Theorem slwhash 19554

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 19540	. . . . 5 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
4	2, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))
5		subgrcl 19063	. . . 4 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		slwhash.3	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
8		slwprm 19539	. . . 4 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)
9	2, 8	syl 17	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℙ)
10	1	grpbn0 18898	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
11	6, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ ∅)
12		hashnncl 14331	. . . . . 6 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
13	7, 12	syl 17	. . . . 5 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
14	11, 13	mpbird 257	. . . 4 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
15	9, 14	pccld 16821	. . 3 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ0)
16		pcdvds 16835	. . . 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 19533	. 2 ⊢ (𝜑 → ∃𝑘 ∈ (SubGrp‘𝐺)(♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
19	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑋 ∈ Fin)
20	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 ∈ (SubGrp‘𝐺))
21		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑘 ∈ (SubGrp‘𝐺))
22		eqid 2729	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
23		eqid 2729	. . . . . . 7 ⊢ (𝐺 ↾s 𝐻) = (𝐺 ↾s 𝐻)
24	23	slwpgp 19543	. . . . . 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 480	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑃 pGrp (𝐺 ↾s 𝐻))
27		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
28		eqid 2729	. . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺)
29	1, 19, 20, 21, 22, 26, 27, 28	sylow2b 19553	. . 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 772	. . . . . 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 480	. . . . . . . . . . . 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 770	. . . . . . . . . . . 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 2729	. . . . . . . . . . . . 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 19182	. . . . . . . . . . . 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 2729	. . . . . . . . . . . 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 19062	. . . . . . . . . . 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 6862	. . . . . . . . 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 19183	. . . . . . . . . . . 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 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 19059	. . . . . . . . . . . . . 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 9212	. . . . . . . . . . . 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 19059	. . . . . . . . . . . . . 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 9212	. . . . . . . . . . . 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 14312	. . . . . . . . . . . 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 257	. . . . . . . . . 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 777	. . . . . . . . . 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 2766	. . . . . . . . 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 2766	. . . . . . . 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 7395	. . . . . . . . 9 ⊢ (𝑛 = (𝑃 pCnt (♯‘𝑋)) → (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
59	58	rspceeqv 3611	. . . . . . . 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 19061	. . . . . . . . 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 2829	. . . . . . . 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 2729	. . . . . . . . 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 19535	. . . . . . . 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 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 19541	. . . . . . 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 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 6862	. . . 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 2764	. . 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 3140	. 2 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
74	18, 73	rexlimddv 3140	1 ⊢ (𝜑 → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   ⊆ wss 3914  ∅c0 4296   class class class wbr 5107   ↦ cmpt 5188  ran crn 5639  ‘cfv 6511  (class class class)co 7387   ≈ cen 8915  Fincfn 8918  ℕcn 12186  ℕ₀cn0 12442  ↑cexp 14026  ♯chash 14295   ∥ cdvds 16222  ℙcprime 16641   pCnt cpc 16807  Basecbs 17179   ↾_s cress 17200  +_gcplusg 17220  Grpcgrp 18865  -_gcsg 18867  SubGrpcsubg 19052   pGrp cpgp 19456   pSyl cslw 19457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-disj 5075  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-omul 8439  df-er 8671  df-ec 8673  df-qs 8677  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-inf 9394  df-oi 9463  df-dju 9854  df-card 9892  df-acn 9895  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-n0 12443  df-xnn0 12516  df-z 12530  df-uz 12794  df-q 12908  df-rp 12952  df-fz 13469  df-fzo 13616  df-fl 13754  df-mod 13832  df-seq 13967  df-exp 14027  df-fac 14239  df-bc 14268  df-hash 14296  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-abs 15202  df-clim 15454  df-sum 15653  df-dvds 16223  df-gcd 16465  df-prm 16642  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 18567  df-sgrp 18646  df-mnd 18662  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-mulg 19000  df-subg 19055  df-eqg 19057  df-ghm 19145  df-ga 19222  df-od 19458  df-pgp 19460  df-slw 19461

This theorem is referenced by:  fislw  19555  sylow2  19556  sylow3lem4  19560