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Theorem pntrsumbnd 26714

Description: A bound on a sum over 𝑅. Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 25-May-2016.)

**Hypothesis**
Ref	Expression
pntrval.r	⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))

**Assertion**
Ref	Expression
pntrsumbnd	⊢ ∃𝑐 ∈ ℝ⁺ ∀𝑚 ∈ ℤ (abs‘Σ𝑛 ∈ (1...𝑚)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐

Distinct variable groups: 𝑚,𝑎,𝑛 𝑚,𝑐,𝑛,𝑅 Allowed substitution hint: 𝑅(𝑎)

Proof of Theorem pntrsumbnd Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssidd 3944	. . . 4 ⊢ (⊤ → ℝ ⊆ ℝ)
2		1red 10976	. . . 4 ⊢ (⊤ → 1 ∈ ℝ)
3		fzfid 13693	. . . . 5 ⊢ ((⊤ ∧ 𝑚 ∈ ℝ) → (1...(⌊‘𝑚)) ∈ Fin)
4		elfznn 13285	. . . . . . 7 ⊢ (𝑛 ∈ (1...(⌊‘𝑚)) → 𝑛 ∈ ℕ)
5	4	adantl 482	. . . . . 6 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ 𝑛 ∈ (1...(⌊‘𝑚))) → 𝑛 ∈ ℕ)
6		nnrp 12741	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
7		pntrval.r	. . . . . . . . . . 11 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
8	7	pntrf 26711	. . . . . . . . . 10 ⊢ 𝑅:ℝ⁺⟶ℝ
9	8	ffvelrni 6960	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (𝑅‘𝑛) ∈ ℝ)
10	6, 9	syl 17	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑅‘𝑛) ∈ ℝ)
11		peano2nn 11985	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
12		nnmulcl 11997	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ ℕ) → (𝑛 · (𝑛 + 1)) ∈ ℕ)
13	11, 12	mpdan 684	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑛 · (𝑛 + 1)) ∈ ℕ)
14	10, 13	nndivred 12027	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℝ)
15	14	recnd 11003	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
16	5, 15	syl 17	. . . . 5 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ 𝑛 ∈ (1...(⌊‘𝑚))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
17	3, 16	fsumcl 15445	. . . 4 ⊢ ((⊤ ∧ 𝑚 ∈ ℝ) → Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
18	7	pntrsumo1 26713	. . . . 5 ⊢ (𝑚 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1)
19	18	a1i 11	. . . 4 ⊢ (⊤ → (𝑚 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1))
20		fzfid 13693	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
21		elfznn 13285	. . . . . . . 8 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
22	21	adantl 482	. . . . . . 7 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
23	22, 15	syl 17	. . . . . 6 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
24	23	abscld 15148	. . . . 5 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
25	20, 24	fsumrecl 15446	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
26	17	adantr 481	. . . . . 6 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
27	26	abscld 15148	. . . . 5 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
28		fzfid 13693	. . . . . 6 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → (1...(⌊‘𝑚)) ∈ Fin)
29	16	adantlr 712	. . . . . . 7 ⊢ ((((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑚))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
30	29	abscld 15148	. . . . . 6 ⊢ ((((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑚))) → (abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
31	28, 30	fsumrecl 15446	. . . . 5 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑚))(abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
32	25	ad2ant2r 744	. . . . 5 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
33	28, 29	fsumabs 15513	. . . . 5 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑚))(abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
34		fzfid 13693	. . . . . 6 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
35	21	adantl 482	. . . . . . . 8 ⊢ ((((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
36	35, 15	syl 17	. . . . . . 7 ⊢ ((((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
37	36	abscld 15148	. . . . . 6 ⊢ ((((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
38	36	absge0d 15156	. . . . . 6 ⊢ ((((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
39		simplr 766	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → 𝑚 ∈ ℝ)
40		simprll 776	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → 𝑥 ∈ ℝ)
41		simprr 770	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → 𝑚 < 𝑥)
42	39, 40, 41	ltled 11123	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → 𝑚 ≤ 𝑥)
43		flword2 13533	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑚 ≤ 𝑥) → (⌊‘𝑥) ∈ (ℤ_≥‘(⌊‘𝑚)))
44	39, 40, 42, 43	syl3anc 1370	. . . . . . 7 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → (⌊‘𝑥) ∈ (ℤ_≥‘(⌊‘𝑚)))
45		fzss2 13296	. . . . . . 7 ⊢ ((⌊‘𝑥) ∈ (ℤ_≥‘(⌊‘𝑚)) → (1...(⌊‘𝑚)) ⊆ (1...(⌊‘𝑥)))
46	44, 45	syl 17	. . . . . 6 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → (1...(⌊‘𝑚)) ⊆ (1...(⌊‘𝑥)))
47	34, 37, 38, 46	fsumless 15508	. . . . 5 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑚))(abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
48	27, 31, 32, 33, 47	letrd 11132	. . . 4 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
49	1, 2, 17, 19, 25, 48	o1bddrp 15251	. . 3 ⊢ (⊤ → ∃𝑐 ∈ ℝ⁺ ∀𝑚 ∈ ℝ (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐)
50	49	mptru 1546	. 2 ⊢ ∃𝑐 ∈ ℝ⁺ ∀𝑚 ∈ ℝ (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐
51		zre 12323	. . . . . 6 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ ℝ)
52	51	imim1i 63	. . . . 5 ⊢ ((𝑚 ∈ ℝ → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐) → (𝑚 ∈ ℤ → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐))
53		flid 13528	. . . . . . . . 9 ⊢ (𝑚 ∈ ℤ → (⌊‘𝑚) = 𝑚)
54	53	oveq2d 7291	. . . . . . . 8 ⊢ (𝑚 ∈ ℤ → (1...(⌊‘𝑚)) = (1...𝑚))
55	54	sumeq1d 15413	. . . . . . 7 ⊢ (𝑚 ∈ ℤ → Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) = Σ𝑛 ∈ (1...𝑚)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
56	55	fveq2d 6778	. . . . . 6 ⊢ (𝑚 ∈ ℤ → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) = (abs‘Σ𝑛 ∈ (1...𝑚)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
57	56	breq1d 5084	. . . . 5 ⊢ (𝑚 ∈ ℤ → ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐 ↔ (abs‘Σ𝑛 ∈ (1...𝑚)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐))
58	52, 57	mpbidi 240	. . . 4 ⊢ ((𝑚 ∈ ℝ → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐) → (𝑚 ∈ ℤ → (abs‘Σ𝑛 ∈ (1...𝑚)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐))
59	58	ralimi2 3084	. . 3 ⊢ (∀𝑚 ∈ ℝ (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐 → ∀𝑚 ∈ ℤ (abs‘Σ𝑛 ∈ (1...𝑚)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐)
60	59	reximi 3178	. 2 ⊢ (∃𝑐 ∈ ℝ⁺ ∀𝑚 ∈ ℝ (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐 → ∃𝑐 ∈ ℝ⁺ ∀𝑚 ∈ ℤ (abs‘Σ𝑛 ∈ (1...𝑚)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐)
61	50, 60	ax-mp 5	1 ⊢ ∃𝑐 ∈ ℝ⁺ ∀𝑚 ∈ ℤ (abs‘Σ𝑛 ∈ (1...𝑚)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ⊤wtru 1540 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 ℕcn 11973 ℤcz 12319 ℤ_≥cuz 12582 ℝ⁺crp 12730 ...cfz 13239 ⌊cfl 13510 abscabs 14945 𝑂(1)co1 15195 Σcsu 15397 ψcchp 26242

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-o1 15199 df-lo1 15200 df-sum 15398 df-ef 15777 df-e 15778 df-sin 15779 df-cos 15780 df-tan 15781 df-pi 15782 df-dvds 15964 df-gcd 16202 df-prm 16377 df-pc 16538 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-ulm 25536 df-log 25712 df-cxp 25713 df-atan 26017 df-em 26142 df-cht 26246 df-vma 26247 df-chp 26248 df-ppi 26249

This theorem is referenced by: pntrsumbnd2 26715

W3C validator