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Mirrors > Home > MPE Home > Th. List > pntleme | Structured version Visualization version GIF version |
Description: Lemma for pnt 26519. Package up pntlemo 26512 in quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.) |
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) |
Ref | Expression |
---|---|
pntleme | ⊢ (𝜑 → ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3)))) |
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 26501 | . 2 ⊢ (𝜑 → 𝑊 ∈ ℝ+) |
16 | 2 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝐴 ∈ ℝ+) |
17 | 3 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝐵 ∈ ℝ+) |
18 | 4 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝐿 ∈ (0(,)1)) |
19 | 7 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝑈 ∈ ℝ+) |
20 | 8 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝑈 ≤ 𝐴) |
21 | 11 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) |
22 | 12 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) |
23 | 13 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝐶 ∈ ℝ+) |
24 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝑣 ∈ (𝑊[,)+∞)) | |
25 | eqid 2738 | . . . 4 ⊢ ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) | |
26 | eqid 2738 | . . . 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 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈) |
29 | oveq1 7238 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑘 · 𝑦) = (𝐾 · 𝑦)) | |
30 | 29 | breq2d 5079 | . . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦) ↔ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦))) |
31 | 30 | anbi2d 632 | . . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦)) ↔ (𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)))) |
32 | 31 | anbi1d 633 | . . . . . . . 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 3224 | . . . . . . 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 3119 | . . . . . 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 26500 | . . . . . . . . 9 ⊢ (𝜑 → (𝐸 ∈ ℝ+ ∧ 𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+))) |
37 | 36 | simp2d 1145 | . . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℝ+) |
38 | 37 | rpxrd 12653 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℝ*) |
39 | pnfxr 10911 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
40 | 39 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ*) |
41 | 37 | rpred 12652 | . . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℝ) |
42 | 41 | ltpnfd 12737 | . . . . . . 7 ⊢ (𝜑 → 𝐾 < +∞) |
43 | lbico1 13013 | . . . . . . 7 ⊢ ((𝐾 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐾 < +∞) → 𝐾 ∈ (𝐾[,)+∞)) | |
44 | 38, 40, 42, 43 | syl3anc 1373 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝐾[,)+∞)) |
45 | 34, 35, 44 | rspcdva 3551 | . . . . 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 484 | . . . 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 484 | . . . 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 26512 | . . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → (abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3)))) |
50 | 49 | ralrimiva 3106 | . 2 ⊢ (𝜑 → ∀𝑣 ∈ (𝑊[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3)))) |
51 | oveq1 7238 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑤[,)+∞) = (𝑊[,)+∞)) | |
52 | 51 | raleqdv 3337 | . . 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 3549 | . 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 587 | 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 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 ∀wral 3062 ∃wrex 3063 class class class wbr 5067 ↦ cmpt 5149 ‘cfv 6397 (class class class)co 7231 0cc0 10753 1c1 10754 + caddc 10756 · cmul 10758 +∞cpnf 10888 ℝ*cxr 10890 < clt 10891 ≤ cle 10892 − cmin 11086 / cdiv 11513 2c2 11909 3c3 11910 4c4 11911 ;cdc 12317 ℝ+crp 12610 (,)cioo 12959 [,)cico 12961 [,]cicc 12962 ...cfz 13119 ⌊cfl 13389 ↑cexp 13659 abscabs 14821 Σcsu 15273 expce 15647 logclog 25467 ψcchp 25999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-inf2 9280 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 ax-pre-sup 10831 ax-addf 10832 ax-mulf 10833 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-int 4874 df-iun 4920 df-iin 4921 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-se 5524 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-isom 6406 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-of 7487 df-om 7663 df-1st 7779 df-2nd 7780 df-supp 7924 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-1o 8222 df-2o 8223 df-oadd 8226 df-er 8411 df-map 8530 df-pm 8531 df-ixp 8599 df-en 8647 df-dom 8648 df-sdom 8649 df-fin 8650 df-fsupp 9010 df-fi 9051 df-sup 9082 df-inf 9083 df-oi 9150 df-dju 9541 df-card 9579 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-div 11514 df-nn 11855 df-2 11917 df-3 11918 df-4 11919 df-5 11920 df-6 11921 df-7 11922 df-8 11923 df-9 11924 df-n0 12115 df-xnn0 12187 df-z 12201 df-dec 12318 df-uz 12463 df-q 12569 df-rp 12611 df-xneg 12728 df-xadd 12729 df-xmul 12730 df-ioo 12963 df-ioc 12964 df-ico 12965 df-icc 12966 df-fz 13120 df-fzo 13263 df-fl 13391 df-mod 13467 df-seq 13599 df-exp 13660 df-fac 13864 df-bc 13893 df-hash 13921 df-shft 14654 df-cj 14686 df-re 14687 df-im 14688 df-sqrt 14822 df-abs 14823 df-limsup 15056 df-clim 15073 df-rlim 15074 df-sum 15274 df-ef 15653 df-e 15654 df-sin 15655 df-cos 15656 df-tan 15657 df-pi 15658 df-dvds 15840 df-gcd 16078 df-prm 16253 df-pc 16414 df-struct 16724 df-sets 16741 df-slot 16759 df-ndx 16769 df-base 16785 df-ress 16809 df-plusg 16839 df-mulr 16840 df-starv 16841 df-sca 16842 df-vsca 16843 df-ip 16844 df-tset 16845 df-ple 16846 df-ds 16848 df-unif 16849 df-hom 16850 df-cco 16851 df-rest 16951 df-topn 16952 df-0g 16970 df-gsum 16971 df-topgen 16972 df-pt 16973 df-prds 16976 df-xrs 17031 df-qtop 17036 df-imas 17037 df-xps 17039 df-mre 17113 df-mrc 17114 df-acs 17116 df-mgm 18138 df-sgrp 18187 df-mnd 18198 df-submnd 18243 df-mulg 18513 df-cntz 18735 df-cmn 19196 df-psmet 20379 df-xmet 20380 df-met 20381 df-bl 20382 df-mopn 20383 df-fbas 20384 df-fg 20385 df-cnfld 20388 df-top 21815 df-topon 21832 df-topsp 21854 df-bases 21867 df-cld 21940 df-ntr 21941 df-cls 21942 df-nei 22019 df-lp 22057 df-perf 22058 df-cn 22148 df-cnp 22149 df-haus 22236 df-cmp 22308 df-tx 22483 df-hmeo 22676 df-fil 22767 df-fm 22859 df-flim 22860 df-flf 22861 df-xms 23242 df-ms 23243 df-tms 23244 df-cncf 23799 df-limc 24787 df-dv 24788 df-ulm 25293 df-log 25469 df-atan 25774 df-em 25899 df-vma 26004 df-chp 26005 |
This theorem is referenced by: pntlemp 26515 |
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