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Theorem pntleme 27737

Description: Lemma for pnt 27743. Package up pntlemo 27736 in quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)

**Hypotheses**
Ref	Expression
pntlem1.r	⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
pntlem1.a	⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
pntlem1.b	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
pntlem1.l	⊢ (𝜑 → 𝐿 ∈ (0(,)1))
pntlem1.d	⊢ 𝐷 = (𝐴 + 1)
pntlem1.f	⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)))
pntlem1.u	⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
pntlem1.u2	⊢ (𝜑 → 𝑈 ≤ 𝐴)
pntlem1.e	⊢ 𝐸 = (𝑈 / 𝐷)
pntlem1.k	⊢ 𝐾 = (exp‘(𝐵 / 𝐸))
pntlem1.y	⊢ (𝜑 → (𝑌 ∈ ℝ⁺ ∧ 1 ≤ 𝑌))
pntlem1.x	⊢ (𝜑 → (𝑋 ∈ ℝ⁺ ∧ 𝑌 < 𝑋))
pntlem1.c	⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
pntlem1.w	⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
pntleme.U	⊢ (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈)
pntleme.K	⊢ (𝜑 → ∀𝑘 ∈ (𝐾[,)+∞)∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ⁺ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸))
pntleme.C	⊢ (𝜑 → ∀𝑧 ∈ (1(,)+∞)((((abs‘(𝑅‘𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) / 𝑧) ≤ 𝐶)

**Assertion**
Ref	Expression
pntleme	⊢ (𝜑 → ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))

Distinct variable groups: 𝑧,𝐶 𝑤,𝐹 𝑦,𝑧 𝑢,𝑘,𝑦,𝑧,𝐿 𝑘,𝐾,𝑦,𝑧 𝜑,𝑣 𝑖,𝑘,𝑢,𝑣,𝑤,𝑦,𝑧,𝑅 𝑤,𝑈,𝑧 𝑣,𝑊,𝑤,𝑧 𝑘,𝑋,𝑦,𝑧 𝑖,𝑌,𝑧 𝑘,𝑎,𝑢,𝑣,𝑦,𝑧,𝐸 Allowed substitution hints: 𝜑(𝑦,𝑧,𝑤,𝑢,𝑖,𝑘,𝑎) 𝐴(𝑦,𝑧,𝑤,𝑣,𝑢,𝑖,𝑘,𝑎) 𝐵(𝑦,𝑧,𝑤,𝑣,𝑢,𝑖,𝑘,𝑎) 𝐶(𝑦,𝑤,𝑣,𝑢,𝑖,𝑘,𝑎) 𝐷(𝑦,𝑧,𝑤,𝑣,𝑢,𝑖,𝑘,𝑎) 𝑅(𝑎) 𝑈(𝑦,𝑣,𝑢,𝑖,𝑘,𝑎) 𝐸(𝑤,𝑖) 𝐹(𝑦,𝑧,𝑣,𝑢,𝑖,𝑘,𝑎) 𝐾(𝑤,𝑣,𝑢,𝑖,𝑎) 𝐿(𝑤,𝑣,𝑖,𝑎) 𝑊(𝑦,𝑢,𝑖,𝑘,𝑎) 𝑋(𝑤,𝑣,𝑢,𝑖,𝑎) 𝑌(𝑦,𝑤,𝑣,𝑢,𝑘,𝑎)

**Proof of Theorem pntleme**
Step	Hyp	Ref	Expression
1		pntlem1.r	. . 3 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
2		pntlem1.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
3		pntlem1.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
4		pntlem1.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ (0(,)1))
5		pntlem1.d	. . 3 ⊢ 𝐷 = (𝐴 + 1)
6		pntlem1.f	. . 3 ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)))
7		pntlem1.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
8		pntlem1.u2	. . 3 ⊢ (𝜑 → 𝑈 ≤ 𝐴)
9		pntlem1.e	. . 3 ⊢ 𝐸 = (𝑈 / 𝐷)
10		pntlem1.k	. . 3 ⊢ 𝐾 = (exp‘(𝐵 / 𝐸))
11		pntlem1.y	. . 3 ⊢ (𝜑 → (𝑌 ∈ ℝ⁺ ∧ 1 ≤ 𝑌))
12		pntlem1.x	. . 3 ⊢ (𝜑 → (𝑋 ∈ ℝ⁺ ∧ 𝑌 < 𝑋))
13		pntlem1.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
14		pntlem1.w	. . 3 ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	pntlema 27725	. 2 ⊢ (𝜑 → 𝑊 ∈ ℝ⁺)
16	2	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝐴 ∈ ℝ⁺)
17	3	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝐵 ∈ ℝ⁺)
18	4	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝐿 ∈ (0(,)1))
19	7	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝑈 ∈ ℝ⁺)
20	8	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝑈 ≤ 𝐴)
21	11	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → (𝑌 ∈ ℝ⁺ ∧ 1 ≤ 𝑌))
22	12	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → (𝑋 ∈ ℝ⁺ ∧ 𝑌 < 𝑋))
23	13	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝐶 ∈ ℝ⁺)
24		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝑣 ∈ (𝑊[,)+∞))
25		eqid 2769	. . . 4 ⊢ ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)
26		eqid 2769	. . . 4 ⊢ (⌊‘(((log‘𝑣) / (log‘𝐾)) / 2)) = (⌊‘(((log‘𝑣) / (log‘𝐾)) / 2))
27		pntleme.U	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈)
28	27	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈)
29		oveq1 7418	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑘 · 𝑦) = (𝐾 · 𝑦))
30	29	breq2d 5125	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦) ↔ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)))
31	30	anbi2d 641	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦)) ↔ (𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦))))
32	31	anbi1d 642	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸) ↔ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸)))
33	32	rexbidv 3195	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (∃𝑧 ∈ ℝ⁺ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸) ↔ ∃𝑧 ∈ ℝ⁺ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸)))
34	33	ralbidv 3194	. . . . . 6 ⊢ (𝑘 = 𝐾 → (∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ⁺ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸) ↔ ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ⁺ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸)))
35		pntleme.K	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ (𝐾[,)+∞)∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ⁺ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸))
36	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	pntlemc 27724	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ∈ ℝ⁺ ∧ 𝐾 ∈ ℝ⁺ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺)))
37	36	simp2d 1159	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
38	37	rpxrd 13060	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℝ^*)
39		pnfxr 11262	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
40	39	a1i 11	. . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ^*)
41	37	rpred 13059	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℝ)
42	41	ltpnfd 13145	. . . . . . 7 ⊢ (𝜑 → 𝐾 < +∞)
43		lbico1 13426	. . . . . . 7 ⊢ ((𝐾 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐾 < +∞) → 𝐾 ∈ (𝐾[,)+∞))
44	38, 40, 42, 43	syl3anc 1396	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝐾[,)+∞))
45	34, 35, 44	rspcdva 3591	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ⁺ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸))
46	45	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ⁺ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸))
47		pntleme.C	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ (1(,)+∞)((((abs‘(𝑅‘𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) / 𝑧) ≤ 𝐶)
48	47	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → ∀𝑧 ∈ (1(,)+∞)((((abs‘(𝑅‘𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) / 𝑧) ≤ 𝐶)
49	1, 16, 17, 18, 5, 6, 19, 20, 9, 10, 21, 22, 23, 14, 24, 25, 26, 28, 46, 48	pntlemo 27736	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → (abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))
50	49	ralrimiva 3163	. 2 ⊢ (𝜑 → ∀𝑣 ∈ (𝑊[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))
51		oveq1 7418	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑤[,)+∞) = (𝑊[,)+∞))
52	51	raleqdv 3329	. . 3 ⊢ (𝑤 = 𝑊 → (∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3))) ↔ ∀𝑣 ∈ (𝑊[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3)))))
53	52	rspcev 3590	. 2 ⊢ ((𝑊 ∈ ℝ⁺ ∧ ∀𝑣 ∈ (𝑊[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3)))) → ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))
54	15, 50, 53	syl2anc 595	1 ⊢ (𝜑 → ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∀wral 3085 ∃wrex 3095 class class class wbr 5113 ↦ cmpt 5196 ‘cfv 6537 (class class class)co 7411 0cc0 11099 1c1 11100 + caddc 11102 · cmul 11104 +∞cpnf 11239 ℝ^*cxr 11241 < clt 11242 ≤ cle 11243 − cmin 11440 / cdiv 11870 2c2 12294 3c3 12295 4c4 12296 cdc 12710 ℝ⁺crp 13015 (,)cioo 13371 [,)cico 13373 [,]cicc 13374 ...cfz 13534 ⌊cfl 13822 ↑cexp 14096 abscabs 15284 Σcsu 15736 expce 16114 logclog 26684 ψcchp 27222

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9609 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-oadd 8456 df-er 8693 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-fi 9370 df-sup 9401 df-inf 9402 df-oi 9471 df-dju 9886 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-xnn0 12577 df-z 12591 df-dec 12711 df-uz 12862 df-q 12972 df-rp 13016 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-ioo 13375 df-ioc 13376 df-ico 13377 df-icc 13378 df-fz 13535 df-fzo 13682 df-fl 13824 df-mod 13902 df-seq 14037 df-exp 14097 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15103 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-limsup 15521 df-clim 15538 df-rlim 15539 df-sum 15737 df-ef 16120 df-e 16121 df-sin 16122 df-cos 16123 df-tan 16124 df-pi 16125 df-dvds 16310 df-gcd 16552 df-prm 16729 df-pc 16896 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-hom 17333 df-cco 17334 df-rest 17474 df-topn 17475 df-0g 17493 df-gsum 17494 df-topgen 17495 df-pt 17496 df-prds 17499 df-xrs 17555 df-qtop 17560 df-imas 17561 df-xps 17563 df-mre 17637 df-mrc 17638 df-acs 17640 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-submnd 18841 df-mulg 19133 df-cntz 19386 df-cmn 19851 df-psmet 21482 df-xmet 21483 df-met 21484 df-bl 21485 df-mopn 21486 df-fbas 21487 df-fg 21488 df-cnfld 21491 df-top 23019 df-topon 23036 df-topsp 23058 df-bases 23071 df-cld 23144 df-ntr 23145 df-cls 23146 df-nei 23223 df-lp 23261 df-perf 23262 df-cn 23352 df-cnp 23353 df-haus 23440 df-cmp 23512 df-tx 23687 df-hmeo 23880 df-fil 23971 df-fm 24063 df-flim 24064 df-flf 24065 df-xms 24445 df-ms 24446 df-tms 24447 df-cncf 25005 df-limc 25993 df-dv 25994 df-ulm 26505 df-log 26686 df-atan 26997 df-em 27122 df-vma 27227 df-chp 27228

This theorem is referenced by: pntlemp 27739

W3C validator