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Theorem pgpfi 17941
 Description: The converse to pgpfi1 17931. A finite group is a 𝑃-group iff it has size some power of 𝑃. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypothesis
Ref Expression
pgpfi.1 𝑋 = (Base‘𝐺)
Assertion
Ref Expression
pgpfi ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛))))
Distinct variable groups:   𝑛,𝐺   𝑃,𝑛   𝑛,𝑋

Proof of Theorem pgpfi
Dummy variables 𝑔 𝑚 𝑝 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pgpfi.1 . . . 4 𝑋 = (Base‘𝐺)
2 eqid 2621 . . . 4 (od‘𝐺) = (od‘𝐺)
31, 2ispgp 17928 . . 3 (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚)))
4 simprl 793 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → 𝑃 ∈ ℙ)
51grpbn0 17372 . . . . . . . . . . 11 (𝐺 ∈ Grp → 𝑋 ≠ ∅)
65ad2antrr 761 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → 𝑋 ≠ ∅)
7 hashnncl 13097 . . . . . . . . . . 11 (𝑋 ∈ Fin → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
87ad2antlr 762 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
96, 8mpbird 247 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → (#‘𝑋) ∈ ℕ)
104, 9pccld 15479 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → (𝑃 pCnt (#‘𝑋)) ∈ ℕ0)
1110nn0red 11296 . . . . . . . . . . . . . . 15 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → (𝑃 pCnt (#‘𝑋)) ∈ ℝ)
1211leidd 10538 . . . . . . . . . . . . . 14 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → (𝑃 pCnt (#‘𝑋)) ≤ (𝑃 pCnt (#‘𝑋)))
1310nn0zd 11424 . . . . . . . . . . . . . . 15 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → (𝑃 pCnt (#‘𝑋)) ∈ ℤ)
14 pcid 15501 . . . . . . . . . . . . . . 15 ((𝑃 ∈ ℙ ∧ (𝑃 pCnt (#‘𝑋)) ∈ ℤ) → (𝑃 pCnt (𝑃↑(𝑃 pCnt (#‘𝑋)))) = (𝑃 pCnt (#‘𝑋)))
154, 13, 14syl2anc 692 . . . . . . . . . . . . . 14 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → (𝑃 pCnt (𝑃↑(𝑃 pCnt (#‘𝑋)))) = (𝑃 pCnt (#‘𝑋)))
1612, 15breqtrrd 4641 . . . . . . . . . . . . 13 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → (𝑃 pCnt (#‘𝑋)) ≤ (𝑃 pCnt (𝑃↑(𝑃 pCnt (#‘𝑋)))))
1716ad2antrr 761 . . . . . . . . . . . 12 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑃 pCnt (#‘𝑋)) ≤ (𝑃 pCnt (𝑃↑(𝑃 pCnt (#‘𝑋)))))
18 simpr 477 . . . . . . . . . . . . 13 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
1918oveq1d 6619 . . . . . . . . . . . 12 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑝 pCnt (#‘𝑋)) = (𝑃 pCnt (#‘𝑋)))
2018oveq1d 6619 . . . . . . . . . . . 12 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑝 pCnt (𝑃↑(𝑃 pCnt (#‘𝑋)))) = (𝑃 pCnt (𝑃↑(𝑃 pCnt (#‘𝑋)))))
2117, 19, 203brtr4d 4645 . . . . . . . . . . 11 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑝 pCnt (#‘𝑋)) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt (#‘𝑋)))))
22 simp-4l 805 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) → 𝐺 ∈ Grp)
23 simplr 791 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → 𝑋 ∈ Fin)
2423ad2antrr 761 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) → 𝑋 ∈ Fin)
25 simplr 791 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) → 𝑝 ∈ ℙ)
26 simpr 477 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) → 𝑝 ∥ (#‘𝑋))
271, 2odcau 17940 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) → ∃𝑔𝑋 ((od‘𝐺)‘𝑔) = 𝑝)
2822, 24, 25, 26, 27syl31anc 1326 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) → ∃𝑔𝑋 ((od‘𝐺)‘𝑔) = 𝑝)
2925adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) ∧ (𝑔𝑋 ∧ ((od‘𝐺)‘𝑔) = 𝑝)) → 𝑝 ∈ ℙ)
30 prmz 15313 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
31 iddvds 14919 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ∈ ℤ → 𝑝𝑝)
3229, 30, 313syl 18 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) ∧ (𝑔𝑋 ∧ ((od‘𝐺)‘𝑔) = 𝑝)) → 𝑝𝑝)
33 simprr 795 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) ∧ (𝑔𝑋 ∧ ((od‘𝐺)‘𝑔) = 𝑝)) → ((od‘𝐺)‘𝑔) = 𝑝)
3432, 33breqtrrd 4641 . . . . . . . . . . . . . . . . . . 19 ((((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) ∧ (𝑔𝑋 ∧ ((od‘𝐺)‘𝑔) = 𝑝)) → 𝑝 ∥ ((od‘𝐺)‘𝑔))
35 simplrr 800 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) → ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))
36 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑔 → ((od‘𝐺)‘𝑥) = ((od‘𝐺)‘𝑔))
3736eqeq1d 2623 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑔 → (((od‘𝐺)‘𝑥) = (𝑃𝑚) ↔ ((od‘𝐺)‘𝑔) = (𝑃𝑚)))
3837rexbidv 3045 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑔 → (∃𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚) ↔ ∃𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑔) = (𝑃𝑚)))
3938rspccva 3294 . . . . . . . . . . . . . . . . . . . . . 22 ((∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚) ∧ 𝑔𝑋) → ∃𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑔) = (𝑃𝑚))
4035, 39sylan 488 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑔𝑋) → ∃𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑔) = (𝑃𝑚))
4140ad2ant2r 782 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) ∧ (𝑔𝑋 ∧ ((od‘𝐺)‘𝑔) = 𝑝)) → ∃𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑔) = (𝑃𝑚))
424ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) ∧ (𝑔𝑋 ∧ ((od‘𝐺)‘𝑔) = 𝑝)) → 𝑃 ∈ ℙ)
43 prmnn 15312 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
4429, 43syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) ∧ (𝑔𝑋 ∧ ((od‘𝐺)‘𝑔) = 𝑝)) → 𝑝 ∈ ℕ)
4533, 44eqeltrd 2698 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) ∧ (𝑔𝑋 ∧ ((od‘𝐺)‘𝑔) = 𝑝)) → ((od‘𝐺)‘𝑔) ∈ ℕ)
46 pcprmpw 15511 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃 ∈ ℙ ∧ ((od‘𝐺)‘𝑔) ∈ ℕ) → (∃𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑔) = (𝑃𝑚) ↔ ((od‘𝐺)‘𝑔) = (𝑃↑(𝑃 pCnt ((od‘𝐺)‘𝑔)))))
4742, 45, 46syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) ∧ (𝑔𝑋 ∧ ((od‘𝐺)‘𝑔) = 𝑝)) → (∃𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑔) = (𝑃𝑚) ↔ ((od‘𝐺)‘𝑔) = (𝑃↑(𝑃 pCnt ((od‘𝐺)‘𝑔)))))
4841, 47mpbid 222 . . . . . . . . . . . . . . . . . . 19 ((((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) ∧ (𝑔𝑋 ∧ ((od‘𝐺)‘𝑔) = 𝑝)) → ((od‘𝐺)‘𝑔) = (𝑃↑(𝑃 pCnt ((od‘𝐺)‘𝑔))))
4934, 48breqtrd 4639 . . . . . . . . . . . . . . . . . 18 ((((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) ∧ (𝑔𝑋 ∧ ((od‘𝐺)‘𝑔) = 𝑝)) → 𝑝 ∥ (𝑃↑(𝑃 pCnt ((od‘𝐺)‘𝑔))))
5042, 45pccld 15479 . . . . . . . . . . . . . . . . . . 19 ((((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) ∧ (𝑔𝑋 ∧ ((od‘𝐺)‘𝑔) = 𝑝)) → (𝑃 pCnt ((od‘𝐺)‘𝑔)) ∈ ℕ0)
51 prmdvdsexpr 15353 . . . . . . . . . . . . . . . . . . 19 ((𝑝 ∈ ℙ ∧ 𝑃 ∈ ℙ ∧ (𝑃 pCnt ((od‘𝐺)‘𝑔)) ∈ ℕ0) → (𝑝 ∥ (𝑃↑(𝑃 pCnt ((od‘𝐺)‘𝑔))) → 𝑝 = 𝑃))
5229, 42, 50, 51syl3anc 1323 . . . . . . . . . . . . . . . . . 18 ((((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) ∧ (𝑔𝑋 ∧ ((od‘𝐺)‘𝑔) = 𝑝)) → (𝑝 ∥ (𝑃↑(𝑃 pCnt ((od‘𝐺)‘𝑔))) → 𝑝 = 𝑃))
5349, 52mpd 15 . . . . . . . . . . . . . . . . 17 ((((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) ∧ (𝑔𝑋 ∧ ((od‘𝐺)‘𝑔) = 𝑝)) → 𝑝 = 𝑃)
5428, 53rexlimddv 3028 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ∥ (#‘𝑋)) → 𝑝 = 𝑃)
5554ex 450 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (#‘𝑋) → 𝑝 = 𝑃))
5655necon3ad 2803 . . . . . . . . . . . . . 14 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) → (𝑝𝑃 → ¬ 𝑝 ∥ (#‘𝑋)))
5756imp 445 . . . . . . . . . . . . 13 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → ¬ 𝑝 ∥ (#‘𝑋))
58 simplr 791 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → 𝑝 ∈ ℙ)
599ad2antrr 761 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → (#‘𝑋) ∈ ℕ)
60 pceq0 15499 . . . . . . . . . . . . . 14 ((𝑝 ∈ ℙ ∧ (#‘𝑋) ∈ ℕ) → ((𝑝 pCnt (#‘𝑋)) = 0 ↔ ¬ 𝑝 ∥ (#‘𝑋)))
6158, 59, 60syl2anc 692 . . . . . . . . . . . . 13 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → ((𝑝 pCnt (#‘𝑋)) = 0 ↔ ¬ 𝑝 ∥ (#‘𝑋)))
6257, 61mpbird 247 . . . . . . . . . . . 12 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → (𝑝 pCnt (#‘𝑋)) = 0)
63 prmnn 15312 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
6463ad2antrl 763 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → 𝑃 ∈ ℕ)
6564, 10nnexpcld 12970 . . . . . . . . . . . . . . 15 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → (𝑃↑(𝑃 pCnt (#‘𝑋))) ∈ ℕ)
6665ad2antrr 761 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → (𝑃↑(𝑃 pCnt (#‘𝑋))) ∈ ℕ)
6758, 66pccld 15479 . . . . . . . . . . . . 13 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → (𝑝 pCnt (𝑃↑(𝑃 pCnt (#‘𝑋)))) ∈ ℕ0)
6867nn0ge0d 11298 . . . . . . . . . . . 12 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → 0 ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt (#‘𝑋)))))
6962, 68eqbrtrd 4635 . . . . . . . . . . 11 (((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → (𝑝 pCnt (#‘𝑋)) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt (#‘𝑋)))))
7021, 69pm2.61dane 2877 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (#‘𝑋)) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt (#‘𝑋)))))
7170ralrimiva 2960 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → ∀𝑝 ∈ ℙ (𝑝 pCnt (#‘𝑋)) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt (#‘𝑋)))))
72 hashcl 13087 . . . . . . . . . . . 12 (𝑋 ∈ Fin → (#‘𝑋) ∈ ℕ0)
7372ad2antlr 762 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → (#‘𝑋) ∈ ℕ0)
7473nn0zd 11424 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → (#‘𝑋) ∈ ℤ)
7565nnzd 11425 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → (𝑃↑(𝑃 pCnt (#‘𝑋))) ∈ ℤ)
76 pc2dvds 15507 . . . . . . . . . 10 (((#‘𝑋) ∈ ℤ ∧ (𝑃↑(𝑃 pCnt (#‘𝑋))) ∈ ℤ) → ((#‘𝑋) ∥ (𝑃↑(𝑃 pCnt (#‘𝑋))) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (#‘𝑋)) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt (#‘𝑋))))))
7774, 75, 76syl2anc 692 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → ((#‘𝑋) ∥ (𝑃↑(𝑃 pCnt (#‘𝑋))) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (#‘𝑋)) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt (#‘𝑋))))))
7871, 77mpbird 247 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → (#‘𝑋) ∥ (𝑃↑(𝑃 pCnt (#‘𝑋))))
79 oveq2 6612 . . . . . . . . . 10 (𝑛 = (𝑃 pCnt (#‘𝑋)) → (𝑃𝑛) = (𝑃↑(𝑃 pCnt (#‘𝑋))))
8079breq2d 4625 . . . . . . . . 9 (𝑛 = (𝑃 pCnt (#‘𝑋)) → ((#‘𝑋) ∥ (𝑃𝑛) ↔ (#‘𝑋) ∥ (𝑃↑(𝑃 pCnt (#‘𝑋)))))
8180rspcev 3295 . . . . . . . 8 (((𝑃 pCnt (#‘𝑋)) ∈ ℕ0 ∧ (#‘𝑋) ∥ (𝑃↑(𝑃 pCnt (#‘𝑋)))) → ∃𝑛 ∈ ℕ0 (#‘𝑋) ∥ (𝑃𝑛))
8210, 78, 81syl2anc 692 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → ∃𝑛 ∈ ℕ0 (#‘𝑋) ∥ (𝑃𝑛))
83 pcprmpw2 15510 . . . . . . . . 9 ((𝑃 ∈ ℙ ∧ (#‘𝑋) ∈ ℕ) → (∃𝑛 ∈ ℕ0 (#‘𝑋) ∥ (𝑃𝑛) ↔ (#‘𝑋) = (𝑃↑(𝑃 pCnt (#‘𝑋)))))
84 pcprmpw 15511 . . . . . . . . 9 ((𝑃 ∈ ℙ ∧ (#‘𝑋) ∈ ℕ) → (∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛) ↔ (#‘𝑋) = (𝑃↑(𝑃 pCnt (#‘𝑋)))))
8583, 84bitr4d 271 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ (#‘𝑋) ∈ ℕ) → (∃𝑛 ∈ ℕ0 (#‘𝑋) ∥ (𝑃𝑛) ↔ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛)))
864, 9, 85syl2anc 692 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → (∃𝑛 ∈ ℕ0 (#‘𝑋) ∥ (𝑃𝑛) ↔ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛)))
8782, 86mpbid 222 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛))
884, 87jca 554 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛)))
89883adantr2 1219 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚))) → (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛)))
9089ex 450 . . 3 ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → ((𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥𝑋𝑚 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑚)) → (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛))))
913, 90syl5bi 232 . 2 ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 → (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛))))
921pgpfi1 17931 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ ∧ 𝑛 ∈ ℕ0) → ((#‘𝑋) = (𝑃𝑛) → 𝑃 pGrp 𝐺))
93923expia 1264 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → (𝑛 ∈ ℕ0 → ((#‘𝑋) = (𝑃𝑛) → 𝑃 pGrp 𝐺)))
9493rexlimdv 3023 . . . 4 ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → (∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛) → 𝑃 pGrp 𝐺))
9594expimpd 628 . . 3 (𝐺 ∈ Grp → ((𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛)) → 𝑃 pGrp 𝐺))
9695adantr 481 . 2 ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → ((𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛)) → 𝑃 pGrp 𝐺))
9791, 96impbid 202 1 ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  ∃wrex 2908  ∅c0 3891   class class class wbr 4613  ‘cfv 5847  (class class class)co 6604  Fincfn 7899  0cc0 9880   ≤ cle 10019  ℕcn 10964  ℕ0cn0 11236  ℤcz 11321  ↑cexp 12800  #chash 13057   ∥ cdvds 14907  ℙcprime 15309   pCnt cpc 15465  Basecbs 15781  Grpcgrp 17343  odcod 17865   pGrp cpgp 17867 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-ga 17644  df-od 17869  df-pgp 17871 This theorem is referenced by:  pgpfi2  17942  sylow2alem2  17954  slwhash  17960  fislw  17961
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