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Theorem slwhash 18033
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 18019 . . . . 5 (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
42, 3syl 17 . . . 4 (𝜑𝐻 ∈ (SubGrp‘𝐺))
5 subgrcl 17593 . . . 4 (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
64, 5syl 17 . . 3 (𝜑𝐺 ∈ Grp)
7 slwhash.3 . . 3 (𝜑𝑋 ∈ Fin)
8 slwprm 18018 . . . 4 (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)
92, 8syl 17 . . 3 (𝜑𝑃 ∈ ℙ)
101grpbn0 17445 . . . . . 6 (𝐺 ∈ Grp → 𝑋 ≠ ∅)
116, 10syl 17 . . . . 5 (𝜑𝑋 ≠ ∅)
12 hashnncl 13152 . . . . . 6 (𝑋 ∈ Fin → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
137, 12syl 17 . . . . 5 (𝜑 → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
1411, 13mpbird 247 . . . 4 (𝜑 → (#‘𝑋) ∈ ℕ)
159, 14pccld 15549 . . 3 (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℕ0)
16 pcdvds 15562 . . . 4 ((𝑃 ∈ ℙ ∧ (#‘𝑋) ∈ ℕ) → (𝑃↑(𝑃 pCnt (#‘𝑋))) ∥ (#‘𝑋))
179, 14, 16syl2anc 693 . . 3 (𝜑 → (𝑃↑(𝑃 pCnt (#‘𝑋))) ∥ (#‘𝑋))
181, 6, 7, 9, 15, 17sylow1 18012 . 2 (𝜑 → ∃𝑘 ∈ (SubGrp‘𝐺)(#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
197adantr 481 . . . 4 ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝑋 ∈ Fin)
204adantr 481 . . . 4 ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐻 ∈ (SubGrp‘𝐺))
21 simprl 794 . . . 4 ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝑘 ∈ (SubGrp‘𝐺))
22 eqid 2621 . . . 4 (+g𝐺) = (+g𝐺)
23 eqid 2621 . . . . . . 7 (𝐺s 𝐻) = (𝐺s 𝐻)
2423slwpgp 18022 . . . . . 6 (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp (𝐺s 𝐻))
252, 24syl 17 . . . . 5 (𝜑𝑃 pGrp (𝐺s 𝐻))
2625adantr 481 . . . 4 ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝑃 pGrp (𝐺s 𝐻))
27 simprr 796 . . . 4 ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
28 eqid 2621 . . . 4 (-g𝐺) = (-g𝐺)
291, 19, 20, 21, 22, 26, 27, 28sylow2b 18032 . . 3 ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → ∃𝑔𝑋 𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))
30 simprr 796 . . . . . 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 762 . . . . . . . 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 762 . . . . . . . 8 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → (𝑃 pCnt (#‘𝑋)) ∈ ℕ0)
3421adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → 𝑘 ∈ (SubGrp‘𝐺))
35 simprl 794 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → 𝑔𝑋)
36 eqid 2621 . . . . . . . . . . . . 13 (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) = (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))
371, 22, 28, 36conjsubg 17686 . . . . . . . . . . . 12 ((𝑘 ∈ (SubGrp‘𝐺) ∧ 𝑔𝑋) → ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) ∈ (SubGrp‘𝐺))
3834, 35, 37syl2anc 693 . . . . . . . . . . 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 2621 . . . . . . . . . . . 12 (𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))) = (𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))
4039subgbas 17592 . . . . . . . . . . 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 6193 . . . . . . . . 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 17687 . . . . . . . . . . . 12 ((𝑘 ∈ (SubGrp‘𝐺) ∧ 𝑔𝑋) → 𝑘 ≈ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))
4434, 35, 43syl2anc 693 . . . . . . . . . . 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 762 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → 𝑋 ∈ Fin)
461subgss 17589 . . . . . . . . . . . . . 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 8177 . . . . . . . . . . . . 13 ((𝑋 ∈ Fin ∧ 𝑘𝑋) → 𝑘 ∈ Fin)
4945, 47, 48syl2anc 693 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑔𝑋𝐻 ⊆ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) → 𝑘 ∈ Fin)
501subgss 17589 . . . . . . . . . . . . . 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 8177 . . . . . . . . . . . . 13 ((𝑋 ∈ Fin ∧ ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) ⊆ 𝑋) → ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)) ∈ Fin)
5345, 51, 52syl2anc 693 . . . . . . . . . . . 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 13130 . . . . . . . . . . . 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 693 . . . . . . . . . . 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 247 . . . . . . . . . 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 801 . . . . . . . . . 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 2657 . . . . . . . . 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 2657 . . . . . . . 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 6655 . . . . . . . . . 10 (𝑛 = (𝑃 pCnt (#‘𝑋)) → (𝑃𝑛) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
6160eqeq2d 2631 . . . . . . . . 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 3307 . . . . . . . 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 693 . . . . . . 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 17591 . . . . . . . . 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 2701 . . . . . . . 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 2621 . . . . . . . . 9 (Base‘(𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔)))) = (Base‘(𝐺s ran (𝑥𝑘 ↦ ((𝑔(+g𝐺)𝑥)(-g𝐺)𝑔))))
6867pgpfi 18014 . . . . . . . 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 693 . . . . . . 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 957 . . . . . 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 18020 . . . . . . 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 693 . . . . . 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 956 . . . . 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 6193 . . . 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 2655 . . 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 3033 . 2 ((𝜑 ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
7718, 76rexlimddv 3033 1 (𝜑 → (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  wne 2793  wrex 2912  wss 3572  c0 3913   class class class wbr 4651  cmpt 4727  ran crn 5113  cfv 5886  (class class class)co 6647  cen 7949  Fincfn 7952  cn 11017  0cn0 11289  cexp 12855  #chash 13112  cdvds 14977  cprime 15379   pCnt cpc 15535  Basecbs 15851  s cress 15852  +gcplusg 15935  Grpcgrp 17416  -gcsg 17418  SubGrpcsubg 17582   pGrp cpgp 17940   pSyl cslw 17941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-inf2 8535  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010  ax-pre-sup 10011
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-disj 4619  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-2o 7558  df-oadd 7561  df-omul 7562  df-er 7739  df-ec 7741  df-qs 7745  df-map 7856  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-sup 8345  df-inf 8346  df-oi 8412  df-card 8762  df-acn 8765  df-cda 8987  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-div 10682  df-nn 11018  df-2 11076  df-3 11077  df-n0 11290  df-xnn0 11361  df-z 11375  df-uz 11685  df-q 11786  df-rp 11830  df-fz 12324  df-fzo 12462  df-fl 12588  df-mod 12664  df-seq 12797  df-exp 12856  df-fac 13056  df-bc 13085  df-hash 13113  df-cj 13833  df-re 13834  df-im 13835  df-sqrt 13969  df-abs 13970  df-clim 14213  df-sum 14411  df-dvds 14978  df-gcd 15211  df-prm 15380  df-pc 15536  df-ndx 15854  df-slot 15855  df-base 15857  df-sets 15858  df-ress 15859  df-plusg 15948  df-0g 16096  df-mgm 17236  df-sgrp 17278  df-mnd 17289  df-submnd 17330  df-grp 17419  df-minusg 17420  df-sbg 17421  df-mulg 17535  df-subg 17585  df-eqg 17587  df-ghm 17652  df-ga 17717  df-od 17942  df-pgp 17944  df-slw 17945
This theorem is referenced by:  fislw  18034  sylow2  18035  sylow3lem4  18039
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