ((𝑘 ∈ Fin ∧ 𝐻 ⊆ 𝑘 ∧ 𝐻 ≈ 𝑘) → 𝐻 = 𝑘)
92	16, 17, 90, 91	syl3anc 1318	. . . . . 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)
93	92	expr 641	. . . . 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))
94		eqid 2610	. . . . . . . . . . . . 13 ⊢ (𝐺 ↾s 𝐻) = (𝐺 ↾s 𝐻)
95	94	subgbas 17421	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 = (Base‘(𝐺 ↾s 𝐻)))
96	95	ad2antrl 760	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐻 = (Base‘(𝐺 ↾s 𝐻)))
97	96	fveq2d 6107	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (#‘𝐻) = (#‘(Base‘(𝐺 ↾s 𝐻))))
98		simprr 792	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
99	97, 98	eqtr3d 2646	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (#‘(Base‘(𝐺 ↾s 𝐻))) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
100		oveq2 6557	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑃 pCnt (#‘𝑋)) → (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
101	100	eqeq2d 2620	. . . . . . . . . 10 ⊢ (𝑛 = (𝑃 pCnt (#‘𝑋)) → ((#‘(Base‘(𝐺 ↾s 𝐻))) = (𝑃↑𝑛) ↔ (#‘(Base‘(𝐺 ↾s 𝐻))) = (𝑃↑(𝑃 pCnt (#‘𝑋)))))
102	101	rspcev 3282	. . . . . . . . 9 ⊢ (((𝑃 pCnt (#‘𝑋)) ∈ ℕ0 ∧ (#‘(Base‘(𝐺 ↾s 𝐻))) = (𝑃↑(𝑃 pCnt (#‘𝑋)))) → ∃𝑛 ∈ ℕ0 (#‘(Base‘(𝐺 ↾s 𝐻))) = (𝑃↑𝑛))
103	52, 99, 102	syl2anc 691	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → ∃𝑛 ∈ ℕ0 (#‘(Base‘(𝐺 ↾s 𝐻))) = (𝑃↑𝑛))
104	94	subggrp 17420	. . . . . . . . . 10 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝐻) ∈ Grp)
105	104	ad2antrl 760	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (𝐺 ↾s 𝐻) ∈ Grp)
106	96, 84	eqeltrrd 2689	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (Base‘(𝐺 ↾s 𝐻)) ∈ Fin)
107		eqid 2610	. . . . . . . . . 10 ⊢ (Base‘(𝐺 ↾s 𝐻)) = (Base‘(𝐺 ↾s 𝐻))
108	107	pgpfi 17843	. . . . . . . . 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))))
109	105, 106, 108	syl2anc 691	. . . . . . . 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))))
110	8, 103, 109	mpbir2and 959	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝑃 pGrp (𝐺 ↾s 𝐻))
111	110	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))
112		oveq2 6557	. . . . . . . 8 ⊢ (𝐻 = 𝑘 → (𝐺 ↾s 𝐻) = (𝐺 ↾s 𝑘))
113	112	breq2d 4595	. . . . . . 7 ⊢ (𝐻 = 𝑘 → (𝑃 pGrp (𝐺 ↾s 𝐻) ↔ 𝑃 pGrp (𝐺 ↾s 𝑘)))
114		eqimss 3620	. . . . . . . 8 ⊢ (𝐻 = 𝑘 → 𝐻 ⊆ 𝑘)
115	114	biantrurd 528	. . . . . . 7 ⊢ (𝐻 = 𝑘 → (𝑃 pGrp (𝐺 ↾s 𝑘) ↔ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘))))
116	113, 115	bitrd 267	. . . . . 6 ⊢ (𝐻 = 𝑘 → (𝑃 pGrp (𝐺 ↾s 𝐻) ↔ (𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘))))
117	111, 116	syl5ibcom 234	. . . . 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))))
118	93, 117	impbid 201	. . . 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))
119	118	ralrimiva 2949	. . 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))
120		isslw 17846	. . 3 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)))
121	8, 9, 119, 120	syl3anbrc 1239	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐻 ∈ (𝑃 pSyl 𝐺))
122	7, 121	impbida 873	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549   ≈ cen 7838   ≼ cdom 7839  Fincfn 7841   ≤ cle 9954  ℕcn 10897  ℕ₀cn0 11169  ℤcz 11254  ℤ_≥cuz 11563  ↑cexp 12722  #chash 12979   ∥ cdvds 14821  ℙcprime 15223   pCnt cpc 15379  Basecbs 15695   ↾_s cress 15696  0_gc0g 15923  Grpcgrp 17245  SubGrpcsubg 17411   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-eqg 17416  df-ghm 17481  df-ga 17546  df-od 17771  df-pgp 17773  df-slw 17774

This theorem is referenced by:  sylow3lem1  17865