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Theorem pcprmpw2 16206
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 765 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝐴 ∈ ℕ)
21nnnn0d 11943 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝐴 ∈ ℕ0)
3 prmnn 16006 . . . . . . 7 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
43ad2antrr 722 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝑃 ∈ ℕ)
5 pccl 16174 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃 pCnt 𝐴) ∈ ℕ0)
65adantr 481 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃 pCnt 𝐴) ∈ ℕ0)
74, 6nnexpcld 13594 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
87nnnn0d 11943 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ0)
96nn0red 11944 . . . . . . . . . . 11 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃 pCnt 𝐴) ∈ ℝ)
109leidd 11194 . . . . . . . . . 10 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐴))
11 simpll 763 . . . . . . . . . . 11 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝑃 ∈ ℙ)
126nn0zd 12073 . . . . . . . . . . 11 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃 pCnt 𝐴) ∈ ℤ)
13 pcid 16197 . . . . . . . . . . 11 ((𝑃 ∈ ℙ ∧ (𝑃 pCnt 𝐴) ∈ ℤ) → (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))) = (𝑃 pCnt 𝐴))
1411, 12, 13syl2anc 584 . . . . . . . . . 10 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))) = (𝑃 pCnt 𝐴))
1510, 14breqtrrd 5085 . . . . . . . . 9 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
1615ad2antrr 722 . . . . . . . 8 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
17 simpr 485 . . . . . . . . 9 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
1817oveq1d 7160 . . . . . . . 8 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑝 pCnt 𝐴) = (𝑃 pCnt 𝐴))
1917oveq1d 7160 . . . . . . . 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 5089 . . . . . . 7 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
21 simplrr 774 . . . . . . . . . . . . 13 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝐴 ∥ (𝑃𝑛))
22 prmz 16007 . . . . . . . . . . . . . . 15 (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
2322adantl 482 . . . . . . . . . . . . . 14 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℤ)
241adantr 481 . . . . . . . . . . . . . . 15 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℕ)
2524nnzd 12074 . . . . . . . . . . . . . 14 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℤ)
26 simprl 767 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝑛 ∈ ℕ0)
274, 26nnexpcld 13594 . . . . . . . . . . . . . . . 16 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃𝑛) ∈ ℕ)
2827adantr 481 . . . . . . . . . . . . . . 15 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑃𝑛) ∈ ℕ)
2928nnzd 12074 . . . . . . . . . . . . . 14 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑃𝑛) ∈ ℤ)
30 dvdstr 15634 . . . . . . . . . . . . . 14 ((𝑝 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝑃𝑛) ∈ ℤ) → ((𝑝𝐴𝐴 ∥ (𝑃𝑛)) → 𝑝 ∥ (𝑃𝑛)))
3123, 25, 29, 30syl3anc 1363 . . . . . . . . . . . . 13 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → ((𝑝𝐴𝐴 ∥ (𝑃𝑛)) → 𝑝 ∥ (𝑃𝑛)))
3221, 31mpan2d 690 . . . . . . . . . . . 12 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝𝐴𝑝 ∥ (𝑃𝑛)))
33 simpr 485 . . . . . . . . . . . . 13 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
3411adantr 481 . . . . . . . . . . . . 13 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑃 ∈ ℙ)
35 simplrl 773 . . . . . . . . . . . . 13 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑛 ∈ ℕ0)
36 prmdvdsexpr 16049 . . . . . . . . . . . . 13 ((𝑝 ∈ ℙ ∧ 𝑃 ∈ ℙ ∧ 𝑛 ∈ ℕ0) → (𝑝 ∥ (𝑃𝑛) → 𝑝 = 𝑃))
3733, 34, 35, 36syl3anc 1363 . . . . . . . . . . . 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 3026 . . . . . . . . . 10 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝𝑃 → ¬ 𝑝𝐴))
4039imp 407 . . . . . . . . 9 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → ¬ 𝑝𝐴)
41 simplr 765 . . . . . . . . . 10 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → 𝑝 ∈ ℙ)
421ad2antrr 722 . . . . . . . . . 10 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → 𝐴 ∈ ℕ)
43 pceq0 16195 . . . . . . . . . 10 ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ((𝑝 pCnt 𝐴) = 0 ↔ ¬ 𝑝𝐴))
4441, 42, 43syl2anc 584 . . . . . . . . 9 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → ((𝑝 pCnt 𝐴) = 0 ↔ ¬ 𝑝𝐴))
4540, 44mpbird 258 . . . . . . . 8 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → (𝑝 pCnt 𝐴) = 0)
467ad2antrr 722 . . . . . . . . . 10 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
4741, 46pccld 16175 . . . . . . . . 9 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ0)
4847nn0ge0d 11946 . . . . . . . 8 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → 0 ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
4945, 48eqbrtrd 5079 . . . . . . 7 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
5020, 49pm2.61dane 3101 . . . . . 6 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
5150ralrimiva 3179 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
521nnzd 12074 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝐴 ∈ ℤ)
537nnzd 12074 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ)
54 pc2dvds 16203 . . . . . 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 584 . . . . 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 258 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝐴 ∥ (𝑃↑(𝑃 pCnt 𝐴)))
57 pcdvds 16188 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
5857adantr 481 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
59 dvdseq 15652 . . . 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 834 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝐴 = (𝑃↑(𝑃 pCnt 𝐴)))
6160rexlimdvaa 3282 . 2 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃𝑛) → 𝐴 = (𝑃↑(𝑃 pCnt 𝐴))))
623adantr 481 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → 𝑃 ∈ ℕ)
6362, 5nnexpcld 13594 . . . . . 6 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
6463nnzd 12074 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ)
65 iddvds 15611 . . . . 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 7153 . . . . . 6 (𝑛 = (𝑃 pCnt 𝐴) → (𝑃𝑛) = (𝑃↑(𝑃 pCnt 𝐴)))
6867breq2d 5069 . . . . 5 (𝑛 = (𝑃 pCnt 𝐴) → ((𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃𝑛) ↔ (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑(𝑃 pCnt 𝐴))))
6968rspcev 3620 . . . 4 (((𝑃 pCnt 𝐴) ∈ ℕ0 ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑(𝑃 pCnt 𝐴))) → ∃𝑛 ∈ ℕ0 (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃𝑛))
705, 66, 69syl2anc 584 . . 3 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ0 (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃𝑛))
71 breq1 5060 . . . 4 (𝐴 = (𝑃↑(𝑃 pCnt 𝐴)) → (𝐴 ∥ (𝑃𝑛) ↔ (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃𝑛)))
7271rexbidv 3294 . . 3 (𝐴 = (𝑃↑(𝑃 pCnt 𝐴)) → (∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃𝑛) ↔ ∃𝑛 ∈ ℕ0 (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃𝑛)))
7370, 72syl5ibrcom 248 . 2 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝐴 = (𝑃↑(𝑃 pCnt 𝐴)) → ∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃𝑛)))
7461, 73impbid 213 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 207  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136   class class class wbr 5057  (class class class)co 7145  0cc0 10525  cle 10664  cn 11626  0cn0 11885  cz 11969  cexp 13417  cdvds 15595  cprime 16003   pCnt cpc 16161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-fz 12881  df-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-dvds 15596  df-gcd 15832  df-prm 16004  df-pc 16162
This theorem is referenced by:  pcprmpw  16207  dvdsprmpweq  16208  pgpfi1  18649  pgpfi  18659  sylow2alem2  18672  lt6abl  18944  pgpfac1lem3a  19127  dvdsppwf1o  25690
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