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Theorem pclem 16856

Description: - Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.)

**Hypothesis**
Ref	Expression
pclem.1	⊢ 𝐴 = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑁}

**Assertion**
Ref	Expression
pclem	⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝑛,𝑦,𝑁 𝑃,𝑛,𝑥,𝑦 Allowed substitution hint: 𝐴(𝑛)

**Proof of Theorem pclem**
Step	Hyp	Ref	Expression
1		pclem.1	. . . . 5 ⊢ 𝐴 = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑁}
2	1	ssrab3 4057	. . . 4 ⊢ 𝐴 ⊆ ℕ₀
3		nn0ssz 12609	. . . 4 ⊢ ℕ₀ ⊆ ℤ
4	2, 3	sstri 3968	. . 3 ⊢ 𝐴 ⊆ ℤ
5	4	a1i 11	. 2 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝐴 ⊆ ℤ)
6		0nn0 12514	. . . . 5 ⊢ 0 ∈ ℕ₀
7	6	a1i 11	. . . 4 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 0 ∈ ℕ₀)
8		eluzelcn 12862	. . . . . . 7 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℂ)
9	8	adantr 480	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑃 ∈ ℂ)
10	9	exp0d 14156	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃↑0) = 1)
11		1dvds 16288	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 1 ∥ 𝑁)
12	11	ad2antrl 728	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 1 ∥ 𝑁)
13	10, 12	eqbrtrd 5141	. . . 4 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃↑0) ∥ 𝑁)
14		oveq2 7411	. . . . . 6 ⊢ (𝑛 = 0 → (𝑃↑𝑛) = (𝑃↑0))
15	14	breq1d 5129	. . . . 5 ⊢ (𝑛 = 0 → ((𝑃↑𝑛) ∥ 𝑁 ↔ (𝑃↑0) ∥ 𝑁))
16	15, 1	elrab2 3674	. . . 4 ⊢ (0 ∈ 𝐴 ↔ (0 ∈ ℕ₀ ∧ (𝑃↑0) ∥ 𝑁))
17	7, 13, 16	sylanbrc 583	. . 3 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 0 ∈ 𝐴)
18	17	ne0d 4317	. 2 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝐴 ≠ ∅)
19		nnssz 12608	. . 3 ⊢ ℕ ⊆ ℤ
20		zcn 12591	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
21	20	abscld 15453	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℝ)
22	21	ad2antrl 728	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (abs‘𝑁) ∈ ℝ)
23		eluzelre 12861	. . . . . 6 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℝ)
24	23	adantr 480	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑃 ∈ ℝ)
25		eluz2gt1 12934	. . . . . 6 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 1 < 𝑃)
26	25	adantr 480	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 1 < 𝑃)
27		expnbnd 14248	. . . . 5 ⊢ (((abs‘𝑁) ∈ ℝ ∧ 𝑃 ∈ ℝ ∧ 1 < 𝑃) → ∃𝑥 ∈ ℕ (abs‘𝑁) < (𝑃↑𝑥))
28	22, 24, 26, 27	syl3anc 1373	. . . 4 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℕ (abs‘𝑁) < (𝑃↑𝑥))
29		simprr 772	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
30		oveq2 7411	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑦 → (𝑃↑𝑛) = (𝑃↑𝑦))
31	30	breq1d 5129	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑦 → ((𝑃↑𝑛) ∥ 𝑁 ↔ (𝑃↑𝑦) ∥ 𝑁))
32	31, 1	elrab2 3674	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ ℕ₀ ∧ (𝑃↑𝑦) ∥ 𝑁))
33	29, 32	sylib 218	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑦 ∈ ℕ₀ ∧ (𝑃↑𝑦) ∥ 𝑁))
34	33	simprd 495	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∥ 𝑁)
35		eluz2nn 12896	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℕ)
36	35	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑃 ∈ ℕ)
37	33	simpld 494	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℕ₀)
38	36, 37	nnexpcld 14261	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℕ)
39	38	nnzd 12613	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℤ)
40		simplrl 776	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑁 ∈ ℤ)
41		simplrr 777	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑁 ≠ 0)
42		dvdsleabs 16328	. . . . . . . . . . . 12 ⊢ (((𝑃↑𝑦) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → ((𝑃↑𝑦) ∥ 𝑁 → (𝑃↑𝑦) ≤ (abs‘𝑁)))
43	39, 40, 41, 42	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((𝑃↑𝑦) ∥ 𝑁 → (𝑃↑𝑦) ≤ (abs‘𝑁)))
44	34, 43	mpd 15	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ≤ (abs‘𝑁))
45	38	nnred 12253	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℝ)
46	22	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (abs‘𝑁) ∈ ℝ)
47	23	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑃 ∈ ℝ)
48		nnnn0 12506	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ₀)
49	48	ad2antrl 728	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℕ₀)
50	47, 49	reexpcld 14179	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑥) ∈ ℝ)
51		lelttr 11323	. . . . . . . . . . 11 ⊢ (((𝑃↑𝑦) ∈ ℝ ∧ (abs‘𝑁) ∈ ℝ ∧ (𝑃↑𝑥) ∈ ℝ) → (((𝑃↑𝑦) ≤ (abs‘𝑁) ∧ (abs‘𝑁) < (𝑃↑𝑥)) → (𝑃↑𝑦) < (𝑃↑𝑥)))
52	45, 46, 50, 51	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (((𝑃↑𝑦) ≤ (abs‘𝑁) ∧ (abs‘𝑁) < (𝑃↑𝑥)) → (𝑃↑𝑦) < (𝑃↑𝑥)))
53	44, 52	mpand 695	. . . . . . . . 9 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((abs‘𝑁) < (𝑃↑𝑥) → (𝑃↑𝑦) < (𝑃↑𝑥)))
54	37	nn0zd 12612	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℤ)
55		nnz 12607	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℤ)
56	55	ad2antrl 728	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℤ)
57	25	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 1 < 𝑃)
58	47, 54, 56, 57	ltexp2d 14267	. . . . . . . . 9 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑦 < 𝑥 ↔ (𝑃↑𝑦) < (𝑃↑𝑥)))
59	53, 58	sylibrd 259	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((abs‘𝑁) < (𝑃↑𝑥) → 𝑦 < 𝑥))
60	37	nn0red 12561	. . . . . . . . 9 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℝ)
61		nnre 12245	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
62	61	ad2antrl 728	. . . . . . . . 9 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℝ)
63		ltle 11321	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 < 𝑥 → 𝑦 ≤ 𝑥))
64	60, 62, 63	syl2anc 584	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑦 < 𝑥 → 𝑦 ≤ 𝑥))
65	59, 64	syld 47	. . . . . . 7 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((abs‘𝑁) < (𝑃↑𝑥) → 𝑦 ≤ 𝑥))
66	65	anassrs 467	. . . . . 6 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ 𝐴) → ((abs‘𝑁) < (𝑃↑𝑥) → 𝑦 ≤ 𝑥))
67	66	ralrimdva 3140	. . . . 5 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℕ) → ((abs‘𝑁) < (𝑃↑𝑥) → ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
68	67	reximdva 3153	. . . 4 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (∃𝑥 ∈ ℕ (abs‘𝑁) < (𝑃↑𝑥) → ∃𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
69	28, 68	mpd 15	. . 3 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
70		ssrexv 4028	. . 3 ⊢ (ℕ ⊆ ℤ → (∃𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
71	19, 69, 70	mpsyl 68	. 2 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
72	5, 18, 71	3jca 1128	1 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 {crab 3415 ⊆ wss 3926 ∅c0 4308 class class class wbr 5119 ‘cfv 6530 (class class class)co 7403 ℂcc 11125 ℝcr 11126 0cc0 11127 1c1 11128 < clt 11267 ≤ cle 11268 ℕcn 12238 2c2 12293 ℕ₀cn0 12499 ℤcz 12586 ℤ_≥cuz 12850 ↑cexp 14077 abscabs 15251 ∥ cdvds 16270

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7860 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8717 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9452 df-inf 9453 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-n0 12500 df-z 12587 df-uz 12851 df-rp 13007 df-fl 13807 df-seq 14018 df-exp 14078 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-dvds 16271

This theorem is referenced by: pcprecl 16857 pcprendvds 16858 pcpremul 16861

W3C validator