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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  cn 10897  0cn0 11169  cexp 12722  #chash 12979  cdvds 14821  cprime 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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