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Theorem pntleme 27546

Description: Lemma for pnt 27552. Package up pntlemo 27545 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 27534	. 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 2731	. . . 4 ⊢ ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)
26		eqid 2731	. . . 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 7353	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑘 · 𝑦) = (𝐾 · 𝑦))
30	29	breq2d 5101	. . . . . . . . . 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 3156	. . . . . . 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 3155	. . . . . 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 27533	. . . . . . . . 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 12935	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℝ^*)
39		pnfxr 11166	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
40	39	a1i 11	. . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ^*)
41	37	rpred 12934	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℝ)
42	41	ltpnfd 13020	. . . . . . 7 ⊢ (𝜑 → 𝐾 < +∞)
43		lbico1 13300	. . . . . . 7 ⊢ ((𝐾 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐾 < +∞) → 𝐾 ∈ (𝐾[,)+∞))
44	38, 40, 42, 43	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝐾[,)+∞))
45	34, 35, 44	rspcdva 3573	. . . . 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 27545	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → (abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))
50	49	ralrimiva 3124	. 2 ⊢ (𝜑 → ∀𝑣 ∈ (𝑊[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))
51		oveq1 7353	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑤[,)+∞) = (𝑊[,)+∞))
52	51	raleqdv 3292	. . 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 3572	. 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 1541 ∈ wcel 2111 ∀wral 3047 ∃wrex 3056 class class class wbr 5089 ↦ cmpt 5170 ‘cfv 6481 (class class class)co 7346 0cc0 11006 1c1 11007 + caddc 11009 · cmul 11011 +∞cpnf 11143 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 − cmin 11344 / cdiv 11774 2c2 12180 3c3 12181 4c4 12182 cdc 12588 ℝ⁺crp 12890 (,)cioo 13245 [,)cico 13247 [,]cicc 13248 ...cfz 13407 ⌊cfl 13694 ↑cexp 13968 abscabs 15141 Σcsu 15593 expce 15968 logclog 26490 ψcchp 27030

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-xnn0 12455 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ioc 13250 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-e 15975 df-sin 15976 df-cos 15977 df-tan 15978 df-pi 15979 df-dvds 16164 df-gcd 16406 df-prm 16583 df-pc 16749 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-fbas 21288 df-fg 21289 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-nei 23013 df-lp 23051 df-perf 23052 df-cn 23142 df-cnp 23143 df-haus 23230 df-cmp 23302 df-tx 23477 df-hmeo 23670 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798 df-limc 25794 df-dv 25795 df-ulm 26313 df-log 26492 df-atan 26804 df-em 26930 df-vma 27035 df-chp 27036

This theorem is referenced by: pntlemp 27548

W3C validator