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Theorem fislw 17961
 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 477 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → 𝐻 ∈ (𝑃 pSyl 𝐺))
2 slwsubg 17946 . . . 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 1063 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → 𝑋 ∈ Fin)
64, 5, 1slwhash 17960 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
73, 6jca 554 . 2 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝐻 ∈ (𝑃 pSyl 𝐺)) → (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋)))))
8 simpl3 1064 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝑃 ∈ ℙ)
9 simprl 793 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐻 ∈ (SubGrp‘𝐺))
10 simpl2 1063 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝑋 ∈ Fin)
1110adantr 481 . . . . . . . 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 793 . . . . . . . . 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 17516 . . . . . . . . 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 8124 . . . . . . . 8 ((𝑋 ∈ Fin ∧ 𝑘𝑋) → 𝑘 ∈ Fin)
1611, 14, 15syl2anc 692 . . . . . . 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 803 . . . . . . 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 7945 . . . . . . . . 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 804 . . . . . . . . . . . . . . 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 2621 . . . . . . . . . . . . . . . . . 18 (𝐺s 𝑘) = (𝐺s 𝑘)
2221subggrp 17518 . . . . . . . . . . . . . . . . 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 17519 . . . . . . . . . . . . . . . . . 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 2699 . . . . . . . . . . . . . . . 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 2621 . . . . . . . . . . . . . . . . 17 (Base‘(𝐺s 𝑘)) = (Base‘(𝐺s 𝑘))
2827pgpfi 17941 . . . . . . . . . . . . . . . 16 (((𝐺s 𝑘) ∈ Grp ∧ (Base‘(𝐺s 𝑘)) ∈ Fin) → (𝑃 pGrp (𝐺s 𝑘) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘(Base‘(𝐺s 𝑘))) = (𝑃𝑛))))
2923, 26, 28syl2anc 692 . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . 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 15312 . . . . . . . . . . . . 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 10979 . . . . . . . . . . 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 11007 . . . . . . . . . . 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 2621 . . . . . . . . . . . . . . . . . 18 (0g𝐺) = (0g𝐺)
3736subg0cl 17523 . . . . . . . . . . . . . . . . 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 3897 . . . . . . . . . . . . . . . 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 13097 . . . . . . . . . . . . . . . 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 15479 . . . . . . . . . . . . 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 11424 . . . . . . . . . . . 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 1062 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐺 ∈ Grp)
474grpbn0 17372 . . . . . . . . . . . . . . . . 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 13097 . . . . . . . . . . . . . . . . 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 15479 . . . . . . . . . . . . . 14 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (𝑃 pCnt (#‘𝑋)) ∈ ℕ0)
5352adantr 481 . . . . . . . . . . . . 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 11424 . . . . . . . . . . . 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)
554lagsubg 17577 . . . . . . . . . . . . . . 15 ((𝑘 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (#‘𝑘) ∥ (#‘𝑋))
5612, 11, 55syl2anc 692 . . . . . . . . . . . . . 14 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (#‘𝑘) ∥ (#‘𝑋))
5743nnzd 11425 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (#‘𝑘) ∈ ℤ)
5851adantr 481 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (#‘𝑋) ∈ ℕ)
5958nnzd 11425 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (#‘𝑋) ∈ ℤ)
60 pc2dvds 15507 . . . . . . . . . . . . . . 15 (((#‘𝑘) ∈ ℤ ∧ (#‘𝑋) ∈ ℤ) → ((#‘𝑘) ∥ (#‘𝑋) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (#‘𝑘)) ≤ (𝑝 pCnt (#‘𝑋))))
6157, 59, 60syl2anc 692 . . . . . . . . . . . . . 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))))
6256, 61mpbid 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)))
63 oveq1 6611 . . . . . . . . . . . . . . 15 (𝑝 = 𝑃 → (𝑝 pCnt (#‘𝑘)) = (𝑃 pCnt (#‘𝑘)))
64 oveq1 6611 . . . . . . . . . . . . . . 15 (𝑝 = 𝑃 → (𝑝 pCnt (#‘𝑋)) = (𝑃 pCnt (#‘𝑋)))
6563, 64breq12d 4626 . . . . . . . . . . . . . 14 (𝑝 = 𝑃 → ((𝑝 pCnt (#‘𝑘)) ≤ (𝑝 pCnt (#‘𝑋)) ↔ (𝑃 pCnt (#‘𝑘)) ≤ (𝑃 pCnt (#‘𝑋))))
6665rspcv 3291 . . . . . . . . . . . . 13 (𝑃 ∈ ℙ → (∀𝑝 ∈ ℙ (𝑝 pCnt (#‘𝑘)) ≤ (𝑝 pCnt (#‘𝑋)) → (𝑃 pCnt (#‘𝑘)) ≤ (𝑃 pCnt (#‘𝑋))))
6731, 62, 66sylc 65 . . . . . . . . . . . 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)))
68 eluz2 11637 . . . . . . . . . . . 12 ((𝑃 pCnt (#‘𝑋)) ∈ (ℤ‘(𝑃 pCnt (#‘𝑘))) ↔ ((𝑃 pCnt (#‘𝑘)) ∈ ℤ ∧ (𝑃 pCnt (#‘𝑋)) ∈ ℤ ∧ (𝑃 pCnt (#‘𝑘)) ≤ (𝑃 pCnt (#‘𝑋))))
6945, 54, 67, 68syl3anbrc 1244 . . . . . . . . . . 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))))
7034, 35, 69leexp2ad 12981 . . . . . . . . . 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))))
7130simprd 479 . . . . . . . . . . . 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))
7225fveq2d 6152 . . . . . . . . . . . . . 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))))
7372eqeq1d 2623 . . . . . . . . . . . . 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)))
7473rexbidv 3045 . . . . . . . . . . . 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)))
7571, 74mpbird 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))
76 pcprmpw 15511 . . . . . . . . . . . 12 ((𝑃 ∈ ℙ ∧ (#‘𝑘) ∈ ℕ) → (∃𝑛 ∈ ℕ0 (#‘𝑘) = (𝑃𝑛) ↔ (#‘𝑘) = (𝑃↑(𝑃 pCnt (#‘𝑘)))))
7731, 43, 76syl2anc 692 . . . . . . . . . . 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)))))
7875, 77mpbid 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))))
79 simplrr 800 . . . . . . . . . 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))))
8070, 78, 793brtr4d 4645 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → (#‘𝑘) ≤ (#‘𝐻))
814subgss 17516 . . . . . . . . . . . . 13 (𝐻 ∈ (SubGrp‘𝐺) → 𝐻𝑋)
8281ad2antrl 763 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐻𝑋)
83 ssfi 8124 . . . . . . . . . . . 12 ((𝑋 ∈ Fin ∧ 𝐻𝑋) → 𝐻 ∈ Fin)
8410, 82, 83syl2anc 692 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐻 ∈ Fin)
8584adantr 481 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝐻 ∈ Fin)
86 hashdom 13108 . . . . . . . . . 10 ((𝑘 ∈ Fin ∧ 𝐻 ∈ Fin) → ((#‘𝑘) ≤ (#‘𝐻) ↔ 𝑘𝐻))
8716, 85, 86syl2anc 692 . . . . . . . . 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))
8880, 87mpbid 222 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝑘𝐻)
89 sbth 8024 . . . . . . . 8 ((𝐻𝑘𝑘𝐻) → 𝐻𝑘)
9019, 88, 89syl2anc 692 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝐻𝑘)
91 fisseneq 8115 . . . . . . 7 ((𝑘 ∈ Fin ∧ 𝐻𝑘𝐻𝑘) → 𝐻 = 𝑘)
9216, 17, 90, 91syl3anc 1323 . . . . . 6 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘)))) → 𝐻 = 𝑘)
9392expr 642 . . . . 5 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → ((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) → 𝐻 = 𝑘))
94 eqid 2621 . . . . . . . . . . . . 13 (𝐺s 𝐻) = (𝐺s 𝐻)
9594subgbas 17519 . . . . . . . . . . . 12 (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 = (Base‘(𝐺s 𝐻)))
9695ad2antrl 763 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐻 = (Base‘(𝐺s 𝐻)))
9796fveq2d 6152 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (#‘𝐻) = (#‘(Base‘(𝐺s 𝐻))))
98 simprr 795 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
9997, 98eqtr3d 2657 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (#‘(Base‘(𝐺s 𝐻))) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
100 oveq2 6612 . . . . . . . . . . 11 (𝑛 = (𝑃 pCnt (#‘𝑋)) → (𝑃𝑛) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
101100eqeq2d 2631 . . . . . . . . . 10 (𝑛 = (𝑃 pCnt (#‘𝑋)) → ((#‘(Base‘(𝐺s 𝐻))) = (𝑃𝑛) ↔ (#‘(Base‘(𝐺s 𝐻))) = (𝑃↑(𝑃 pCnt (#‘𝑋)))))
102101rspcev 3295 . . . . . . . . 9 (((𝑃 pCnt (#‘𝑋)) ∈ ℕ0 ∧ (#‘(Base‘(𝐺s 𝐻))) = (𝑃↑(𝑃 pCnt (#‘𝑋)))) → ∃𝑛 ∈ ℕ0 (#‘(Base‘(𝐺s 𝐻))) = (𝑃𝑛))
10352, 99, 102syl2anc 692 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → ∃𝑛 ∈ ℕ0 (#‘(Base‘(𝐺s 𝐻))) = (𝑃𝑛))
10494subggrp 17518 . . . . . . . . . 10 (𝐻 ∈ (SubGrp‘𝐺) → (𝐺s 𝐻) ∈ Grp)
105104ad2antrl 763 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (𝐺s 𝐻) ∈ Grp)
10696, 84eqeltrrd 2699 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → (Base‘(𝐺s 𝐻)) ∈ Fin)
107 eqid 2621 . . . . . . . . . 10 (Base‘(𝐺s 𝐻)) = (Base‘(𝐺s 𝐻))
108107pgpfi 17941 . . . . . . . . 9 (((𝐺s 𝐻) ∈ Grp ∧ (Base‘(𝐺s 𝐻)) ∈ Fin) → (𝑃 pGrp (𝐺s 𝐻) ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘(Base‘(𝐺s 𝐻))) = (𝑃𝑛))))
109105, 106, 108syl2anc 692 . . . . . . . 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))))
1108, 103, 109mpbir2and 956 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝑃 pGrp (𝐺s 𝐻))
111110adantr 481 . . . . . 6 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) ∧ 𝑘 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺s 𝐻))
112 oveq2 6612 . . . . . . . 8 (𝐻 = 𝑘 → (𝐺s 𝐻) = (𝐺s 𝑘))
113112breq2d 4625 . . . . . . 7 (𝐻 = 𝑘 → (𝑃 pGrp (𝐺s 𝐻) ↔ 𝑃 pGrp (𝐺s 𝑘)))
114 eqimss 3636 . . . . . . . 8 (𝐻 = 𝑘𝐻𝑘)
115114biantrurd 529 . . . . . . 7 (𝐻 = 𝑘 → (𝑃 pGrp (𝐺s 𝑘) ↔ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘))))
116113, 115bitrd 268 . . . . . 6 (𝐻 = 𝑘 → (𝑃 pGrp (𝐺s 𝐻) ↔ (𝐻𝑘𝑃 pGrp (𝐺s 𝑘))))
117111, 116syl5ibcom 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))))
11893, 117impbid 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))
119118ralrimiva 2960 . . 3 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘))
120 isslw 17944 . . 3 (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))
1218, 9, 119, 120syl3anbrc 1244 . 2 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))) → 𝐻 ∈ (𝑃 pSyl 𝐺))
1227, 121impbida 876 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 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  ∃wrex 2908   ⊆ wss 3555  ∅c0 3891   class class class wbr 4613  ‘cfv 5847  (class class class)co 6604   ≈ cen 7896   ≼ cdom 7897  Fincfn 7899   ≤ cle 10019  ℕcn 10964  ℕ0cn0 11236  ℤcz 11321  ℤ≥cuz 11631  ↑cexp 12800  #chash 13057   ∥ cdvds 14907  ℙcprime 15309   pCnt cpc 15465  Basecbs 15781   ↾s cress 15782  0gc0g 16021  Grpcgrp 17343  SubGrpcsubg 17509   pGrp cpgp 17867   pSyl cslw 17868 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-disj 4584  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-er 7687  df-ec 7689  df-qs 7693  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-acn 8712  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-fz 12269  df-fzo 12407  df-fl 12533  df-mod 12609  df-seq 12742  df-exp 12801  df-fac 13001  df-bc 13030  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-sum 14351  df-dvds 14908  df-gcd 15141  df-prm 15310  df-pc 15466  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-grp 17346  df-minusg 17347  df-sbg 17348  df-mulg 17462  df-subg 17512  df-eqg 17514  df-ghm 17579  df-ga 17644  df-od 17869  df-pgp 17871  df-slw 17872 This theorem is referenced by:  sylow3lem1  17963
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