Theorem slwhash 18801

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 18787	. . . . 5 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
4	2, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))
5		subgrcl 18336	. . . 4 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		slwhash.3	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
8		slwprm 18786	. . . 4 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)
9	2, 8	syl 17	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℙ)
10	1	grpbn0 18184	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
11	6, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ ∅)
12		hashnncl 13762	. . . . . 6 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
13	7, 12	syl 17	. . . . 5 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
14	11, 13	mpbird 260	. . . 4 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
15	9, 14	pccld 16227	. . 3 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ0)
16		pcdvds 16240	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝑋) ∈ ℕ) → (𝑃↑(𝑃 pCnt (♯‘𝑋))) ∥ (♯‘𝑋))
17	9, 14, 16	syl2anc 588	. . 3 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt (♯‘𝑋))) ∥ (♯‘𝑋))
18	1, 6, 7, 9, 15, 17	sylow1 18780	. 2 ⊢ (𝜑 → ∃𝑘 ∈ (SubGrp‘𝐺)(♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
19	7	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑋 ∈ Fin)
20	4	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 ∈ (SubGrp‘𝐺))
21		simprl 771	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑘 ∈ (SubGrp‘𝐺))
22		eqid 2759	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
23		eqid 2759	. . . . . . 7 ⊢ (𝐺 ↾s 𝐻) = (𝐺 ↾s 𝐻)
24	23	slwpgp 18790	. . . . . 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 485	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑃 pGrp (𝐺 ↾s 𝐻))
27		simprr 773	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
28		eqid 2759	. . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺)
29	1, 19, 20, 21, 22, 26, 27, 28	sylow2b 18800	. . 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 773	. . . . . 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 726	. . . . . . . 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 726	. . . . . . . 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 485	. . . . . . . . . . . 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 771	. . . . . . . . . . . 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 2759	. . . . . . . . . . . . 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 18442	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (SubGrp‘𝐺) ∧ 𝑔 ∈ 𝑋) → ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)) ∈ (SubGrp‘𝐺))
38	34, 35, 37	syl2anc 588	. . . . . . . . . . 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 2759	. . . . . . . . . . . 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 18335	. . . . . . . . . . 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 6655	. . . . . . . . 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 18443	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (SubGrp‘𝐺) ∧ 𝑔 ∈ 𝑋) → 𝑘 ≈ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))
44	34, 35, 43	syl2anc 588	. . . . . . . . . . 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 726	. . . . . . . . . . . . 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 18332	. . . . . . . . . . . . . 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 8755	. . . . . . . . . . . 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 18332	. . . . . . . . . . . . . 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 8755	. . . . . . . . . . . 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 13742	. . . . . . . . . . . 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 588	. . . . . . . . . . 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 260	. . . . . . . . . 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 778	. . . . . . . . . 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 2796	. . . . . . . . 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 2796	. . . . . . . 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 7151	. . . . . . . . 9 ⊢ (𝑛 = (𝑃 pCnt (♯‘𝑋)) → (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
59	58	rspceeqv 3554	. . . . . . . 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 588	. . . . . . 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 18334	. . . . . . . . 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 2852	. . . . . . . 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 2759	. . . . . . . . 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 18782	. . . . . . . 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 588	. . . . . . 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 713	. . . . . 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 18788	. . . . . . 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 588	. . . . . 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 712	. . . . 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 6655	. . . 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 2794	. . 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 3213	. 2 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
74	18, 73	rexlimddv 3213	1 ⊢ (𝜑 → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112   ≠ wne 2949  ∃wrex 3069   ⊆ wss 3854  ∅c0 4221   class class class wbr 5025   ↦ cmpt 5105  ran crn 5518  ‘cfv 6328  (class class class)co 7143   ≈ cen 8517  Fincfn 8520  ℕcn 11659  ℕ₀cn0 11919  ↑cexp 13464  ♯chash 13725   ∥ cdvds 15640  ℙcprime 16052   pCnt cpc 16213  Basecbs 16526   ↾_s cress 16527  +_gcplusg 16608  Grpcgrp 18154  -_gcsg 18156  SubGrpcsubg 18325   pGrp cpgp 18706   pSyl cslw 18707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5149  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452  ax-inf2 9122  ax-cnex 10616  ax-resscn 10617  ax-1cn 10618  ax-icn 10619  ax-addcl 10620  ax-addrcl 10621  ax-mulcl 10622  ax-mulrcl 10623  ax-mulcom 10624  ax-addass 10625  ax-mulass 10626  ax-distr 10627  ax-i2m1 10628  ax-1ne0 10629  ax-1rid 10630  ax-rnegex 10631  ax-rrecex 10632  ax-cnre 10633  ax-pre-lttri 10634  ax-pre-lttrn 10635  ax-pre-ltadd 10636  ax-pre-mulgt0 10637  ax-pre-sup 10638

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-nel 3054  df-ral 3073  df-rex 3074  df-reu 3075  df-rmo 3076  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-tp 4520  df-op 4522  df-uni 4792  df-int 4832  df-iun 4878  df-disj 4991  df-br 5026  df-opab 5088  df-mpt 5106  df-tr 5132  df-id 5423  df-eprel 5428  df-po 5436  df-so 5437  df-fr 5476  df-se 5477  df-we 5478  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7101  df-ov 7146  df-oprab 7147  df-mpo 7148  df-om 7573  df-1st 7686  df-2nd 7687  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-2o 8106  df-oadd 8109  df-omul 8110  df-er 8292  df-ec 8294  df-qs 8298  df-map 8411  df-en 8521  df-dom 8522  df-sdom 8523  df-fin 8524  df-sup 8924  df-inf 8925  df-oi 8992  df-dju 9348  df-card 9386  df-acn 9389  df-pnf 10700  df-mnf 10701  df-xr 10702  df-ltxr 10703  df-le 10704  df-sub 10895  df-neg 10896  df-div 11321  df-nn 11660  df-2 11722  df-3 11723  df-n0 11920  df-xnn0 11992  df-z 12006  df-uz 12268  df-q 12374  df-rp 12416  df-fz 12925  df-fzo 13068  df-fl 13196  df-mod 13272  df-seq 13404  df-exp 13465  df-fac 13669  df-bc 13698  df-hash 13726  df-cj 14491  df-re 14492  df-im 14493  df-sqrt 14627  df-abs 14628  df-clim 14878  df-sum 15076  df-dvds 15641  df-gcd 15879  df-prm 16053  df-pc 16214  df-ndx 16529  df-slot 16530  df-base 16532  df-sets 16533  df-ress 16534  df-plusg 16621  df-0g 16758  df-mgm 17903  df-sgrp 17952  df-mnd 17963  df-submnd 18008  df-grp 18157  df-minusg 18158  df-sbg 18159  df-mulg 18277  df-subg 18328  df-eqg 18330  df-ghm 18408  df-ga 18472  df-od 18708  df-pgp 18710  df-slw 18711

This theorem is referenced by:  fislw  18802  sylow2  18803  sylow3lem4  18807