Theorem slwhash 19406

Description: A sylow subgroup has cardinality equal to the maximum power of 𝑃 dividing the group. (Contributed by Mario Carneiro, 18-Jan-2015.)

Hypotheses

Ref	Expression
fislw.1	⊢ 𝑋 = (Base‘𝐺)
slwhash.3	⊢ (𝜑 → 𝑋 ∈ Fin)
slwhash.4	⊢ (𝜑 → 𝐻 ∈ (𝑃 pSyl 𝐺))

Assertion

Ref	Expression
slwhash	⊢ (𝜑 → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))

Proof of Theorem slwhash

Dummy variables 𝑔 𝑘 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fislw.1	. . 3 ⊢ 𝑋 = (Base‘𝐺)
2		slwhash.4	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (𝑃 pSyl 𝐺))
3		slwsubg 19392	. . . . 5 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
4	2, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))
5		subgrcl 18933	. . . 4 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		slwhash.3	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
8		slwprm 19391	. . . 4 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)
9	2, 8	syl 17	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℙ)
10	1	grpbn0 18779	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
11	6, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ ∅)
12		hashnncl 14266	. . . . . 6 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
13	7, 12	syl 17	. . . . 5 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
14	11, 13	mpbird 256	. . . 4 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
15	9, 14	pccld 16722	. . 3 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ0)
16		pcdvds 16736	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝑋) ∈ ℕ) → (𝑃↑(𝑃 pCnt (♯‘𝑋))) ∥ (♯‘𝑋))
17	9, 14, 16	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt (♯‘𝑋))) ∥ (♯‘𝑋))
18	1, 6, 7, 9, 15, 17	sylow1 19385	. 2 ⊢ (𝜑 → ∃𝑘 ∈ (SubGrp‘𝐺)(♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
19	7	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑋 ∈ Fin)
20	4	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 ∈ (SubGrp‘𝐺))
21		simprl 769	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑘 ∈ (SubGrp‘𝐺))
22		eqid 2736	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
23		eqid 2736	. . . . . . 7 ⊢ (𝐺 ↾s 𝐻) = (𝐺 ↾s 𝐻)
24	23	slwpgp 19395	. . . . . 6 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp (𝐺 ↾s 𝐻))
25	2, 24	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝐻))
26	25	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑃 pGrp (𝐺 ↾s 𝐻))
27		simprr 771	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
28		eqid 2736	. . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺)
29	1, 19, 20, 21, 22, 26, 27, 28	sylow2b 19405	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))
30		simprr 771	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))
31	2	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → 𝐻 ∈ (𝑃 pSyl 𝐺))
32	31, 8	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → 𝑃 ∈ ℙ)
33	15	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ0)
34	21	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → 𝑘 ∈ (SubGrp‘𝐺))
35		simprl 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → 𝑔 ∈ 𝑋)
36		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)) = (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))
37	1, 22, 28, 36	conjsubg 19040	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (SubGrp‘𝐺) ∧ 𝑔 ∈ 𝑋) → ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)) ∈ (SubGrp‘𝐺))
38	34, 35, 37	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)) ∈ (SubGrp‘𝐺))
39		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))) = (𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))
40	39	subgbas 18932	. . . . . . . . . . 11 ⊢ (ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)) ∈ (SubGrp‘𝐺) → ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)) = (Base‘(𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))))
41	38, 40	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)) = (Base‘(𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))))
42	41	fveq2d 6846	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → (♯‘ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))) = (♯‘(Base‘(𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))))))
43	1, 22, 28, 36	conjsubgen 19041	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (SubGrp‘𝐺) ∧ 𝑔 ∈ 𝑋) → 𝑘 ≈ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))
44	34, 35, 43	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → 𝑘 ≈ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))
45	7	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → 𝑋 ∈ Fin)
46	1	subgss 18929	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (SubGrp‘𝐺) → 𝑘 ⊆ 𝑋)
47	34, 46	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → 𝑘 ⊆ 𝑋)
48	45, 47	ssfid 9211	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → 𝑘 ∈ Fin)
49	1	subgss 18929	. . . . . . . . . . . . . 14 ⊢ (ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)) ∈ (SubGrp‘𝐺) → ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)) ⊆ 𝑋)
50	38, 49	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)) ⊆ 𝑋)
51	45, 50	ssfid 9211	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)) ∈ Fin)
52		hashen 14247	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ Fin ∧ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)) ∈ Fin) → ((♯‘𝑘) = (♯‘ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))) ↔ 𝑘 ≈ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))))
53	48, 51, 52	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → ((♯‘𝑘) = (♯‘ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))) ↔ 𝑘 ≈ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))))
54	44, 53	mpbird 256	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → (♯‘𝑘) = (♯‘ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))))
55		simplrr 776	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
56	54, 55	eqtr3d 2778	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → (♯‘ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
57	42, 56	eqtr3d 2778	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → (♯‘(Base‘(𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))))) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
58		oveq2 7365	. . . . . . . . 9 ⊢ (𝑛 = (𝑃 pCnt (♯‘𝑋)) → (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
59	58	rspceeqv 3595	. . . . . . . 8 ⊢ (((𝑃 pCnt (♯‘𝑋)) ∈ ℕ0 ∧ (♯‘(Base‘(𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))))) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) → ∃𝑛 ∈ ℕ0 (♯‘(Base‘(𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))))) = (𝑃↑𝑛))
60	33, 57, 59	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → ∃𝑛 ∈ ℕ0 (♯‘(Base‘(𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))))) = (𝑃↑𝑛))
61	39	subggrp 18931	. . . . . . . . 9 ⊢ (ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)) ∈ (SubGrp‘𝐺) → (𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))) ∈ Grp)
62	38, 61	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → (𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))) ∈ Grp)
63	41, 51	eqeltrrd 2839	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → (Base‘(𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) ∈ Fin)
64		eqid 2736	. . . . . . . . 9 ⊢ (Base‘(𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) = (Base‘(𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))))
65	64	pgpfi 19387	. . . . . . . 8 ⊢ (((𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))) ∈ Grp ∧ (Base‘(𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) ∈ Fin) → (𝑃 pGrp (𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘(Base‘(𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))))) = (𝑃↑𝑛))))
66	62, 63, 65	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → (𝑃 pGrp (𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘(Base‘(𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))))) = (𝑃↑𝑛))))
67	32, 60, 66	mpbir2and 711	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → 𝑃 pGrp (𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))))
68	39	slwispgp 19393	. . . . . . 7 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)) ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)) ∧ 𝑃 pGrp (𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) ↔ 𝐻 = ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))))
69	31, 38, 68	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → ((𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)) ∧ 𝑃 pGrp (𝐺 ↾s ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) ↔ 𝐻 = ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))))
70	30, 67, 69	mpbi2and 710	. . . . 5 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → 𝐻 = ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))
71	70	fveq2d 6846	. . . 4 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → (♯‘𝐻) = (♯‘ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔))))
72	71, 56	eqtrd 2776	. . 3 ⊢ (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑔 ∈ 𝑋 ∧ 𝐻 ⊆ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))) → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
73	29, 72	rexlimddv 3158	. 2 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
74	18, 73	rexlimddv 3158	1 ⊢ (𝜑 → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3073   ⊆ wss 3910  ∅c0 4282   class class class wbr 5105   ↦ cmpt 5188  ran crn 5634  ‘cfv 6496  (class class class)co 7357   ≈ cen 8880  Fincfn 8883  ℕcn 12153  ℕ₀cn0 12413  ↑cexp 13967  ♯chash 14230   ∥ cdvds 16136  ℙcprime 16547   pCnt cpc 16708  Basecbs 17083   ↾_s cress 17112  +_gcplusg 17133  Grpcgrp 18748  -_gcsg 18750  SubGrpcsubg 18922   pGrp cpgp 19308   pSyl cslw 19309

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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-omul 8417  df-er 8648  df-ec 8650  df-qs 8654  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-dvds 16137  df-gcd 16375  df-prm 16548  df-pc 16709  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-eqg 18927  df-ghm 19006  df-ga 19070  df-od 19310  df-pgp 19312  df-slw 19313

This theorem is referenced by:  fislw  19407  sylow2  19408  sylow3lem4  19412