Theorem slwhash 19777

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 19763	. . . . 5 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
4	2, 3	syl 18	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))
5		subgrcl 19280	. . . 4 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
6	4, 5	syl 18	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		slwhash.3	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
8		slwprm 19762	. . . 4 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)
9	2, 8	syl 18	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℙ)
10	1	grpbn0 19116	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
11	6, 10	syl 18	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ ∅)
12		hashnncl 14455	. . . . . 6 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
13	7, 12	syl 18	. . . . 5 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
14	11, 13	mpbird 260	. . . 4 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
15	9, 14	pccld 16967	. . 3 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ0)
16		pcdvds 16981	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝑋) ∈ ℕ) → (𝑃↑(𝑃 pCnt (♯‘𝑋))) ∥ (♯‘𝑋))
17	9, 14, 16	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt (♯‘𝑋))) ∥ (♯‘𝑋))
18	1, 6, 7, 9, 15, 17	sylow1 19756	. 2 ⊢ (𝜑 → ∃𝑘 ∈ (SubGrp‘𝐺)(♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
19	7	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑋 ∈ Fin)
20	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 ∈ (SubGrp‘𝐺))
21		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑘 ∈ (SubGrp‘𝐺))
22		eqid 2760	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
23		eqid 2760	. . . . . . 7 ⊢ (𝐺 ↾s 𝐻) = (𝐺 ↾s 𝐻)
24	23	slwpgp 19766	. . . . . 6 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp (𝐺 ↾s 𝐻))
25	2, 24	syl 18	. . . . 5 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝐻))
26	25	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑃 pGrp (𝐺 ↾s 𝐻))
27		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
28		eqid 2760	. . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺)
29	1, 19, 20, 21, 22, 26, 27, 28	sylow2b 19776	. . 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 785	. . . . . 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 739	. . . . . . . 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 18	. . . . . . 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 739	. . . . . . . 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 486	. . . . . . . . . . . 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 783	. . . . . . . . . . . 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 2760	. . . . . . . . . . . . 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 19403	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (SubGrp‘𝐺) ∧ 𝑔 ∈ 𝑋) → ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)) ∈ (SubGrp‘𝐺))
38	34, 35, 37	syl2anc 596	. . . . . . . . . . 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 2760	. . . . . . . . . . . 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 19279	. . . . . . . . . . 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 18	. . . . . . . . . 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 6885	. . . . . . . . 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 19404	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (SubGrp‘𝐺) ∧ 𝑔 ∈ 𝑋) → 𝑘 ≈ ran (𝑥 ∈ 𝑘 ↦ ((𝑔(+g‘𝐺)𝑥)(-g‘𝐺)𝑔)))
44	34, 35, 43	syl2anc 596	. . . . . . . . . . 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 739	. . . . . . . . . . . . 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 19276	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (SubGrp‘𝐺) → 𝑘 ⊆ 𝑋)
47	34, 46	syl 18	. . . . . . . . . . . . 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 9246	. . . . . . . . . . . 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 19276	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 9246	. . . . . . . . . . . 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 14436	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 790	. . . . . . . . . 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 2797	. . . . . . . . 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 2797	. . . . . . . 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 7424	. . . . . . . . 9 ⊢ (𝑛 = (𝑃 pCnt (♯‘𝑋)) → (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
59	58	rspceeqv 3599	. . . . . . . 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 596	. . . . . . 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 19278	. . . . . . . . 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 18	. . . . . . . 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 2861	. . . . . . . 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 2760	. . . . . . . . 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 19758	. . . . . . . 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 596	. . . . . . 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 726	. . . . . 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 19764	. . . . . . 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 596	. . . . . 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 725	. . . . 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 6885	. . . 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 2795	. . 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 3169	. 2 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
74	18, 73	rexlimddv 3169	1 ⊢ (𝜑 → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103   ↦ cmpt 5186  ran crn 5656  ‘cfv 6535  (class class class)co 7416   ≈ cen 8956  Fincfn 8959  ℕcn 12282  ℕ₀cn0 12553  ↑cexp 14150  ♯chash 14419   ∥ cdvds 16367  ℙcprime 16786   pCnt cpc 16953  Basecbs 17326   ↾_s cress 17347  +_gcplusg 17367  Grpcgrp 19083  -_gcsg 19085  SubGrpcsubg 19269   pGrp cpgp 19679   pSyl cslw 19680

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742  ax-inf2 9627  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226  ax-pre-sup 11227

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-disj 5071  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8462  df-2o 8463  df-oadd 8466  df-omul 8467  df-er 8703  df-ec 8705  df-qs 8709  df-map 8835  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-sup 9419  df-inf 9420  df-oi 9489  df-dju 9931  df-card 9969  df-acn 9972  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11492  df-neg 11493  df-div 11921  df-nn 12283  df-2 12352  df-3 12353  df-n0 12554  df-xnn0 12627  df-z 12641  df-uz 12913  df-q 13023  df-rp 13068  df-fz 13587  df-fzo 13735  df-fl 13878  df-mod 13956  df-seq 14091  df-exp 14151  df-fac 14363  df-bc 14392  df-hash 14420  df-cj 15211  df-re 15212  df-im 15213  df-sqrt 15347  df-abs 15348  df-clim 15600  df-sum 15799  df-dvds 16368  df-gcd 16610  df-prm 16787  df-pc 16954  df-sets 17281  df-slot 17299  df-ndx 17311  df-base 17327  df-ress 17348  df-plusg 17380  df-0g 17551  df-mgm 18755  df-sgrp 18847  df-mnd 18863  df-submnd 18918  df-grp 19086  df-minusg 19087  df-sbg 19088  df-mulg 19217  df-subg 19272  df-eqg 19274  df-ghm 19367  df-ga 19443  df-od 19681  df-pgp 19683  df-slw 19684

This theorem is used by:  fislw  19778  sylow2  19779  sylow3lem4  19783