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Theorem slwhash 17862
 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 (𝜑𝑋 ∈ Fin)
slwhash.4 (𝜑𝐻 ∈ (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.
StepHypRef Expression
1 fislw.1 . . 3 𝑋 = (Base‘𝐺)
2 slwhash.4 . . . . 5 (𝜑𝐻 ∈ (𝑃 pSyl 𝐺))
3 slwsubg 17848 . . . . 5 (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
42, 3syl 17 . . . 4 (𝜑𝐻 ∈ (SubGrp‘𝐺))
5 subgrcl 17422 . . . 4 (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
64, 5syl 17 . . 3 (𝜑𝐺 ∈ Grp)
7 slwhash.3 . . 3 (𝜑𝑋 ∈ Fin)
8 slwprm 17847 . . . 4 (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)
92, 8syl 17 . . 3 (𝜑𝑃 ∈ ℙ)
101grpbn0 17274 . . . . . 6 (𝐺 ∈ Grp → 𝑋 ≠ ∅)
116, 10syl 17 . . . . 5 (𝜑𝑋 ≠ ∅)
12 hashnncl 13018 . . . . . 6 (𝑋 ∈ Fin → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
137, 12syl 17 . . . . 5 (𝜑 → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
1411, 13mpbird 246 . . . 4 (𝜑 → (#‘𝑋) ∈ ℕ)
159, 14pccld 15393 . . 3 (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℕ0)
16 pcdvds 15406 . . . 4 ((𝑃 ∈ ℙ ∧ (#‘𝑋) ∈ ℕ) → (𝑃↑(𝑃 pCnt (#‘𝑋))) ∥ (#‘𝑋))
179, 14, 16syl2anc 691 . . 3 (𝜑 → (𝑃↑(𝑃 pCnt (#‘𝑋))) ∥ (#‘𝑋))
181, 6, 7, 9, 15, 17sylow1 17841 . 2 (𝜑 → ∃𝑘 ∈ (SubGrp‘𝐺)(#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
197adantr 480 . . . 4 ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝑋 ∈ Fin)
204adantr 480 . . . 4 ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐻 ∈ (SubGrp‘𝐺))
21 simprl 790 . . . 4 ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝑘 ∈ (SubGrp‘𝐺))
22 eqid 2610 . . . 4 (+g𝐺) = (+g𝐺)
23 eqid 2610 . . . . . . 7 (𝐺s 𝐻) = (𝐺s 𝐻)
2423slwpgp 17851 . . . . . 6 (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp (𝐺s 𝐻))
252, 24syl 17 . . . . 5 (𝜑𝑃 pGrp (𝐺s 𝐻))
2625adantr 480 . . . 4 ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝑃 pGrp (𝐺s 𝐻))
27 simprr 792 . . . 4 ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
28 eqid 2610 . . . 4 (-g𝐺) = (-g𝐺)
291, 19, 20, 21, 22, 26, 27, 28sylow2b 17861 . . 3 ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → ∃𝑔𝑋 𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))
30 simprr 792 . . . . . 6 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → 𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))
312ad2antrr 758 . . . . . . . 8 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → 𝐻 ∈ (𝑃 pSyl 𝐺))
3231, 8syl 17 . . . . . . 7 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → 𝑃 ∈ ℙ)
3315ad2antrr 758 . . . . . . . 8 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → (𝑃 pCnt (#‘𝑋)) ∈ ℕ0)
3421adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → 𝑘 ∈ (SubGrp‘𝐺))
35 simprl 790 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → 𝑔𝑋)
36 eqid 2610 . . . . . . . . . . . . 13 (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) = (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))
371, 22, 28, 36conjsubg 17515 . . . . . . . . . . . 12 ((𝑘 ∈ (SubGrp‘𝐺) ∧ 𝑔𝑋) → ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) ∈ (SubGrp‘𝐺))
3834, 35, 37syl2anc 691 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) ∈ (SubGrp‘𝐺))
39 eqid 2610 . . . . . . . . . . . 12 (𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))) = (𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))
4039subgbas 17421 . . . . . . . . . . 11 (ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) ∈ (SubGrp‘𝐺) → ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) = (Base‘(𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))))
4138, 40syl 17 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) = (Base‘(𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))))
4241fveq2d 6107 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → (#‘ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))) = (#‘(Base‘(𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))))
431, 22, 28, 36conjsubgen 17516 . . . . . . . . . . . 12 ((𝑘 ∈ (SubGrp‘𝐺) ∧ 𝑔𝑋) → 𝑘 ≈ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))
4434, 35, 43syl2anc 691 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → 𝑘 ≈ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))
457ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → 𝑋 ∈ Fin)
461subgss 17418 . . . . . . . . . . . . . 14 (𝑘 ∈ (SubGrp‘𝐺) → 𝑘𝑋)
4734, 46syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → 𝑘𝑋)
48 ssfi 8065 . . . . . . . . . . . . 13 ((𝑋 ∈ Fin ∧ 𝑘𝑋) → 𝑘 ∈ Fin)
4945, 47, 48syl2anc 691 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → 𝑘 ∈ Fin)
501subgss 17418 . . . . . . . . . . . . . 14 (ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) ∈ (SubGrp‘𝐺) → ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) ⊆ 𝑋)
5138, 50syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) ⊆ 𝑋)
52 ssfi 8065 . . . . . . . . . . . . 13 ((𝑋 ∈ Fin ∧ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) ⊆ 𝑋) → ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) ∈ Fin)
5345, 51, 52syl2anc 691 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) ∈ Fin)
54 hashen 12997 . . . . . . . . . . . 12 ((𝑘 ∈ Fin ∧ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) ∈ Fin) → ((#‘𝑘) = (#‘ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))) ↔ 𝑘 ≈ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))
5549, 53, 54syl2anc 691 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → ((#‘𝑘) = (#‘ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))) ↔ 𝑘 ≈ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))
5644, 55mpbird 246 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → (#‘𝑘) = (#‘ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))
57 simplrr 797 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
5856, 57eqtr3d 2646 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → (#‘ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
5942, 58eqtr3d 2646 . . . . . . . 8 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → (#‘(Base‘(𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
60 oveq2 6557 . . . . . . . . . 10 (𝑛 = (𝑃 pCnt (#‘𝑋)) → (𝑃𝑛) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
6160eqeq2d 2620 . . . . . . . . 9 (𝑛 = (𝑃 pCnt (#‘𝑋)) → ((#‘(Base‘(𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))) = (𝑃𝑛) ↔ (#‘(Base‘(𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))) = (𝑃↑(𝑃 pCnt (#‘𝑋)))))
6261rspcev 3282 . . . . . . . 8 (((𝑃 pCnt (#‘𝑋)) ∈ ℕ0 ∧ (#‘(Base‘(𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))) = (𝑃↑(𝑃 pCnt (#‘𝑋)))) → ∃𝑛 ∈ ℕ0 (#‘(Base‘(𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))) = (𝑃𝑛))
6333, 59, 62syl2anc 691 . . . . . . 7 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → ∃𝑛 ∈ ℕ0 (#‘(Base‘(𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))) = (𝑃𝑛))
6439subggrp 17420 . . . . . . . . 9 (ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) ∈ (SubGrp‘𝐺) → (𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))) ∈ Grp)
6538, 64syl 17 . . . . . . . 8 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → (𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))) ∈ Grp)
6641, 53eqeltrrd 2689 . . . . . . . 8 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → (Base‘(𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) ∈ Fin)
67 eqid 2610 . . . . . . . . 9 (Base‘(𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) = (Base‘(𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))
6867pgpfi 17843 . . . . . . . 8 (((𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))) ∈ Grp ∧ (Base‘(𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) ∈ Fin) → (𝑃 pGrp (𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘(Base‘(𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))) = (𝑃𝑛))))
6965, 66, 68syl2anc 691 . . . . . . 7 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → (𝑃 pGrp (𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘(Base‘(𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))) = (𝑃𝑛))))
7032, 63, 69mpbir2and 959 . . . . . 6 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → 𝑃 pGrp (𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))
7139slwispgp 17849 . . . . . . 7 ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) ∧ 𝑃 pGrp (𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) ↔ 𝐻 = ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))
7231, 38, 71syl2anc 691 . . . . . 6 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → ((𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) ∧ 𝑃 pGrp (𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) ↔ 𝐻 = ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))
7330, 70, 72mpbi2and 958 . . . . 5 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → 𝐻 = ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))
7473fveq2d 6107 . . . 4 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → (#‘𝐻) = (#‘ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))
7574, 58eqtrd 2644 . . 3 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
7629, 75rexlimddv 3017 . 2 ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
7718, 76rexlimddv 3017 1 (𝜑 → (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ≈ cen 7838  Fincfn 7841  ℕcn 10897  ℕ0cn0 11169  ↑cexp 12722  #chash 12979   ∥ cdvds 14821  ℙcprime 15223   pCnt cpc 15379  Basecbs 15695   ↾s cress 15696  +gcplusg 15768  Grpcgrp 17245  -gcsg 17247  SubGrpcsubg 17411   pGrp cpgp 17769   pSyl cslw 17770 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-dvds 14822  df-gcd 15055  df-prm 15224  df-pc 15380  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-eqg 17416  df-ghm 17481  df-ga 17546  df-od 17771  df-pgp 17773  df-slw 17774 This theorem is referenced by:  fislw  17863  sylow2  17864  sylow3lem4  17868
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