Theorem slwhash 19503

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 19489	. . . . 5 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
4	2, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))
5		subgrcl 19010	. . . 4 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		slwhash.3	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
8		slwprm 19488	. . . 4 ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)
9	2, 8	syl 17	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℙ)
10	1	grpbn0 18845	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
11	6, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ ∅)
12		hashnncl 14273	. . . . . 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 16762	. . 3 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ0)
16		pcdvds 16776	. . . 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 19482	. 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 19492	. . . . . 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 19502	. . 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 19129	. . . . . . . . . . . 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 19009	. . . . . . . . . . 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 6826	. . . . . . . . 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 19130	. . . . . . . . . . . 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 19006	. . . . . . . . . . . . . 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 9158	. . . . . . . . . . . 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 19006	. . . . . . . . . . . . . 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 9158	. . . . . . . . . . . 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 14254	. . . . . . . . . . . 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 7357	. . . . . . . . 9 ⊢ (𝑛 = (𝑃 pCnt (♯‘𝑋)) → (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
59	58	rspceeqv 3600	. . . . . . . 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 19008	. . . . . . . . 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 19484	. . . . . . . 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 19490	. . . . . . 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 6826	. . . 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 3136	. 2 ⊢ ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
74	18, 73	rexlimddv 3136	1 ⊢ (𝜑 → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   ⊆ wss 3903  ∅c0 4284   class class class wbr 5092   ↦ cmpt 5173  ran crn 5620  ‘cfv 6482  (class class class)co 7349   ≈ cen 8869  Fincfn 8872  ℕcn 12128  ℕ₀cn0 12384  ↑cexp 13968  ♯chash 14237   ∥ cdvds 16163  ℙcprime 16582   pCnt cpc 16748  Basecbs 17120   ↾_s cress 17141  +_gcplusg 17161  Grpcgrp 18812  -_gcsg 18814  SubGrpcsubg 18999   pGrp cpgp 19405   pSyl cslw 19406

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-disj 5060  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-omul 8393  df-er 8625  df-ec 8627  df-qs 8631  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-inf 9333  df-oi 9402  df-dju 9797  df-card 9835  df-acn 9838  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-xnn0 12458  df-z 12472  df-uz 12736  df-q 12850  df-rp 12894  df-fz 13411  df-fzo 13558  df-fl 13696  df-mod 13774  df-seq 13909  df-exp 13969  df-fac 14181  df-bc 14210  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-sum 15594  df-dvds 16164  df-gcd 16406  df-prm 16583  df-pc 16749  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-0g 17345  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-submnd 18658  df-grp 18815  df-minusg 18816  df-sbg 18817  df-mulg 18947  df-subg 19002  df-eqg 19004  df-ghm 19092  df-ga 19169  df-od 19407  df-pgp 19409  df-slw 19410

This theorem is referenced by:  fislw  19504  sylow2  19505  sylow3lem4  19509