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Theorem pcprmpw2 15521
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 791 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝐴 ∈ ℕ)
21nnnn0d 11303 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝐴 ∈ ℕ0)
3 prmnn 15323 . . . . . . 7 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
43ad2antrr 761 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝑃 ∈ ℕ)
5 pccl 15489 . . . . . . 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 12978 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
87nnnn0d 11303 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ0)
96nn0red 11304 . . . . . . . . . . 11 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃 pCnt 𝐴) ∈ ℝ)
109leidd 10546 . . . . . . . . . 10 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐴))
11 simpll 789 . . . . . . . . . . 11 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝑃 ∈ ℙ)
126nn0zd 11432 . . . . . . . . . . 11 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃 pCnt 𝐴) ∈ ℤ)
13 pcid 15512 . . . . . . . . . . 11 ((𝑃 ∈ ℙ ∧ (𝑃 pCnt 𝐴) ∈ ℤ) → (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))) = (𝑃 pCnt 𝐴))
1411, 12, 13syl2anc 692 . . . . . . . . . 10 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))) = (𝑃 pCnt 𝐴))
1510, 14breqtrrd 4646 . . . . . . . . 9 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
1615ad2antrr 761 . . . . . . . 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 6625 . . . . . . . 8 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑝 pCnt 𝐴) = (𝑃 pCnt 𝐴))
1917oveq1d 6625 . . . . . . . 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 4650 . . . . . . 7 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
21 simplrr 800 . . . . . . . . . . . . 13 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝐴 ∥ (𝑃𝑛))
22 prmz 15324 . . . . . . . . . . . . . . 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 11433 . . . . . . . . . . . . . 14 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℤ)
26 simprl 793 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝑛 ∈ ℕ0)
274, 26nnexpcld 12978 . . . . . . . . . . . . . . . 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 11433 . . . . . . . . . . . . . 14 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑃𝑛) ∈ ℤ)
30 dvdstr 14953 . . . . . . . . . . . . . 14 ((𝑝 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝑃𝑛) ∈ ℤ) → ((𝑝𝐴𝐴 ∥ (𝑃𝑛)) → 𝑝 ∥ (𝑃𝑛)))
3123, 25, 29, 30syl3anc 1323 . . . . . . . . . . . . 13 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → ((𝑝𝐴𝐴 ∥ (𝑃𝑛)) → 𝑝 ∥ (𝑃𝑛)))
3221, 31mpan2d 709 . . . . . . . . . . . 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 481 . . . . . . . . . . . . 13 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑃 ∈ ℙ)
35 simplrl 799 . . . . . . . . . . . . 13 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑛 ∈ ℕ0)
36 prmdvdsexpr 15364 . . . . . . . . . . . . 13 ((𝑝 ∈ ℙ ∧ 𝑃 ∈ ℙ ∧ 𝑛 ∈ ℕ0) → (𝑝 ∥ (𝑃𝑛) → 𝑝 = 𝑃))
3733, 34, 35, 36syl3anc 1323 . . . . . . . . . . . 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 2803 . . . . . . . . . 10 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝𝑃 → ¬ 𝑝𝐴))
4039imp 445 . . . . . . . . 9 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → ¬ 𝑝𝐴)
41 simplr 791 . . . . . . . . . 10 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → 𝑝 ∈ ℙ)
421ad2antrr 761 . . . . . . . . . 10 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → 𝐴 ∈ ℕ)
43 pceq0 15510 . . . . . . . . . 10 ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ((𝑝 pCnt 𝐴) = 0 ↔ ¬ 𝑝𝐴))
4441, 42, 43syl2anc 692 . . . . . . . . 9 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → ((𝑝 pCnt 𝐴) = 0 ↔ ¬ 𝑝𝐴))
4540, 44mpbird 247 . . . . . . . 8 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → (𝑝 pCnt 𝐴) = 0)
467ad2antrr 761 . . . . . . . . . 10 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
4741, 46pccld 15490 . . . . . . . . 9 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ0)
4847nn0ge0d 11306 . . . . . . . 8 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → 0 ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
4945, 48eqbrtrd 4640 . . . . . . 7 (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝𝑃) → (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
5020, 49pm2.61dane 2877 . . . . . 6 ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
5150ralrimiva 2961 . . . . 5 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
521nnzd 11433 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝐴 ∈ ℤ)
537nnzd 11433 . . . . . 6 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ)
54 pc2dvds 15518 . . . . . 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 692 . . . . 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 247 . . . 4 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝐴 ∥ (𝑃↑(𝑃 pCnt 𝐴)))
57 pcdvds 15503 . . . . 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 14971 . . . 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 1324 . . 3 (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0𝐴 ∥ (𝑃𝑛))) → 𝐴 = (𝑃↑(𝑃 pCnt 𝐴)))
6160rexlimdvaa 3026 . 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 12978 . . . . . 6 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
6463nnzd 11433 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ)
65 iddvds 14930 . . . . 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 6618 . . . . . 6 (𝑛 = (𝑃 pCnt 𝐴) → (𝑃𝑛) = (𝑃↑(𝑃 pCnt 𝐴)))
6867breq2d 4630 . . . . 5 (𝑛 = (𝑃 pCnt 𝐴) → ((𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃𝑛) ↔ (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑(𝑃 pCnt 𝐴))))
6968rspcev 3298 . . . 4 (((𝑃 pCnt 𝐴) ∈ ℕ0 ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑(𝑃 pCnt 𝐴))) → ∃𝑛 ∈ ℕ0 (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃𝑛))
705, 66, 69syl2anc 692 . . 3 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ0 (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃𝑛))
71 breq1 4621 . . . 4 (𝐴 = (𝑃↑(𝑃 pCnt 𝐴)) → (𝐴 ∥ (𝑃𝑛) ↔ (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃𝑛)))
7271rexbidv 3046 . . 3 (𝐴 = (𝑃↑(𝑃 pCnt 𝐴)) → (∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃𝑛) ↔ ∃𝑛 ∈ ℕ0 (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃𝑛)))
7370, 72syl5ibrcom 237 . 2 ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝐴 = (𝑃↑(𝑃 pCnt 𝐴)) → ∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃𝑛)))
7461, 73impbid 202 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 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908   class class class wbr 4618  (class class class)co 6610  0cc0 9888  cle 10027  cn 10972  0cn0 11244  cz 11329  cexp 12808  cdvds 14918  cprime 15320   pCnt cpc 15476
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-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-sup 8300  df-inf 8301  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-n0 11245  df-z 11330  df-uz 11640  df-q 11741  df-rp 11785  df-fz 12277  df-fl 12541  df-mod 12617  df-seq 12750  df-exp 12809  df-cj 13781  df-re 13782  df-im 13783  df-sqrt 13917  df-abs 13918  df-dvds 14919  df-gcd 15152  df-prm 15321  df-pc 15477
This theorem is referenced by:  pcprmpw  15522  dvdsprmpweq  15523  pgpfi1  17942  pgpfi  17952  sylow2alem2  17965  lt6abl  18228  pgpfac1lem3a  18407  dvdsppwf1o  24829
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