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Theorem pcprmpw2 16072
Description: Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.)
Assertion
Ref Expression
pcprmpw2 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃𝑛) ↔ 𝐴 = (𝑃↑(𝑃 pCnt 𝐴))))
Distinct variable groups:   𝐴,𝑛   𝑃,𝑛

Proof of Theorem pcprmpw2
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 simplr 757 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝐴 ∈ ℕ)
21nnnn0d 11765 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝐴 ∈ ℕ0)
3 prmnn 15872 . . . . . . 7 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
43ad2antrr 714 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝑃 ∈ ℕ)
5 pccl 16040 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃 pCnt 𝐴) ∈ ℕ0)
65adantr 473 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃 pCnt 𝐴) ∈ ℕ0)
74, 6nnexpcld 13419 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
87nnnn0d 11765 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ0)
96nn0red 11766 . . . . . . . . . . 11 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃 pCnt 𝐴) ∈ ℝ)
109leidd 11005 . . . . . . . . . 10 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐴))
11 simpll 755 . . . . . . . . . . 11 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝑃 ∈ ℙ)
126nn0zd 11896 . . . . . . . . . . 11 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃 pCnt 𝐴) ∈ ℤ)
13 pcid 16063 . . . . . . . . . . 11 ((𝑃 ∈ ℙ ∧ (𝑃 pCnt 𝐴) ∈ ℤ) → (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))) = (𝑃 pCnt 𝐴))
1411, 12, 13syl2anc 576 . . . . . . . . . 10 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))) = (𝑃 pCnt 𝐴))
1510, 14breqtrrd 4953 . . . . . . . . 9 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
1615ad2antrr 714 . . . . . . . 8 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
17 simpr 477 . . . . . . . . 9 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
1817oveq1d 6989 . . . . . . . 8 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑝 pCnt 𝐴) = (𝑃 pCnt 𝐴))
1917oveq1d 6989 . . . . . . . 8 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))) = (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
2016, 18, 193brtr4d 4957 . . . . . . 7 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
21 simplrr 766 . . . . . . . . . . . . 13 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝐴 ∥ (𝑃𝑛))
22 prmz 15873 . . . . . . . . . . . . . . 15 (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
2322adantl 474 . . . . . . . . . . . . . 14 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℤ)
241adantr 473 . . . . . . . . . . . . . . 15 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℕ)
2524nnzd 11897 . . . . . . . . . . . . . 14 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℤ)
26 simprl 759 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝑛 ∈ ℕ0)
274, 26nnexpcld 13419 . . . . . . . . . . . . . . . 16 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃𝑛) ∈ ℕ)
2827adantr 473 . . . . . . . . . . . . . . 15 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑃𝑛) ∈ ℕ)
2928nnzd 11897 . . . . . . . . . . . . . 14 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑃𝑛) ∈ ℤ)
30 dvdstr 15504 . . . . . . . . . . . . . 14 ((𝑝 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝑃𝑛) ∈ ℤ) → ((𝑝𝐴𝐴 ∥ (𝑃𝑛)) → 𝑝 ∥ (𝑃𝑛)))
3123, 25, 29, 30syl3anc 1352 . . . . . . . . . . . . 13 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → ((𝑝𝐴𝐴 ∥ (𝑃𝑛)) → 𝑝 ∥ (𝑃𝑛)))
3221, 31mpan2d 682 . . . . . . . . . . . 12 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝𝐴𝑝 ∥ (𝑃𝑛)))
33 simpr 477 . . . . . . . . . . . . 13 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
3411adantr 473 . . . . . . . . . . . . 13 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑃 ∈ ℙ)
35 simplrl 765 . . . . . . . . . . . . 13 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑛 ∈ ℕ0)
36 prmdvdsexpr 15915 . . . . . . . . . . . . 13 ((𝑝 ∈ ℙ ∧ 𝑃 ∈ ℙ ∧ 𝑛 ∈ ℕ0) → (𝑝 ∥ (𝑃𝑛) → 𝑝 = 𝑃))
3733, 34, 35, 36syl3anc 1352 . . . . . . . . . . . 12 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝑃𝑛) → 𝑝 = 𝑃))
3832, 37syld 47 . . . . . . . . . . 11 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝𝐴𝑝 = 𝑃))
3938necon3ad 2973 . . . . . . . . . 10 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝𝑃 → ¬ 𝑝𝐴))
4039imp 398 . . . . . . . . 9 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → ¬ 𝑝𝐴)
41 simplr 757 . . . . . . . . . 10 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → 𝑝 ∈ ℙ)
421ad2antrr 714 . . . . . . . . . 10 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → 𝐴 ∈ ℕ)
43 pceq0 16061 . . . . . . . . . 10 ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ((𝑝 pCnt 𝐴) = 0 ↔ ¬ 𝑝𝐴))
4441, 42, 43syl2anc 576 . . . . . . . . 9 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → ((𝑝 pCnt 𝐴) = 0 ↔ ¬ 𝑝𝐴))
4540, 44mpbird 249 . . . . . . . 8 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → (𝑝 pCnt 𝐴) = 0)
467ad2antrr 714 . . . . . . . . . 10 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
4741, 46pccld 16041 . . . . . . . . 9 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ0)
4847nn0ge0d 11768 . . . . . . . 8 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → 0 ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
4945, 48eqbrtrd 4947 . . . . . . 7 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
5020, 49pm2.61dane 3048 . . . . . 6 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
5150ralrimiva 3125 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
521nnzd 11897 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝐴 ∈ ℤ)
537nnzd 11897 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ)
54 pc2dvds 16069 . . . . . 6 ((𝐴 ∈ ℤ ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ) → (𝐴 ∥ (𝑃↑(𝑃 pCnt 𝐴)) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴)))))
5552, 53, 54syl2anc 576 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝐴 ∥ (𝑃↑(𝑃 pCnt 𝐴)) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴)))))
5651, 55mpbird 249 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝐴 ∥ (𝑃↑(𝑃 pCnt 𝐴)))
57 pcdvds 16054 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
5857adantr 473 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
59 dvdseq 15522 . . . 4 (((𝐴 ∈ ℕ0 ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ0) ∧ (𝐴 ∥ (𝑃↑(𝑃 pCnt 𝐴)) ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)) → 𝐴 = (𝑃↑(𝑃 pCnt 𝐴)))
602, 8, 56, 58, 59syl22anc 827 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝐴 = (𝑃↑(𝑃 pCnt 𝐴)))
6160rexlimdvaa 3223 . 2 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃𝑛) → 𝐴 = (𝑃↑(𝑃 pCnt 𝐴))))
623adantr 473 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → 𝑃 ∈ ℕ)
6362, 5nnexpcld 13419 . . . . . 6 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
6463nnzd 11897 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ)
65 iddvds 15481 . . . . 5 ((𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ → (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑(𝑃 pCnt 𝐴)))
6664, 65syl 17 . . . 4 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑(𝑃 pCnt 𝐴)))
67 oveq2 6982 . . . . . 6 (𝑛 = (𝑃 pCnt 𝐴) → (𝑃𝑛) = (𝑃↑(𝑃 pCnt 𝐴)))
6867breq2d 4937 . . . . 5 (𝑛 = (𝑃 pCnt 𝐴) → ((𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃𝑛) ↔ (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑(𝑃 pCnt 𝐴))))
6968rspcev 3528 . . . 4 (((𝑃 pCnt 𝐴) ∈ ℕ0 ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑(𝑃 pCnt 𝐴))) → ∃𝑛 ∈ ℕ0 (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃𝑛))
705, 66, 69syl2anc 576 . . 3 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ0 (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃𝑛))
71 breq1 4928 . . . 4 (𝐴 = (𝑃↑(𝑃 pCnt 𝐴)) → (𝐴 ∥ (𝑃𝑛) ↔ (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃𝑛)))
7271rexbidv 3235 . . 3 (𝐴 = (𝑃↑(𝑃 pCnt 𝐴)) → (∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃𝑛) ↔ ∃𝑛 ∈ ℕ0 (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃𝑛)))
7370, 72syl5ibrcom 239 . 2 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝐴 = (𝑃↑(𝑃 pCnt 𝐴)) → ∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃𝑛)))
7461, 73impbid 204 1 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃𝑛) ↔ 𝐴 = (𝑃↑(𝑃 pCnt 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387   = wceq 1508  wcel 2051  wne 2960  wral 3081  wrex 3082   class class class wbr 4925  (class class class)co 6974  0cc0 10333  cle 10473  cn 11437  0cn0 11705  cz 11791  cexp 13242  cdvds 15465  cprime 15869   pCnt cpc 16027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410  ax-pre-sup 10411
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-nel 3067  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-2o 7904  df-er 8087  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-sup 8699  df-inf 8700  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-div 11097  df-nn 11438  df-2 11501  df-3 11502  df-n0 11706  df-z 11792  df-uz 12057  df-q 12161  df-rp 12203  df-fz 12707  df-fl 12975  df-mod 13051  df-seq 13183  df-exp 13243  df-cj 14317  df-re 14318  df-im 14319  df-sqrt 14453  df-abs 14454  df-dvds 15466  df-gcd 15702  df-prm 15870  df-pc 16028
This theorem is referenced by:  pcprmpw  16073  dvdsprmpweq  16074  pgpfi1  18493  pgpfi  18503  sylow2alem2  18516  lt6abl  18781  pgpfac1lem3a  18960  dvdsppwf1o  25480
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