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Theorem pclem 16710

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 4040	. . . 4 ⊢ 𝐴 ⊆ ℕ₀
3		nn0ssz 12522	. . . 4 ⊢ ℕ₀ ⊆ ℤ
4	2, 3	sstri 3953	. . 3 ⊢ 𝐴 ⊆ ℤ
5	4	a1i 11	. 2 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝐴 ⊆ ℤ)
6		0nn0 12428	. . . . 5 ⊢ 0 ∈ ℕ₀
7	6	a1i 11	. . . 4 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 0 ∈ ℕ₀)
8		eluzelcn 12775	. . . . . . 7 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℂ)
9	8	adantr 481	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑃 ∈ ℂ)
10	9	exp0d 14045	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃↑0) = 1)
11		1dvds 16153	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 1 ∥ 𝑁)
12	11	ad2antrl 726	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 1 ∥ 𝑁)
13	10, 12	eqbrtrd 5127	. . . 4 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃↑0) ∥ 𝑁)
14		oveq2 7365	. . . . . 6 ⊢ (𝑛 = 0 → (𝑃↑𝑛) = (𝑃↑0))
15	14	breq1d 5115	. . . . 5 ⊢ (𝑛 = 0 → ((𝑃↑𝑛) ∥ 𝑁 ↔ (𝑃↑0) ∥ 𝑁))
16	15, 1	elrab2 3648	. . . 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 4295	. 2 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝐴 ≠ ∅)
19		nnssz 12521	. . 3 ⊢ ℕ ⊆ ℤ
20		zcn 12504	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
21	20	abscld 15321	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℝ)
22	21	ad2antrl 726	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (abs‘𝑁) ∈ ℝ)
23		eluzelre 12774	. . . . . 6 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℝ)
24	23	adantr 481	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑃 ∈ ℝ)
25		eluz2gt1 12845	. . . . . 6 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 1 < 𝑃)
26	25	adantr 481	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 1 < 𝑃)
27		expnbnd 14135	. . . . 5 ⊢ (((abs‘𝑁) ∈ ℝ ∧ 𝑃 ∈ ℝ ∧ 1 < 𝑃) → ∃𝑥 ∈ ℕ (abs‘𝑁) < (𝑃↑𝑥))
28	22, 24, 26, 27	syl3anc 1371	. . . 4 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℕ (abs‘𝑁) < (𝑃↑𝑥))
29		simprr 771	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
30		oveq2 7365	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑦 → (𝑃↑𝑛) = (𝑃↑𝑦))
31	30	breq1d 5115	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑦 → ((𝑃↑𝑛) ∥ 𝑁 ↔ (𝑃↑𝑦) ∥ 𝑁))
32	31, 1	elrab2 3648	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ ℕ₀ ∧ (𝑃↑𝑦) ∥ 𝑁))
33	29, 32	sylib 217	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑦 ∈ ℕ₀ ∧ (𝑃↑𝑦) ∥ 𝑁))
34	33	simprd 496	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∥ 𝑁)
35		eluz2nn 12809	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℕ)
36	35	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑃 ∈ ℕ)
37	33	simpld 495	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℕ₀)
38	36, 37	nnexpcld 14148	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℕ)
39	38	nnzd 12526	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℤ)
40		simplrl 775	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑁 ∈ ℤ)
41		simplrr 776	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑁 ≠ 0)
42		dvdsleabs 16193	. . . . . . . . . . . 12 ⊢ (((𝑃↑𝑦) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → ((𝑃↑𝑦) ∥ 𝑁 → (𝑃↑𝑦) ≤ (abs‘𝑁)))
43	39, 40, 41, 42	syl3anc 1371	. . . . . . . . . . 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 12168	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℝ)
46	22	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (abs‘𝑁) ∈ ℝ)
47	23	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑃 ∈ ℝ)
48		nnnn0 12420	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ₀)
49	48	ad2antrl 726	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℕ₀)
50	47, 49	reexpcld 14068	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑥) ∈ ℝ)
51		lelttr 11245	. . . . . . . . . . 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 1371	. . . . . . . . . 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 693	. . . . . . . . 9 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((abs‘𝑁) < (𝑃↑𝑥) → (𝑃↑𝑦) < (𝑃↑𝑥)))
54	37	nn0zd 12525	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℤ)
55		nnz 12520	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℤ)
56	55	ad2antrl 726	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℤ)
57	25	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 1 < 𝑃)
58	47, 54, 56, 57	ltexp2d 14154	. . . . . . . . 9 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑦 < 𝑥 ↔ (𝑃↑𝑦) < (𝑃↑𝑥)))
59	53, 58	sylibrd 258	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((abs‘𝑁) < (𝑃↑𝑥) → 𝑦 < 𝑥))
60	37	nn0red 12474	. . . . . . . . 9 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℝ)
61		nnre 12160	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
62	61	ad2antrl 726	. . . . . . . . 9 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℝ)
63		ltle 11243	. . . . . . . . 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 468	. . . . . 6 ⊢ ((((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ 𝐴) → ((abs‘𝑁) < (𝑃↑𝑥) → 𝑦 ≤ 𝑥))
67	66	ralrimdva 3151	. . . . 5 ⊢ (((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℕ) → ((abs‘𝑁) < (𝑃↑𝑥) → ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
68	67	reximdva 3165	. . . 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 4011	. . 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 {crab 3407 ⊆ wss 3910 ∅c0 4282 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 < clt 11189 ≤ cle 11190 ℕcn 12153 2c2 12208 ℕ₀cn0 12413 ℤcz 12499 ℤ_≥cuz 12763 ↑cexp 13967 abscabs 15119 ∥ cdvds 16136

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-fl 13697 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-dvds 16137

This theorem is referenced by: pcprecl 16711 pcprendvds 16712 pcpremul 16715

W3C validator