pntleme - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pntleme		Structured version Visualization version GIF version

Theorem pntleme 27495

Description: Lemma for pnt 27501. Package up pntlemo 27494 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 27483	. 2 ⊢ (𝜑 → 𝑊 ∈ ℝ⁺)
16	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝐴 ∈ ℝ⁺)
17	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝐵 ∈ ℝ⁺)
18	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝐿 ∈ (0(,)1))
19	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝑈 ∈ ℝ⁺)
20	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝑈 ≤ 𝐴)
21	11	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → (𝑌 ∈ ℝ⁺ ∧ 1 ≤ 𝑌))
22	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → (𝑋 ∈ ℝ⁺ ∧ 𝑌 < 𝑋))
23	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝐶 ∈ ℝ⁺)
24		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝑣 ∈ (𝑊[,)+∞))
25		eqid 2729	. . . 4 ⊢ ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)
26		eqid 2729	. . . 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈)
29		oveq1 7376	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑘 · 𝑦) = (𝐾 · 𝑦))
30	29	breq2d 5114	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦) ↔ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)))
31	30	anbi2d 630	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦)) ↔ (𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦))))
32	31	anbi1d 631	. . . . . . . 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 3157	. . . . . . 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 3156	. . . . . 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 27482	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ∈ ℝ⁺ ∧ 𝐾 ∈ ℝ⁺ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺)))
37	36	simp2d 1143	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
38	37	rpxrd 12972	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℝ^*)
39		pnfxr 11204	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
40	39	a1i 11	. . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ^*)
41	37	rpred 12971	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℝ)
42	41	ltpnfd 13057	. . . . . . 7 ⊢ (𝜑 → 𝐾 < +∞)
43		lbico1 13337	. . . . . . 7 ⊢ ((𝐾 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐾 < +∞) → 𝐾 ∈ (𝐾[,)+∞))
44	38, 40, 42, 43	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝐾[,)+∞))
45	34, 35, 44	rspcdva 3586	. . . . 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 480	. . . 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 480	. . . 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 27494	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → (abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))
50	49	ralrimiva 3125	. 2 ⊢ (𝜑 → ∀𝑣 ∈ (𝑊[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))
51		oveq1 7376	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑤[,)+∞) = (𝑊[,)+∞))
52	51	raleqdv 3296	. . 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 3585	. 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 584	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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 class class class wbr 5102 ↦ cmpt 5183 ‘cfv 6499 (class class class)co 7369 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 +∞cpnf 11181 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 − cmin 11381 / cdiv 11811 2c2 12217 3c3 12218 4c4 12219 cdc 12625 ℝ⁺crp 12927 (,)cioo 13282 [,)cico 13284 [,]cicc 13285 ...cfz 13444 ⌊cfl 13728 ↑cexp 14002 abscabs 15176 Σcsu 15628 expce 16003 logclog 26439 ψcchp 26979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15629 df-ef 16009 df-e 16010 df-sin 16011 df-cos 16012 df-tan 16013 df-pi 16014 df-dvds 16199 df-gcd 16441 df-prm 16618 df-pc 16784 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-lp 22999 df-perf 23000 df-cn 23090 df-cnp 23091 df-haus 23178 df-cmp 23250 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cncf 24747 df-limc 25743 df-dv 25744 df-ulm 26262 df-log 26441 df-atan 26753 df-em 26879 df-vma 26984 df-chp 26985

This theorem is referenced by: pntlemp 27497

W3C validator