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Theorem pntrsumbnd 27625

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 4019	. . . 4 ⊢ (⊤ → ℝ ⊆ ℝ)
2		1red 11260	. . . 4 ⊢ (⊤ → 1 ∈ ℝ)
3		fzfid 14011	. . . . 5 ⊢ ((⊤ ∧ 𝑚 ∈ ℝ) → (1...(⌊‘𝑚)) ∈ Fin)
4		elfznn 13590	. . . . . . 7 ⊢ (𝑛 ∈ (1...(⌊‘𝑚)) → 𝑛 ∈ ℕ)
5	4	adantl 481	. . . . . 6 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ 𝑛 ∈ (1...(⌊‘𝑚))) → 𝑛 ∈ ℕ)
6		nnrp 13044	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
7		pntrval.r	. . . . . . . . . . 11 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
8	7	pntrf 27622	. . . . . . . . . 10 ⊢ 𝑅:ℝ⁺⟶ℝ
9	8	ffvelcdmi 7103	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (𝑅‘𝑛) ∈ ℝ)
10	6, 9	syl 17	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑅‘𝑛) ∈ ℝ)
11		peano2nn 12276	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
12		nnmulcl 12288	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ ℕ) → (𝑛 · (𝑛 + 1)) ∈ ℕ)
13	11, 12	mpdan 687	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑛 · (𝑛 + 1)) ∈ ℕ)
14	10, 13	nndivred 12318	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℝ)
15	14	recnd 11287	. . . . . 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 15766	. . . 4 ⊢ ((⊤ ∧ 𝑚 ∈ ℝ) → Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
18	7	pntrsumo1 27624	. . . . 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 14011	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
21		elfznn 13590	. . . . . . . 8 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
22	21	adantl 481	. . . . . . 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 15472	. . . . 5 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
25	20, 24	fsumrecl 15767	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
26	17	adantr 480	. . . . . 6 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
27	26	abscld 15472	. . . . 5 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
28		fzfid 14011	. . . . . 6 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → (1...(⌊‘𝑚)) ∈ Fin)
29	16	adantlr 715	. . . . . . 7 ⊢ ((((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑚))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
30	29	abscld 15472	. . . . . 6 ⊢ ((((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑚))) → (abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
31	28, 30	fsumrecl 15767	. . . . 5 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑚))(abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
32	25	ad2ant2r 747	. . . . 5 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
33	28, 29	fsumabs 15834	. . . . 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 14011	. . . . . 6 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
35	21	adantl 481	. . . . . . . 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 15472	. . . . . 6 ⊢ ((((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
38	36	absge0d 15480	. . . . . 6 ⊢ ((((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
39		simplr 769	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → 𝑚 ∈ ℝ)
40		simprll 779	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → 𝑥 ∈ ℝ)
41		simprr 773	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → 𝑚 < 𝑥)
42	39, 40, 41	ltled 11407	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → 𝑚 ≤ 𝑥)
43		flword2 13850	. . . . . . . 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 13601	. . . . . . 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 15829	. . . . 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 11416	. . . 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 15575	. . 3 ⊢ (⊤ → ∃𝑐 ∈ ℝ⁺ ∀𝑚 ∈ ℝ (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐)
50	49	mptru 1544	. 2 ⊢ ∃𝑐 ∈ ℝ⁺ ∀𝑚 ∈ ℝ (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐
51		zre 12615	. . . . . 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 13845	. . . . . . . . 9 ⊢ (𝑚 ∈ ℤ → (⌊‘𝑚) = 𝑚)
54	53	oveq2d 7447	. . . . . . . 8 ⊢ (𝑚 ∈ ℤ → (1...(⌊‘𝑚)) = (1...𝑚))
55	54	sumeq1d 15733	. . . . . . 7 ⊢ (𝑚 ∈ ℤ → Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) = Σ𝑛 ∈ (1...𝑚)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
56	55	fveq2d 6911	. . . . . 6 ⊢ (𝑚 ∈ ℤ → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) = (abs‘Σ𝑛 ∈ (1...𝑚)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
57	56	breq1d 5158	. . . . 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 241	. . . 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 3076	. . 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 3082	. 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 395 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 1c1 11154 + caddc 11156 · cmul 11158 < clt 11293 ≤ cle 11294 − cmin 11490 / cdiv 11918 ℕcn 12264 ℤcz 12611 ℤ_≥cuz 12876 ℝ⁺crp 13032 ...cfz 13544 ⌊cfl 13827 abscabs 15270 𝑂(1)co1 15519 Σcsu 15719 ψcchp 27151

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-o1 15523 df-lo1 15524 df-sum 15720 df-ef 16100 df-e 16101 df-sin 16102 df-cos 16103 df-tan 16104 df-pi 16105 df-dvds 16288 df-gcd 16529 df-prm 16706 df-pc 16871 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-ulm 26435 df-log 26613 df-cxp 26614 df-atan 26925 df-em 27051 df-cht 27155 df-vma 27156 df-chp 27157 df-ppi 27158

This theorem is referenced by: pntrsumbnd2 27626

W3C validator