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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  cn 10964  0cn0 11236  cz 11321  cuz 11631  cexp 12800  #chash 13057  cdvds 14907  cprime 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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