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Theorem pntrsumbnd 27950

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 3936	. . . 4 ⊢ (⊤ → ℝ ⊆ ℝ)
2		1red 11348	. . . 4 ⊢ (⊤ → 1 ∈ ℝ)
3		fzfid 14156	. . . . 5 ⊢ ((⊤ ∧ 𝑚 ∈ ℝ) → (1...(⌊‘𝑚)) ∈ Fin)
4		elfznn 13727	. . . . . . 7 ⊢ (𝑛 ∈ (1...(⌊‘𝑚)) → 𝑛 ∈ ℕ)
5	4	adantl 487	. . . . . 6 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ 𝑛 ∈ (1...(⌊‘𝑚))) → 𝑛 ∈ ℕ)
6		nnrp 13172	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
7		pntrval.r	. . . . . . . . . . 11 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
8	7	pntrf 27947	. . . . . . . . . 10 ⊢ 𝑅:ℝ⁺⟶ℝ
9	8	ffvelcdmi 7115	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (𝑅‘𝑛) ∈ ℝ)
10	6, 9	syl 18	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑅‘𝑛) ∈ ℝ)
11		peano2nn 12387	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
12		nnmulcl 12399	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ ℕ) → (𝑛 · (𝑛 + 1)) ∈ ℕ)
13	11, 12	mpdan 700	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑛 · (𝑛 + 1)) ∈ ℕ)
14	10, 13	nndivred 12432	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℝ)
15	14	recnd 11376	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
16	5, 15	syl 18	. . . . 5 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ 𝑛 ∈ (1...(⌊‘𝑚))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
17	3, 16	fsumcl 15943	. . . 4 ⊢ ((⊤ ∧ 𝑚 ∈ ℝ) → Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
18	7	pntrsumo1 27949	. . . . 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 14156	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
21		elfznn 13727	. . . . . . . 8 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
22	21	adantl 487	. . . . . . 7 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
23	22, 15	syl 18	. . . . . 6 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
24	23	abscld 15650	. . . . 5 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
25	20, 24	fsumrecl 15944	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
26	17	adantr 486	. . . . . 6 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
27	26	abscld 15650	. . . . 5 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
28		fzfid 14156	. . . . . 6 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → (1...(⌊‘𝑚)) ∈ Fin)
29	16	adantlr 728	. . . . . . 7 ⊢ ((((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑚))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
30	29	abscld 15650	. . . . . 6 ⊢ ((((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑚))) → (abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
31	28, 30	fsumrecl 15944	. . . . 5 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑚))(abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
32	25	ad2ant2r 760	. . . . 5 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
33	28, 29	fsumabs 16012	. . . . 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 14156	. . . . . 6 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
35	21	adantl 487	. . . . . . . 8 ⊢ ((((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
36	35, 15	syl 18	. . . . . . 7 ⊢ ((((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
37	36	abscld 15650	. . . . . 6 ⊢ ((((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ ℝ)
38	36	absge0d 15658	. . . . . 6 ⊢ ((((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
39		simplr 781	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → 𝑚 ∈ ℝ)
40		simprll 791	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → 𝑥 ∈ ℝ)
41		simprr 785	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → 𝑚 < 𝑥)
42	39, 40, 41	ltled 11497	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → 𝑚 ≤ 𝑥)
43		flword2 13993	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑚 ≤ 𝑥) → (⌊‘𝑥) ∈ (ℤ_≥‘(⌊‘𝑚)))
44	39, 40, 42, 43	syl3anc 1399	. . . . . . 7 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → (⌊‘𝑥) ∈ (ℤ_≥‘(⌊‘𝑚)))
45		fzss2 13738	. . . . . . 7 ⊢ ((⌊‘𝑥) ∈ (ℤ_≥‘(⌊‘𝑚)) → (1...(⌊‘𝑚)) ⊆ (1...(⌊‘𝑥)))
46	44, 45	syl 18	. . . . . 6 ⊢ (((⊤ ∧ 𝑚 ∈ ℝ) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑚 < 𝑥)) → (1...(⌊‘𝑚)) ⊆ (1...(⌊‘𝑥)))
47	34, 37, 38, 46	fsumless 16007	. . . . 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 11506	. . . 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 15753	. . 3 ⊢ (⊤ → ∃𝑐 ∈ ℝ⁺ ∀𝑚 ∈ ℝ (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐)
50	49	mptru 1577	. 2 ⊢ ∃𝑐 ∈ ℝ⁺ ∀𝑚 ∈ ℝ (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑐
51		zre 12737	. . . . . 6 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ ℝ)
52	51	imim1i 64	. . . . 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 13988	. . . . . . . . 9 ⊢ (𝑚 ∈ ℤ → (⌊‘𝑚) = 𝑚)
54	53	oveq2d 7467	. . . . . . . 8 ⊢ (𝑚 ∈ ℤ → (1...(⌊‘𝑚)) = (1...𝑚))
55	54	sumeq1d 15911	. . . . . . 7 ⊢ (𝑚 ∈ ℤ → Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) = Σ𝑛 ∈ (1...𝑚)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
56	55	fveq2d 6920	. . . . . 6 ⊢ (𝑚 ∈ ℤ → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑚))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) = (abs‘Σ𝑛 ∈ (1...𝑚)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
57	56	breq1d 5118	. . . . 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 244	. . . 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 3098	. . 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 3104	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ⊤wtru 1571 ∈ wcel 2147 ∀wral 3080 ∃wrex 3090 ⊆ wss 3900 class class class wbr 5108 ↦ cmpt 5192 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 1c1 11240 + caddc 11242 · cmul 11244 < clt 11382 ≤ cle 11383 − cmin 11580 / cdiv 12013 ℕcn 12375 ℤcz 12733 ℤ_≥cuz 13005 ℝ⁺crp 13160 ...cfz 13679 ⌊cfl 13970 abscabs 15444 𝑂(1)co1 15697 Σcsu 15897 ψcchp 27477

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-oadd 8518 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-dju 10019 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-xnn0 12720 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ioc 13521 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-fac 14459 df-bc 14488 df-hash 14516 df-shft 15263 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-limsup 15682 df-clim 15699 df-rlim 15700 df-o1 15701 df-lo1 15702 df-sum 15898 df-ef 16277 df-e 16278 df-sin 16279 df-cos 16280 df-tan 16281 df-pi 16282 df-dvds 16467 df-gcd 16709 df-prm 16891 df-pc 17059 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-mulg 19328 df-cntz 19582 df-cmn 20047 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-haus 23688 df-cmp 23760 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-limc 26241 df-dv 26242 df-ulm 26761 df-log 26941 df-cxp 26942 df-atan 27252 df-em 27377 df-cht 27481 df-vma 27482 df-chp 27483 df-ppi 27484

This theorem is used by: pntrsumbnd2 27951

W3C validator