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Theorem fislw 19643
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 484 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → 𝐻 ∈ (𝑃 pSyl 𝐺))
2 slwsubg 19628 . . . 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 1193 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → 𝑋 ∈ Fin)
64, 5, 1slwhash 19642 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
73, 6jca 511 . 2 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))))
8 simpl3 1194 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑃 ∈ ℙ)
9 simprl 771 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 ∈ (SubGrp‘𝐺))
10 simpl2 1193 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑋 ∈ Fin)
1110adantr 480 . . . . . . . 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 771 . . . . . . . . 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 19145 . . . . . . . . 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)
1511, 14ssfid 9301 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝑘 ∈ Fin)
16 simprrl 781 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝐻𝑘)
17 ssdomg 9040 . . . . . . . . 9 (𝑘 ∈ Fin → (𝐻𝑘𝐻𝑘))
1815, 16, 17sylc 65 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝐻𝑘)
19 simprrr 782 . . . . . . . . . . . . . . 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))
20 eqid 2737 . . . . . . . . . . . . . . . . . 18 (𝐺s 𝑘) = (𝐺s 𝑘)
2120subggrp 19147 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (SubGrp‘𝐺) → (𝐺s 𝑘) ∈ Grp)
2212, 21syl 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)
2320subgbas 19148 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (SubGrp‘𝐺) → 𝑘 = (Base‘(𝐺s 𝑘)))
2412, 23syl 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)))
2524, 15eqeltrrd 2842 . . . . . . . . . . . . . . . 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)
26 eqid 2737 . . . . . . . . . . . . . . . . 17 (Base‘(𝐺s 𝑘)) = (Base‘(𝐺s 𝑘))
2726pgpfi 19623 . . . . . . . . . . . . . . . 16 (((𝐺s 𝑘) ∈ Grp ∧ (Base‘(𝐺s 𝑘)) ∈ Fin) → (𝑃 pGrp (𝐺s 𝑘) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘(Base‘(𝐺s 𝑘))) = (𝑃𝑛))))
2822, 25, 27syl2anc 584 . . . . . . . . . . . . . . 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))))
2919, 28mpbid 232 . . . . . . . . . . . . . 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)))
3029simpld 494 . . . . . . . . . . . . 13 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝑃 ∈ ℙ)
31 prmnn 16711 . . . . . . . . . . . . 13 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
3230, 31syl 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)
3332nnred 12281 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝑃 ∈ ℝ)
3432nnge1d 12314 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 1 ≤ 𝑃)
35 eqid 2737 . . . . . . . . . . . . . . . . . 18 (0g𝐺) = (0g𝐺)
3635subg0cl 19152 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑘)
3712, 36syl 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)
3837ne0d 4342 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝑘 ≠ ∅)
39 hashnncl 14405 . . . . . . . . . . . . . . . 16 (𝑘 ∈ Fin → ((♯‘𝑘) ∈ ℕ ↔ 𝑘 ≠ ∅))
4015, 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) ∈ ℕ ↔ 𝑘 ≠ ∅))
4138, 40mpbird 257 . . . . . . . . . . . . . 14 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (♯‘𝑘) ∈ ℕ)
4230, 41pccld 16888 . . . . . . . . . . . . 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)
4342nn0zd 12639 . . . . . . . . . . . 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)
44 simpl1 1192 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐺 ∈ Grp)
454grpbn0 18984 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ Grp → 𝑋 ≠ ∅)
4644, 45syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑋 ≠ ∅)
47 hashnncl 14405 . . . . . . . . . . . . . . . . 17 (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
4810, 47syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
4946, 48mpbird 257 . . . . . . . . . . . . . . 15 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝑋) ∈ ℕ)
508, 49pccld 16888 . . . . . . . . . . . . . 14 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ0)
5150adantr 480 . . . . . . . . . . . . 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)
5251nn0zd 12639 . . . . . . . . . . . 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)
53 oveq1 7438 . . . . . . . . . . . . . 14 (𝑝 = 𝑃 → (𝑝 pCnt (♯‘𝑘)) = (𝑃 pCnt (♯‘𝑘)))
54 oveq1 7438 . . . . . . . . . . . . . 14 (𝑝 = 𝑃 → (𝑝 pCnt (♯‘𝑋)) = (𝑃 pCnt (♯‘𝑋)))
5553, 54breq12d 5156 . . . . . . . . . . . . 13 (𝑝 = 𝑃 → ((𝑝 pCnt (♯‘𝑘)) ≤ (𝑝 pCnt (♯‘𝑋)) ↔ (𝑃 pCnt (♯‘𝑘)) ≤ (𝑃 pCnt (♯‘𝑋))))
564lagsubg 19213 . . . . . . . . . . . . . . 15 ((𝑘 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (♯‘𝑘) ∥ (♯‘𝑋))
5712, 11, 56syl2anc 584 . . . . . . . . . . . . . 14 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (♯‘𝑘) ∥ (♯‘𝑋))
5841nnzd 12640 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (♯‘𝑘) ∈ ℤ)
5949adantr 480 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (♯‘𝑋) ∈ ℕ)
6059nnzd 12640 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (♯‘𝑋) ∈ ℤ)
61 pc2dvds 16917 . . . . . . . . . . . . . . 15 (((♯‘𝑘) ∈ ℤ ∧ (♯‘𝑋) ∈ ℤ) → ((♯‘𝑘) ∥ (♯‘𝑋) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (♯‘𝑘)) ≤ (𝑝 pCnt (♯‘𝑋))))
6258, 60, 61syl2anc 584 . . . . . . . . . . . . . 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))))
6357, 62mpbid 232 . . . . . . . . . . . . 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)))
6455, 63, 30rspcdva 3623 . . . . . . . . . . . 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)))
65 eluz2 12884 . . . . . . . . . . . 12 ((𝑃 pCnt (♯‘𝑋)) ∈ (ℤ‘(𝑃 pCnt (♯‘𝑘))) ↔ ((𝑃 pCnt (♯‘𝑘)) ∈ ℤ ∧ (𝑃 pCnt (♯‘𝑋)) ∈ ℤ ∧ (𝑃 pCnt (♯‘𝑘)) ≤ (𝑃 pCnt (♯‘𝑋))))
6643, 52, 64, 65syl3anbrc 1344 . . . . . . . . . . 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))))
6733, 34, 66leexp2ad 14293 . . . . . . . . . 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))))
6829simprd 495 . . . . . . . . . . . 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))
6924fveqeq2d 6914 . . . . . . . . . . . . 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)))
7069rexbidv 3179 . . . . . . . . . . . 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)))
7168, 70mpbird 257 . . . . . . . . . . 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))
72 pcprmpw 16921 . . . . . . . . . . . 12 ((𝑃 ∈ ℙ ∧ (♯‘𝑘) ∈ ℕ) → (∃𝑛 ∈ ℕ0 (♯‘𝑘) = (𝑃𝑛) ↔ (♯‘𝑘) = (𝑃↑(𝑃 pCnt (♯‘𝑘)))))
7330, 41, 72syl2anc 584 . . . . . . . . . . 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)))))
7471, 73mpbid 232 . . . . . . . . . 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))))
75 simplrr 778 . . . . . . . . . 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))))
7667, 74, 753brtr4d 5175 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (♯‘𝑘) ≤ (♯‘𝐻))
774subgss 19145 . . . . . . . . . . . . 13 (𝐻 ∈ (SubGrp‘𝐺) → 𝐻𝑋)
7877ad2antrl 728 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻𝑋)
7910, 78ssfid 9301 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 ∈ Fin)
8079adantr 480 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝐻 ∈ Fin)
81 hashdom 14418 . . . . . . . . . 10 ((𝑘 ∈ Fin ∧ 𝐻 ∈ Fin) → ((♯‘𝑘) ≤ (♯‘𝐻) ↔ 𝑘𝐻))
8215, 80, 81syl2anc 584 . . . . . . . . 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))
8376, 82mpbid 232 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝑘𝐻)
84 sbth 9133 . . . . . . . 8 ((𝐻𝑘𝑘𝐻) → 𝐻𝑘)
8518, 83, 84syl2anc 584 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝐻𝑘)
86 fisseneq 9293 . . . . . . 7 ((𝑘 ∈ Fin ∧ 𝐻𝑘𝐻𝑘) → 𝐻 = 𝑘)
8715, 16, 85, 86syl3anc 1373 . . . . . 6 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝐻 = 𝑘)
8887expr 456 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → ((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) → 𝐻 = 𝑘))
89 eqid 2737 . . . . . . . . . . . . 13 (𝐺s 𝐻) = (𝐺s 𝐻)
9089subgbas 19148 . . . . . . . . . . . 12 (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 = (Base‘(𝐺s 𝐻)))
9190ad2antrl 728 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 = (Base‘(𝐺s 𝐻)))
9291fveq2d 6910 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝐻) = (♯‘(Base‘(𝐺s 𝐻))))
93 simprr 773 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
9492, 93eqtr3d 2779 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (♯‘(Base‘(𝐺s 𝐻))) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
95 oveq2 7439 . . . . . . . . . 10 (𝑛 = (𝑃 pCnt (♯‘𝑋)) → (𝑃𝑛) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
9695rspceeqv 3645 . . . . . . . . 9 (((𝑃 pCnt (♯‘𝑋)) ∈ ℕ0 ∧ (♯‘(Base‘(𝐺s 𝐻))) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) → ∃𝑛 ∈ ℕ0 (♯‘(Base‘(𝐺s 𝐻))) = (𝑃𝑛))
9750, 94, 96syl2anc 584 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → ∃𝑛 ∈ ℕ0 (♯‘(Base‘(𝐺s 𝐻))) = (𝑃𝑛))
9889subggrp 19147 . . . . . . . . . 10 (𝐻 ∈ (SubGrp‘𝐺) → (𝐺s 𝐻) ∈ Grp)
9998ad2antrl 728 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (𝐺s 𝐻) ∈ Grp)
10091, 79eqeltrrd 2842 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → (Base‘(𝐺s 𝐻)) ∈ Fin)
101 eqid 2737 . . . . . . . . . 10 (Base‘(𝐺s 𝐻)) = (Base‘(𝐺s 𝐻))
102101pgpfi 19623 . . . . . . . . 9 (((𝐺s 𝐻) ∈ Grp ∧ (Base‘(𝐺s 𝐻)) ∈ Fin) → (𝑃 pGrp (𝐺s 𝐻) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘(Base‘(𝐺s 𝐻))) = (𝑃𝑛))))
10399, 100, 102syl2anc 584 . . . . . . . 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))))
1048, 97, 103mpbir2and 713 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝑃 pGrp (𝐺s 𝐻))
105104adantr 480 . . . . . 6 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺s 𝐻))
106 oveq2 7439 . . . . . . . 8 (𝐻 = 𝑘 → (𝐺s 𝐻) = (𝐺s 𝑘))
107106breq2d 5155 . . . . . . 7 (𝐻 = 𝑘 → (𝑃 pGrp (𝐺s 𝐻) ↔ 𝑃 pGrp (𝐺s 𝑘)))
108 eqimss 4042 . . . . . . . 8 (𝐻 = 𝑘𝐻𝑘)
109108biantrurd 532 . . . . . . 7 (𝐻 = 𝑘 → (𝑃 pGrp (𝐺s 𝑘) ↔ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘))))
110107, 109bitrd 279 . . . . . 6 (𝐻 = 𝑘 → (𝑃 pGrp (𝐺s 𝐻) ↔ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘))))
111105, 110syl5ibcom 245 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → (𝐻 = 𝑘 → (𝐻𝑘𝑃 pGrp (𝐺s 𝑘))))
11288, 111impbid 212 . . . 4 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → ((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘))
113112ralrimiva 3146 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘))
114 isslw 19626 . . 3 (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
1158, 9, 113, 114syl3anbrc 1344 . 2 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))) → 𝐻 ∈ (𝑃 pSyl 𝐺))
1167, 115impbida 801 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  wss 3951  c0 4333   class class class wbr 5143  cfv 6561  (class class class)co 7431  cen 8982  cdom 8983  Fincfn 8985  cle 11296  cn 12266  0cn0 12526  cz 12613  cuz 12878  cexp 14102  chash 14369  cdvds 16290  cprime 16708   pCnt cpc 16874  Basecbs 17247  s cress 17274  0gc0g 17484  Grpcgrp 18951  SubGrpcsubg 19138   pGrp cpgp 19544   pSyl cslw 19545
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-acn 9982  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-dvds 16291  df-gcd 16532  df-prm 16709  df-pc 16875  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-eqg 19143  df-ghm 19231  df-ga 19308  df-od 19546  df-pgp 19548  df-slw 19549
This theorem is referenced by:  sylow3lem1  19645
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