Theorem pcprmpw2 12700

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 ∥ (𝑃↑𝑛) ↔ 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.

Step	Hyp	Ref	Expression
1		simplr 528	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → 𝐴 ∈ ℕ)
2	1	nnnn0d 9355	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → 𝐴 ∈ ℕ0)
3		prmnn 12476	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	3	ad2antrr 488	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → 𝑃 ∈ ℕ)
5		pccl 12666	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃 pCnt 𝐴) ∈ ℕ0)
6	5	adantr 276	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃 pCnt 𝐴) ∈ ℕ0)
7	4, 6	nnexpcld 10847	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
8	7	nnnn0d 9355	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ0)
9	6	nn0red 9356	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃 pCnt 𝐴) ∈ ℝ)
10	9	leidd 8594	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐴))
11		simpll 527	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → 𝑃 ∈ ℙ)
12	6	nn0zd 9500	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃 pCnt 𝐴) ∈ ℤ)
13		pcid 12691	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 pCnt 𝐴) ∈ ℤ) → (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))) = (𝑃 pCnt 𝐴))
14	11, 12, 13	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))) = (𝑃 pCnt 𝐴))
15	10, 14	breqtrrd 4075	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
16	15	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
17		simpr 110	. . . . . . . . 9 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
18	17	oveq1d 5966	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑝 pCnt 𝐴) = (𝑃 pCnt 𝐴))
19	17	oveq1d 5966	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))) = (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
20	16, 18, 19	3brtr4d 4079	. . . . . . 7 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
21		simplrr 536	. . . . . . . . . . . . 13 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → 𝐴 ∥ (𝑃↑𝑛))
22		prmz 12477	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
23	22	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℤ)
24	1	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℕ)
25	24	nnzd 9501	. . . . . . . . . . . . . 14 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℤ)
26		simprl 529	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → 𝑛 ∈ ℕ0)
27	4, 26	nnexpcld 10847	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃↑𝑛) ∈ ℕ)
28	27	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑃↑𝑛) ∈ ℕ)
29	28	nnzd 9501	. . . . . . . . . . . . . 14 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑃↑𝑛) ∈ ℤ)
30		dvdstr 12183	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝑃↑𝑛) ∈ ℤ) → ((𝑝 ∥ 𝐴 ∧ 𝐴 ∥ (𝑃↑𝑛)) → 𝑝 ∥ (𝑃↑𝑛)))
31	23, 25, 29, 30	syl3anc 1250	. . . . . . . . . . . . 13 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ 𝐴 ∧ 𝐴 ∥ (𝑃↑𝑛)) → 𝑝 ∥ (𝑃↑𝑛)))
32	21, 31	mpan2d 428	. . . . . . . . . . . 12 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ 𝐴 → 𝑝 ∥ (𝑃↑𝑛)))
33		simpr 110	. . . . . . . . . . . . 13 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
34	11	adantr 276	. . . . . . . . . . . . 13 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑃 ∈ ℙ)
35		simplrl 535	. . . . . . . . . . . . 13 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑛 ∈ ℕ0)
36		prmdvdsexpr 12516	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℙ ∧ 𝑃 ∈ ℙ ∧ 𝑛 ∈ ℕ0) → (𝑝 ∥ (𝑃↑𝑛) → 𝑝 = 𝑃))
37	33, 34, 35, 36	syl3anc 1250	. . . . . . . . . . . 12 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝑃↑𝑛) → 𝑝 = 𝑃))
38	32, 37	syld 45	. . . . . . . . . . 11 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ 𝐴 → 𝑝 = 𝑃))
39	38	necon3ad 2419	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝 ≠ 𝑃 → ¬ 𝑝 ∥ 𝐴))
40	39	imp 124	. . . . . . . . 9 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ≠ 𝑃) → ¬ 𝑝 ∥ 𝐴)
41		simplr 528	. . . . . . . . . 10 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ≠ 𝑃) → 𝑝 ∈ ℙ)
42	1	ad2antrr 488	. . . . . . . . . 10 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ≠ 𝑃) → 𝐴 ∈ ℕ)
43		pceq0 12689	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ((𝑝 pCnt 𝐴) = 0 ↔ ¬ 𝑝 ∥ 𝐴))
44	41, 42, 43	syl2anc 411	. . . . . . . . 9 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ≠ 𝑃) → ((𝑝 pCnt 𝐴) = 0 ↔ ¬ 𝑝 ∥ 𝐴))
45	40, 44	mpbird 167	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ≠ 𝑃) → (𝑝 pCnt 𝐴) = 0)
46	7	ad2antrr 488	. . . . . . . . . 10 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ≠ 𝑃) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
47	41, 46	pccld 12667	. . . . . . . . 9 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ≠ 𝑃) → (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ0)
48	47	nn0ge0d 9358	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ≠ 𝑃) → 0 ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
49	45, 48	eqbrtrd 4069	. . . . . . 7 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ≠ 𝑃) → (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
50		prmz 12477	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
51	50	adantr 276	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → 𝑃 ∈ ℤ)
52	51	ad2antrr 488	. . . . . . . . 9 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑃 ∈ ℤ)
53		zdceq 9455	. . . . . . . . 9 ⊢ ((𝑝 ∈ ℤ ∧ 𝑃 ∈ ℤ) → DECID 𝑝 = 𝑃)
54	23, 52, 53	syl2anc 411	. . . . . . . 8 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → DECID 𝑝 = 𝑃)
55		dcne 2388	. . . . . . . 8 ⊢ (DECID 𝑝 = 𝑃 ↔ (𝑝 = 𝑃 ∨ 𝑝 ≠ 𝑃))
56	54, 55	sylib 122	. . . . . . 7 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝 = 𝑃 ∨ 𝑝 ≠ 𝑃))
57	20, 49, 56	mpjaodan 800	. . . . . 6 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
58	57	ralrimiva 2580	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
59	1	nnzd 9501	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → 𝐴 ∈ ℤ)
60	7	nnzd 9501	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ)
61		pc2dvds 12697	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ) → (𝐴 ∥ (𝑃↑(𝑃 pCnt 𝐴)) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴)))))
62	59, 60, 61	syl2anc 411	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝐴 ∥ (𝑃↑(𝑃 pCnt 𝐴)) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴)))))
63	58, 62	mpbird 167	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → 𝐴 ∥ (𝑃↑(𝑃 pCnt 𝐴)))
64		pcdvds 12682	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
65	64	adantr 276	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
66		dvdseq 12203	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ0) ∧ (𝐴 ∥ (𝑃↑(𝑃 pCnt 𝐴)) ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)) → 𝐴 = (𝑃↑(𝑃 pCnt 𝐴)))
67	2, 8, 63, 65, 66	syl22anc 1251	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → 𝐴 = (𝑃↑(𝑃 pCnt 𝐴)))
68	67	rexlimdvaa 2625	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃↑𝑛) → 𝐴 = (𝑃↑(𝑃 pCnt 𝐴))))
69	3	adantr 276	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → 𝑃 ∈ ℕ)
70	69, 5	nnexpcld 10847	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
71	70	nnzd 9501	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ)
72		iddvds 12159	. . . . 5 ⊢ ((𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ → (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑(𝑃 pCnt 𝐴)))
73	71, 72	syl 14	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑(𝑃 pCnt 𝐴)))
74		oveq2 5959	. . . . . 6 ⊢ (𝑛 = (𝑃 pCnt 𝐴) → (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt 𝐴)))
75	74	breq2d 4059	. . . . 5 ⊢ (𝑛 = (𝑃 pCnt 𝐴) → ((𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑𝑛) ↔ (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑(𝑃 pCnt 𝐴))))
76	75	rspcev 2878	. . . 4 ⊢ (((𝑃 pCnt 𝐴) ∈ ℕ0 ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑(𝑃 pCnt 𝐴))) → ∃𝑛 ∈ ℕ0 (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑𝑛))
77	5, 73, 76	syl2anc 411	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ0 (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑𝑛))
78		breq1 4050	. . . 4 ⊢ (𝐴 = (𝑃↑(𝑃 pCnt 𝐴)) → (𝐴 ∥ (𝑃↑𝑛) ↔ (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑𝑛)))
79	78	rexbidv 2508	. . 3 ⊢ (𝐴 = (𝑃↑(𝑃 pCnt 𝐴)) → (∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃↑𝑛) ↔ ∃𝑛 ∈ ℕ0 (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑𝑛)))
80	77, 79	syl5ibrcom 157	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝐴 = (𝑃↑(𝑃 pCnt 𝐴)) → ∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃↑𝑛)))
81	68, 80	impbid 129	1 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃↑𝑛) ↔ 𝐴 = (𝑃↑(𝑃 pCnt 𝐴))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710  DECID wdc 836   = wceq 1373   ∈ wcel 2177   ≠ wne 2377  ∀wral 2485  ∃wrex 2486   class class class wbr 4047  (class class class)co 5951  0cc0 7932   ≤ cle 8115  ℕcn 9043  ℕ₀cn0 9302  ℤcz 9379  ↑cexp 10690   ∥ cdvds 12142  ℙcprime 12473   pCnt cpc 12651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-isom 5285  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-frec 6484  df-1o 6509  df-2o 6510  df-er 6627  df-en 6835  df-sup 7093  df-inf 7094  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-n0 9303  df-xnn0 9366  df-z 9380  df-uz 9656  df-q 9748  df-rp 9783  df-fz 10138  df-fzo 10272  df-fl 10420  df-mod 10475  df-seqfrec 10600  df-exp 10691  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354  df-dvds 12143  df-gcd 12319  df-prm 12474  df-pc 12652

This theorem is referenced by:  pcprmpw  12701  dvdsprmpweq  12702  dvdsppwf1o  15505