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Theorem fislw 18253
 Description: The sylow subgroups of a finite group are exactly the groups which have cardinality equal to the maximum power of 𝑃 dividing the group. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypothesis
Ref Expression
fislw.1 𝑋 = (Base‘𝐺)
Assertion
Ref Expression
fislw ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))))

Proof of Theorem fislw
Dummy variables 𝑘 𝑛 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 480 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → 𝐻 ∈ (𝑃 pSyl 𝐺))
2 slwsubg 18238 . . . 4 (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
31, 2syl 17 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → 𝐻 ∈ (SubGrp‘𝐺))
4 fislw.1 . . . 4 𝑋 = (Base‘𝐺)
5 simpl2 1227 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → 𝑋 ∈ Fin)
64, 5, 1slwhash 18252 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
73, 6jca 556 . 2 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))))
8 simpl3 1229 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑃 ∈ ℙ)
9 simprl 808 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 ∈ (SubGrp‘𝐺))
10 simpl2 1227 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑋 ∈ Fin)
1110adantr 473 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝑋 ∈ Fin)
12 simprl 808 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝑘 ∈ (SubGrp‘𝐺))
134subgss 17809 . . . . . . . . 9 (𝑘 ∈ (SubGrp‘𝐺) → 𝑘𝑋)
1412, 13syl 17 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝑘𝑋)
15 ssfi 8334 . . . . . . . 8 ((𝑋 ∈ Fin ∧ 𝑘𝑋) → 𝑘 ∈ Fin)
1611, 14, 15syl2anc 693 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝑘 ∈ Fin)
17 simprrl 820 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝐻𝑘)
18 ssdomg 8153 . . . . . . . . 9 (𝑘 ∈ Fin → (𝐻𝑘𝐻𝑘))
1916, 17, 18sylc 65 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝐻𝑘)
20 simprrr 821 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝑃 pGrp (𝐺s 𝑘))
21 eqid 2769 . . . . . . . . . . . . . . . . . 18 (𝐺s 𝑘) = (𝐺s 𝑘)
2221subggrp 17811 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (SubGrp‘𝐺) → (𝐺s 𝑘) ∈ Grp)
2312, 22syl 17 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (𝐺s 𝑘) ∈ Grp)
2421subgbas 17812 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (SubGrp‘𝐺) → 𝑘 = (Base‘(𝐺s 𝑘)))
2512, 24syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝑘 = (Base‘(𝐺s 𝑘)))
2625, 16eqeltrrd 2849 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (Base‘(𝐺s 𝑘)) ∈ Fin)
27 eqid 2769 . . . . . . . . . . . . . . . . 17 (Base‘(𝐺s 𝑘)) = (Base‘(𝐺s 𝑘))
2827pgpfi 18233 . . . . . . . . . . . . . . . 16 (((𝐺s 𝑘) ∈ Grp ∧ (Base‘(𝐺s 𝑘)) ∈ Fin) → (𝑃 pGrp (𝐺s 𝑘) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘(Base‘(𝐺s 𝑘))) = (𝑃𝑛))))
2923, 26, 28syl2anc 693 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (𝑃 pGrp (𝐺s 𝑘) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘(Base‘(𝐺s 𝑘))) = (𝑃𝑛))))
3020, 29mpbid 222 . . . . . . . . . . . . . 14 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘(Base‘(𝐺s 𝑘))) = (𝑃𝑛)))
3130simpld 478 . . . . . . . . . . . . 13 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝑃 ∈ ℙ)
32 prmnn 15601 . . . . . . . . . . . . 13 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
3331, 32syl 17 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝑃 ∈ ℕ)
3433nnred 11235 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝑃 ∈ ℝ)
3533nnge1d 11263 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 1 ≤ 𝑃)
36 eqid 2769 . . . . . . . . . . . . . . . . . 18 (0g𝐺) = (0g𝐺)
3736subg0cl 17816 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑘)
3812, 37syl 17 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (0g𝐺) ∈ 𝑘)
39 ne0i 4066 . . . . . . . . . . . . . . . 16 ((0g𝐺) ∈ 𝑘𝑘 ≠ ∅)
4038, 39syl 17 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝑘 ≠ ∅)
41 hashnncl 13362 . . . . . . . . . . . . . . . 16 (𝑘 ∈ Fin → ((♯‘𝑘) ∈ ℕ ↔ 𝑘 ≠ ∅))
4216, 41syl 17 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → ((♯‘𝑘) ∈ ℕ ↔ 𝑘 ≠ ∅))
4340, 42mpbird 247 . . . . . . . . . . . . . 14 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (♯‘𝑘) ∈ ℕ)
4431, 43pccld 15768 . . . . . . . . . . . . 13 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (𝑃 pCnt (♯‘𝑘)) ∈ ℕ0)
4544nn0zd 11680 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (𝑃 pCnt (♯‘𝑘)) ∈ ℤ)
46 simpl1 1225 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐺 ∈ Grp)
474grpbn0 17665 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ Grp → 𝑋 ≠ ∅)
4846, 47syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑋 ≠ ∅)
49 hashnncl 13362 . . . . . . . . . . . . . . . . 17 (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
5010, 49syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
5148, 50mpbird 247 . . . . . . . . . . . . . . 15 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝑋) ∈ ℕ)
528, 51pccld 15768 . . . . . . . . . . . . . 14 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ0)
5352adantr 473 . . . . . . . . . . . . 13 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ0)
5453nn0zd 11680 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (𝑃 pCnt (♯‘𝑋)) ∈ ℤ)
55 oveq1 6798 . . . . . . . . . . . . . 14 (𝑝 = 𝑃 → (𝑝 pCnt (♯‘𝑘)) = (𝑃 pCnt (♯‘𝑘)))
56 oveq1 6798 . . . . . . . . . . . . . 14 (𝑝 = 𝑃 → (𝑝 pCnt (♯‘𝑋)) = (𝑃 pCnt (♯‘𝑋)))
5755, 56breq12d 4796 . . . . . . . . . . . . 13 (𝑝 = 𝑃 → ((𝑝 pCnt (♯‘𝑘)) ≤ (𝑝 pCnt (♯‘𝑋)) ↔ (𝑃 pCnt (♯‘𝑘)) ≤ (𝑃 pCnt (♯‘𝑋))))
584lagsubg 17870 . . . . . . . . . . . . . . 15 ((𝑘 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (♯‘𝑘) ∥ (♯‘𝑋))
5912, 11, 58syl2anc 693 . . . . . . . . . . . . . 14 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (♯‘𝑘) ∥ (♯‘𝑋))
6043nnzd 11681 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (♯‘𝑘) ∈ ℤ)
6151adantr 473 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (♯‘𝑋) ∈ ℕ)
6261nnzd 11681 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (♯‘𝑋) ∈ ℤ)
63 pc2dvds 15796 . . . . . . . . . . . . . . 15 (((♯‘𝑘) ∈ ℤ ∧ (♯‘𝑋) ∈ ℤ) → ((♯‘𝑘) ∥ (♯‘𝑋) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (♯‘𝑘)) ≤ (𝑝 pCnt (♯‘𝑋))))
6460, 62, 63syl2anc 693 . . . . . . . . . . . . . 14 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → ((♯‘𝑘) ∥ (♯‘𝑋) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (♯‘𝑘)) ≤ (𝑝 pCnt (♯‘𝑋))))
6559, 64mpbid 222 . . . . . . . . . . . . 13 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → ∀𝑝 ∈ ℙ (𝑝 pCnt (♯‘𝑘)) ≤ (𝑝 pCnt (♯‘𝑋)))
6657, 65, 31rspcdva 3463 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (𝑃 pCnt (♯‘𝑘)) ≤ (𝑃 pCnt (♯‘𝑋)))
67 eluz2 11893 . . . . . . . . . . . 12 ((𝑃 pCnt (♯‘𝑋)) ∈ (ℤ‘(𝑃 pCnt (♯‘𝑘))) ↔ ((𝑃 pCnt (♯‘𝑘)) ∈ ℤ ∧ (𝑃 pCnt (♯‘𝑋)) ∈ ℤ ∧ (𝑃 pCnt (♯‘𝑘)) ≤ (𝑃 pCnt (♯‘𝑋))))
6845, 54, 66, 67syl3anbrc 1426 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (𝑃 pCnt (♯‘𝑋)) ∈ (ℤ‘(𝑃 pCnt (♯‘𝑘))))
6934, 35, 68leexp2ad 13248 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (𝑃↑(𝑃 pCnt (♯‘𝑘))) ≤ (𝑃↑(𝑃 pCnt (♯‘𝑋))))
7030simprd 483 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → ∃𝑛 ∈ ℕ0 (♯‘(Base‘(𝐺s 𝑘))) = (𝑃𝑛))
7125fveq2d 6335 . . . . . . . . . . . . . 14 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (♯‘𝑘) = (♯‘(Base‘(𝐺s 𝑘))))
7271eqeq1d 2771 . . . . . . . . . . . . 13 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → ((♯‘𝑘) = (𝑃𝑛) ↔ (♯‘(Base‘(𝐺s 𝑘))) = (𝑃𝑛)))
7372rexbidv 3198 . . . . . . . . . . . 12 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (∃𝑛 ∈ ℕ0 (♯‘𝑘) = (𝑃𝑛) ↔ ∃𝑛 ∈ ℕ0 (♯‘(Base‘(𝐺s 𝑘))) = (𝑃𝑛)))
7470, 73mpbird 247 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → ∃𝑛 ∈ ℕ0 (♯‘𝑘) = (𝑃𝑛))
75 pcprmpw 15800 . . . . . . . . . . . 12 ((𝑃 ∈ ℙ ∧ (♯‘𝑘) ∈ ℕ) → (∃𝑛 ∈ ℕ0 (♯‘𝑘) = (𝑃𝑛) ↔ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑘)))))
7631, 43, 75syl2anc 693 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (∃𝑛 ∈ ℕ0 (♯‘𝑘) = (𝑃𝑛) ↔ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑘)))))
7774, 76mpbid 222 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑘))))
78 simplrr 817 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
7969, 77, 783brtr4d 4815 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (♯‘𝑘) ≤ (♯‘𝐻))
804subgss 17809 . . . . . . . . . . . . 13 (𝐻 ∈ (SubGrp‘𝐺) → 𝐻𝑋)
8180ad2antrl 763 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻𝑋)
82 ssfi 8334 . . . . . . . . . . . 12 ((𝑋 ∈ Fin ∧ 𝐻𝑋) → 𝐻 ∈ Fin)
8310, 81, 82syl2anc 693 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 ∈ Fin)
8483adantr 473 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝐻 ∈ Fin)
85 hashdom 13373 . . . . . . . . . 10 ((𝑘 ∈ Fin ∧ 𝐻 ∈ Fin) → ((♯‘𝑘) ≤ (♯‘𝐻) ↔ 𝑘𝐻))
8616, 84, 85syl2anc 693 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → ((♯‘𝑘) ≤ (♯‘𝐻) ↔ 𝑘𝐻))
8779, 86mpbid 222 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝑘𝐻)
88 sbth 8234 . . . . . . . 8 ((𝐻𝑘𝑘𝐻) → 𝐻𝑘)
8919, 87, 88syl2anc 693 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝐻𝑘)
90 fisseneq 8325 . . . . . . 7 ((𝑘 ∈ Fin ∧ 𝐻𝑘𝐻𝑘) → 𝐻 = 𝑘)
9116, 17, 89, 90syl3anc 1474 . . . . . 6 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝐻 = 𝑘)
9291expr 641 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → ((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) → 𝐻 = 𝑘))
93 eqid 2769 . . . . . . . . . . . . 13 (𝐺s 𝐻) = (𝐺s 𝐻)
9493subgbas 17812 . . . . . . . . . . . 12 (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 = (Base‘(𝐺s 𝐻)))
9594ad2antrl 763 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 = (Base‘(𝐺s 𝐻)))
9695fveq2d 6335 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝐻) = (♯‘(Base‘(𝐺s 𝐻))))
97 simprr 810 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
9896, 97eqtr3d 2805 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘(Base‘(𝐺s 𝐻))) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
99 oveq2 6799 . . . . . . . . . . 11 (𝑛 = (𝑃 pCnt (♯‘𝑋)) → (𝑃𝑛) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
10099eqeq2d 2779 . . . . . . . . . 10 (𝑛 = (𝑃 pCnt (♯‘𝑋)) → ((♯‘(Base‘(𝐺s 𝐻))) = (𝑃𝑛) ↔ (♯‘(Base‘(𝐺s 𝐻))) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))))
101100rspcev 3457 . . . . . . . . 9 (((𝑃 pCnt (♯‘𝑋)) ∈ ℕ0 ∧ (♯‘(Base‘(𝐺s 𝐻))) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) → ∃𝑛 ∈ ℕ0 (♯‘(Base‘(𝐺s 𝐻))) = (𝑃𝑛))
10252, 98, 101syl2anc 693 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → ∃𝑛 ∈ ℕ0 (♯‘(Base‘(𝐺s 𝐻))) = (𝑃𝑛))
10393subggrp 17811 . . . . . . . . . 10 (𝐻 ∈ (SubGrp‘𝐺) → (𝐺s 𝐻) ∈ Grp)
104103ad2antrl 763 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (𝐺s 𝐻) ∈ Grp)
10595, 83eqeltrrd 2849 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (Base‘(𝐺s 𝐻)) ∈ Fin)
106 eqid 2769 . . . . . . . . . 10 (Base‘(𝐺s 𝐻)) = (Base‘(𝐺s 𝐻))
107106pgpfi 18233 . . . . . . . . 9 (((𝐺s 𝐻) ∈ Grp ∧ (Base‘(𝐺s 𝐻)) ∈ Fin) → (𝑃 pGrp (𝐺s 𝐻) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘(Base‘(𝐺s 𝐻))) = (𝑃𝑛))))
108104, 105, 107syl2anc 693 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (𝑃 pGrp (𝐺s 𝐻) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘(Base‘(𝐺s 𝐻))) = (𝑃𝑛))))
1098, 102, 108mpbir2and 992 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑃 pGrp (𝐺s 𝐻))
110109adantr 473 . . . . . 6 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺s 𝐻))
111 oveq2 6799 . . . . . . . 8 (𝐻 = 𝑘 → (𝐺s 𝐻) = (𝐺s 𝑘))
112111breq2d 4795 . . . . . . 7 (𝐻 = 𝑘 → (𝑃 pGrp (𝐺s 𝐻) ↔ 𝑃 pGrp (𝐺s 𝑘)))
113 eqimss 3803 . . . . . . . 8 (𝐻 = 𝑘𝐻𝑘)
114113biantrurd 531 . . . . . . 7 (𝐻 = 𝑘 → (𝑃 pGrp (𝐺s 𝑘) ↔ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘))))
115112, 114bitrd 268 . . . . . 6 (𝐻 = 𝑘 → (𝑃 pGrp (𝐺s 𝐻) ↔ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘))))
116110, 115syl5ibcom 235 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → (𝐻 = 𝑘 → (𝐻𝑘𝑃 pGrp (𝐺s 𝑘))))
11792, 116impbid 202 . . . 4 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → ((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘))
118117ralrimiva 3113 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘))
119 isslw 18236 . . 3 (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
1208, 9, 118, 119syl3anbrc 1426 . 2 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 ∈ (𝑃 pSyl 𝐺))
1217, 120impbida 910 1 ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1069   = wceq 1629   ∈ wcel 2143   ≠ wne 2941  ∀wral 3059  ∃wrex 3060   ⊆ wss 3720  ∅c0 4060   class class class wbr 4783  ‘cfv 6030  (class class class)co 6791   ≈ cen 8104   ≼ cdom 8105  Fincfn 8107   ≤ cle 10275  ℕcn 11220  ℕ0cn0 11492  ℤcz 11577  ℤ≥cuz 11887  ↑cexp 13067  ♯chash 13324   ∥ cdvds 15194  ℙcprime 15598   pCnt cpc 15754  Basecbs 16070   ↾s cress 16071  0gc0g 16314  Grpcgrp 17636  SubGrpcsubg 17802   pGrp cpgp 18159   pSyl cslw 18160 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-rep 4901  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033  ax-un 7094  ax-inf2 8700  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213  ax-pre-sup 10214 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1070  df-3an 1071  df-tru 1632  df-fal 1635  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-nel 3045  df-ral 3064  df-rex 3065  df-reu 3066  df-rmo 3067  df-rab 3068  df-v 3350  df-sbc 3585  df-csb 3680  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-pss 3736  df-nul 4061  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4572  df-int 4609  df-iun 4653  df-disj 4752  df-br 4784  df-opab 4844  df-mpt 4861  df-tr 4884  df-id 5156  df-eprel 5161  df-po 5169  df-so 5170  df-fr 5207  df-se 5208  df-we 5209  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-2o 7712  df-oadd 7715  df-omul 7716  df-er 7894  df-ec 7896  df-qs 7900  df-map 8009  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-sup 8502  df-inf 8503  df-oi 8569  df-card 8963  df-acn 8966  df-cda 9190  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-div 10885  df-nn 11221  df-2 11279  df-3 11280  df-n0 11493  df-xnn0 11564  df-z 11578  df-uz 11888  df-q 11991  df-rp 12035  df-fz 12533  df-fzo 12673  df-fl 12800  df-mod 12876  df-seq 13009  df-exp 13068  df-fac 13268  df-bc 13297  df-hash 13325  df-cj 14050  df-re 14051  df-im 14052  df-sqrt 14186  df-abs 14187  df-clim 14430  df-sum 14628  df-dvds 15195  df-gcd 15431  df-prm 15599  df-pc 15755  df-ndx 16073  df-slot 16074  df-base 16076  df-sets 16077  df-ress 16078  df-plusg 16168  df-0g 16316  df-mgm 17456  df-sgrp 17498  df-mnd 17509  df-submnd 17550  df-grp 17639  df-minusg 17640  df-sbg 17641  df-mulg 17755  df-subg 17805  df-eqg 17807  df-ghm 17872  df-ga 17936  df-od 18161  df-pgp 18163  df-slw 18164 This theorem is referenced by:  sylow3lem1  18255
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