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Mirrors > Home > MPE Home > Th. List > pntleme | Structured version Visualization version GIF version |
Description: Lemma for pnt 26962. Package up pntlemo 26955 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 26944 | . 2 ⊢ (𝜑 → 𝑊 ∈ ℝ+) |
16 | 2 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝐴 ∈ ℝ+) |
17 | 3 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝐵 ∈ ℝ+) |
18 | 4 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝐿 ∈ (0(,)1)) |
19 | 7 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝑈 ∈ ℝ+) |
20 | 8 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝑈 ≤ 𝐴) |
21 | 11 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) |
22 | 12 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) |
23 | 13 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝐶 ∈ ℝ+) |
24 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → 𝑣 ∈ (𝑊[,)+∞)) | |
25 | eqid 2736 | . . . 4 ⊢ ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) | |
26 | eqid 2736 | . . . 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 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈) |
29 | oveq1 7364 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑘 · 𝑦) = (𝐾 · 𝑦)) | |
30 | 29 | breq2d 5117 | . . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦) ↔ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦))) |
31 | 30 | anbi2d 629 | . . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦)) ↔ (𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)))) |
32 | 31 | anbi1d 630 | . . . . . . . 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 3175 | . . . . . . 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 3174 | . . . . . 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 26943 | . . . . . . . . 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 12958 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℝ*) |
39 | pnfxr 11209 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
40 | 39 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ*) |
41 | 37 | rpred 12957 | . . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℝ) |
42 | 41 | ltpnfd 13042 | . . . . . . 7 ⊢ (𝜑 → 𝐾 < +∞) |
43 | lbico1 13318 | . . . . . . 7 ⊢ ((𝐾 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐾 < +∞) → 𝐾 ∈ (𝐾[,)+∞)) | |
44 | 38, 40, 42, 43 | syl3anc 1371 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝐾[,)+∞)) |
45 | 34, 35, 44 | rspcdva 3582 | . . . . 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 481 | . . . 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 481 | . . . 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 26955 | . . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝑊[,)+∞)) → (abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3)))) |
50 | 49 | ralrimiva 3143 | . 2 ⊢ (𝜑 → ∀𝑣 ∈ (𝑊[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3)))) |
51 | oveq1 7364 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑤[,)+∞) = (𝑊[,)+∞)) | |
52 | 51 | raleqdv 3313 | . . 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 3581 | . 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 class class class wbr 5105 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 +∞cpnf 11186 ℝ*cxr 11188 < clt 11189 ≤ cle 11190 − cmin 11385 / cdiv 11812 2c2 12208 3c3 12209 4c4 12210 ;cdc 12618 ℝ+crp 12915 (,)cioo 13264 [,)cico 13266 [,]cicc 13267 ...cfz 13424 ⌊cfl 13695 ↑cexp 13967 abscabs 15119 Σcsu 15570 expce 15944 logclog 25910 ψcchp 26442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-xnn0 12486 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-e 15951 df-sin 15952 df-cos 15953 df-tan 15954 df-pi 15955 df-dvds 16137 df-gcd 16375 df-prm 16548 df-pc 16709 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231 df-ulm 25736 df-log 25912 df-atan 26217 df-em 26342 df-vma 26447 df-chp 26448 |
This theorem is referenced by: pntlemp 26958 |
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