Theorem pcprmpw2 15780

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 809	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → 𝐴 ∈ ℕ)
2	1	nnnn0d 11535	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → 𝐴 ∈ ℕ0)
3		prmnn 15582	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	3	ad2antrr 764	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → 𝑃 ∈ ℕ)
5		pccl 15748	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃 pCnt 𝐴) ∈ ℕ0)
6	5	adantr 472	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃 pCnt 𝐴) ∈ ℕ0)
7	4, 6	nnexpcld 13216	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
8	7	nnnn0d 11535	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ0)
9	6	nn0red 11536	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃 pCnt 𝐴) ∈ ℝ)
10	9	leidd 10778	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐴))
11		simpll 807	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → 𝑃 ∈ ℙ)
12	6	nn0zd 11664	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃 pCnt 𝐴) ∈ ℤ)
13		pcid 15771	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 pCnt 𝐴) ∈ ℤ) → (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))) = (𝑃 pCnt 𝐴))
14	11, 12, 13	syl2anc 696	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))) = (𝑃 pCnt 𝐴))
15	10, 14	breqtrrd 4824	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
16	15	ad2antrr 764	. . . . . . . 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 479	. . . . . . . . 9 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
18	17	oveq1d 6820	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 = 𝑃) → (𝑝 pCnt 𝐴) = (𝑃 pCnt 𝐴))
19	17	oveq1d 6820	. . . . . . . 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 4828	. . . . . . 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 820	. . . . . . . . . . . . 13 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → 𝐴 ∥ (𝑃↑𝑛))
22		prmz 15583	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
23	22	adantl 473	. . . . . . . . . . . . . 14 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℤ)
24	1	adantr 472	. . . . . . . . . . . . . . 15 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℕ)
25	24	nnzd 11665	. . . . . . . . . . . . . 14 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℤ)
26		simprl 811	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → 𝑛 ∈ ℕ0)
27	4, 26	nnexpcld 13216	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃↑𝑛) ∈ ℕ)
28	27	adantr 472	. . . . . . . . . . . . . . 15 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑃↑𝑛) ∈ ℕ)
29	28	nnzd 11665	. . . . . . . . . . . . . 14 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑃↑𝑛) ∈ ℤ)
30		dvdstr 15212	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝑃↑𝑛) ∈ ℤ) → ((𝑝 ∥ 𝐴 ∧ 𝐴 ∥ (𝑃↑𝑛)) → 𝑝 ∥ (𝑃↑𝑛)))
31	23, 25, 29, 30	syl3anc 1473	. . . . . . . . . . . . 13 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ 𝐴 ∧ 𝐴 ∥ (𝑃↑𝑛)) → 𝑝 ∥ (𝑃↑𝑛)))
32	21, 31	mpan2d 712	. . . . . . . . . . . 12 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ 𝐴 → 𝑝 ∥ (𝑃↑𝑛)))
33		simpr 479	. . . . . . . . . . . . 13 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
34	11	adantr 472	. . . . . . . . . . . . 13 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑃 ∈ ℙ)
35		simplrl 819	. . . . . . . . . . . . 13 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → 𝑛 ∈ ℕ0)
36		prmdvdsexpr 15623	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℙ ∧ 𝑃 ∈ ℙ ∧ 𝑛 ∈ ℕ0) → (𝑝 ∥ (𝑃↑𝑛) → 𝑝 = 𝑃))
37	33, 34, 35, 36	syl3anc 1473	. . . . . . . . . . . 12 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝑃↑𝑛) → 𝑝 = 𝑃))
38	32, 37	syld 47	. . . . . . . . . . 11 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ 𝐴 → 𝑝 = 𝑃))
39	38	necon3ad 2937	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝 ≠ 𝑃 → ¬ 𝑝 ∥ 𝐴))
40	39	imp 444	. . . . . . . . 9 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ≠ 𝑃) → ¬ 𝑝 ∥ 𝐴)
41		simplr 809	. . . . . . . . . 10 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ≠ 𝑃) → 𝑝 ∈ ℙ)
42	1	ad2antrr 764	. . . . . . . . . 10 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ≠ 𝑃) → 𝐴 ∈ ℕ)
43		pceq0 15769	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ((𝑝 pCnt 𝐴) = 0 ↔ ¬ 𝑝 ∥ 𝐴))
44	41, 42, 43	syl2anc 696	. . . . . . . . 9 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ≠ 𝑃) → ((𝑝 pCnt 𝐴) = 0 ↔ ¬ 𝑝 ∥ 𝐴))
45	40, 44	mpbird 247	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ≠ 𝑃) → (𝑝 pCnt 𝐴) = 0)
46	7	ad2antrr 764	. . . . . . . . . 10 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ≠ 𝑃) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
47	41, 46	pccld 15749	. . . . . . . . 9 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ≠ 𝑃) → (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))) ∈ ℕ0)
48	47	nn0ge0d 11538	. . . . . . . 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 4818	. . . . . . 7 ⊢ (((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) ∧ 𝑝 ≠ 𝑃) → (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
50	20, 49	pm2.61dane 3011	. . . . . 6 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
51	50	ralrimiva 3096	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴))))
52	1	nnzd 11665	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → 𝐴 ∈ ℤ)
53	7	nnzd 11665	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ)
54		pc2dvds 15777	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ) → (𝐴 ∥ (𝑃↑(𝑃 pCnt 𝐴)) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ (𝑝 pCnt (𝑃↑(𝑃 pCnt 𝐴)))))
55	52, 53, 54	syl2anc 696	. . . . 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)))))
56	51, 55	mpbird 247	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → 𝐴 ∥ (𝑃↑(𝑃 pCnt 𝐴)))
57		pcdvds 15762	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
58	57	adantr 472	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
59		dvdseq 15230	. . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ0) ∧ (𝐴 ∥ (𝑃↑(𝑃 pCnt 𝐴)) ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)) → 𝐴 = (𝑃↑(𝑃 pCnt 𝐴)))
60	2, 8, 56, 58, 59	syl22anc 1474	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) ∧ (𝑛 ∈ ℕ0 ∧ 𝐴 ∥ (𝑃↑𝑛))) → 𝐴 = (𝑃↑(𝑃 pCnt 𝐴)))
61	60	rexlimdvaa 3162	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃↑𝑛) → 𝐴 = (𝑃↑(𝑃 pCnt 𝐴))))
62	3	adantr 472	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → 𝑃 ∈ ℕ)
63	62, 5	nnexpcld 13216	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
64	63	nnzd 11665	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ)
65		iddvds 15189	. . . . 5 ⊢ ((𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ → (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑(𝑃 pCnt 𝐴)))
66	64, 65	syl 17	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑(𝑃 pCnt 𝐴)))
67		oveq2 6813	. . . . . 6 ⊢ (𝑛 = (𝑃 pCnt 𝐴) → (𝑃↑𝑛) = (𝑃↑(𝑃 pCnt 𝐴)))
68	67	breq2d 4808	. . . . 5 ⊢ (𝑛 = (𝑃 pCnt 𝐴) → ((𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑𝑛) ↔ (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑(𝑃 pCnt 𝐴))))
69	68	rspcev 3441	. . . 4 ⊢ (((𝑃 pCnt 𝐴) ∈ ℕ0 ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑(𝑃 pCnt 𝐴))) → ∃𝑛 ∈ ℕ0 (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑𝑛))
70	5, 66, 69	syl2anc 696	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ0 (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑𝑛))
71		breq1 4799	. . . 4 ⊢ (𝐴 = (𝑃↑(𝑃 pCnt 𝐴)) → (𝐴 ∥ (𝑃↑𝑛) ↔ (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑𝑛)))
72	71	rexbidv 3182	. . 3 ⊢ (𝐴 = (𝑃↑(𝑃 pCnt 𝐴)) → (∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃↑𝑛) ↔ ∃𝑛 ∈ ℕ0 (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝑃↑𝑛)))
73	70, 72	syl5ibrcom 237	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝐴 = (𝑃↑(𝑃 pCnt 𝐴)) → ∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃↑𝑛)))
74	61, 73	impbid 202	1 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (∃𝑛 ∈ ℕ0 𝐴 ∥ (𝑃↑𝑛) ↔ 𝐴 = (𝑃↑(𝑃 pCnt 𝐴))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1624   ∈ wcel 2131   ≠ wne 2924  ∀wral 3042  ∃wrex 3043   class class class wbr 4796  (class class class)co 6805  0cc0 10120   ≤ cle 10259  ℕcn 11204  ℕ₀cn0 11476  ℤcz 11561  ↑cexp 13046   ∥ cdvds 15174  ℙcprime 15579   pCnt cpc 15735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197  ax-pre-sup 10198

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-2o 7722  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8505  df-inf 8506  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-div 10869  df-nn 11205  df-2 11263  df-3 11264  df-n0 11477  df-z 11562  df-uz 11872  df-q 11974  df-rp 12018  df-fz 12512  df-fl 12779  df-mod 12855  df-seq 12988  df-exp 13047  df-cj 14030  df-re 14031  df-im 14032  df-sqrt 14166  df-abs 14167  df-dvds 15175  df-gcd 15411  df-prm 15580  df-pc 15736

This theorem is referenced by:  pcprmpw  15781  dvdsprmpweq  15782  pgpfi1  18202  pgpfi  18212  sylow2alem2  18225  lt6abl  18488  pgpfac1lem3a  18667  dvdsppwf1o  25103