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Theorem List for Metamath Proof Explorer - 35101-35200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgg-divcn 35101 Complex number division is a continuous function, when the second argument is nonzero. (Contributed by Mario Carneiro, 12-Aug-2014.) Avoid ax-mulf 11186. (Revised by GG, 16-Mar-2025.)
𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t (ℂ ∖ {0}))        / ∈ ((𝐽 ×t 𝐾) Cn 𝐽)
 
Theoremgg-expcn 35102* The power function on complex numbers, for fixed exponent 𝑁, is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) Avoid ax-mulf 11186. (Revised by GG, 16-Mar-2025.)
𝐽 = (TopOpen‘ℂfld)       (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥𝑁)) ∈ (𝐽 Cn 𝐽))
 
Theoremgg-divccn 35103* Division by a nonzero constant is a continuous operation. (Contributed by Mario Carneiro, 5-May-2014.) Avoid ax-mulf 11186. (Revised by GG, 16-Mar-2025.)
𝐽 = (TopOpen‘ℂfld)       ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℂ ↦ (𝑥 / 𝐴)) ∈ (𝐽 Cn 𝐽))
 
Theoremgg-negcncf 35104* The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.) Avoid ax-mulf 11186. (Revised by GG, 16-Mar-2025.)
𝐹 = (𝑥𝐴 ↦ -𝑥)       (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴cn→ℂ))
 
Theoremgg-iihalf1cn 35105 The first half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.) Avoid ax-mulf 11186. (Revised by GG, 16-Mar-2025.)
𝐽 = ((topGen‘ran (,)) ↾t (0[,](1 / 2)))       (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (𝐽 Cn II)
 
Theoremgg-iihalf2cn 35106 The second half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.) Avoid ax-mulf 11186. (Revised by GG, 16-Mar-2025.)
𝐽 = ((topGen‘ran (,)) ↾t ((1 / 2)[,]1))       (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (𝐽 Cn II)
 
Theoremgg-iimulcn 35107* Multiplication is a continuous function on the unit interval. (Contributed by Mario Carneiro, 8-Jun-2014.) Avoid ax-mulf 11186. (Revised by GG, 16-Mar-2025.)
(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥 · 𝑦)) ∈ ((II ×t II) Cn II)
 
Theoremgg-icchmeo 35108* The natural bijection from [0, 1] to an arbitrary nontrivial closed interval [𝐴, 𝐵] is a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.) Avoid ax-mulf 11186. (Revised by GG, 16-Mar-2025.)
𝐽 = (TopOpen‘ℂfld)    &   𝐹 = (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴)))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (IIHomeo(𝐽t (𝐴[,]𝐵))))
 
Theoremgg-cnrehmeo 35109* The canonical bijection from (ℝ × ℝ) to described in cnref1o 12965 is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if (ℝ × ℝ) is metrized with the l<SUP>2</SUP> norm.) (Contributed by Mario Carneiro, 25-Aug-2014.) Avoid ax-mulf 11186. (Revised by GG, 16-Mar-2025.)
𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    &   𝐽 = (topGen‘ran (,))    &   𝐾 = (TopOpen‘ℂfld)       𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾)
 
Theoremgg-reparphti 35110* Lemma for reparpht 24496. (Contributed by NM, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.) Avoid ax-mulf 11186. (Revised by GG, 16-Mar-2025.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn II))    &   (𝜑 → (𝐺‘0) = 0)    &   (𝜑 → (𝐺‘1) = 1)    &   𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘(((1 − 𝑦) · (𝐺𝑥)) + (𝑦 · 𝑥))))       (𝜑𝐻 ∈ ((𝐹𝐺)(PHtpy‘𝐽)𝐹))
 
Theoremgg-mulcncf 35111* The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.) Avoid ax-mulf 11186. (Revised by GG, 16-Mar-2025.)
(𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝑋cn→ℂ))
 
Theoremgg-dvcnp2 35112 A function is continuous at each point for which it is differentiable. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) Avoid ax-mulf 11186. (Revised by GG, 16-Mar-2025.)
𝐽 = (𝐾t 𝐴)    &   𝐾 = (TopOpen‘ℂfld)       (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵))
 
Theoremgg-dvmulbr 35113 The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmul 25440. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) Avoid ax-mulf 11186. (Revised by GG, 16-Mar-2025.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑌⟶ℂ)    &   (𝜑𝑌𝑆)    &   (𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐾𝑉)    &   (𝜑𝐿𝑉)    &   (𝜑𝐶(𝑆 D 𝐹)𝐾)    &   (𝜑𝐶(𝑆 D 𝐺)𝐿)    &   𝐽 = (TopOpen‘ℂfld)       (𝜑𝐶(𝑆 D (𝐹f · 𝐺))((𝐾 · (𝐺𝐶)) + (𝐿 · (𝐹𝐶))))
 
Theoremgg-dvcobr 35114 The chain rule for derivatives at a point. For the (simpler but more limited) function version, see dvco 25446. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) Avoid ax-mulf 11186. (Revised by GG, 16-Mar-2025.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑌𝑋)    &   (𝜑𝑌𝑇)    &   (𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑇 ⊆ ℂ)    &   (𝜑𝐾𝑉)    &   (𝜑𝐿𝑉)    &   (𝜑 → (𝐺𝐶)(𝑆 D 𝐹)𝐾)    &   (𝜑𝐶(𝑇 D 𝐺)𝐿)    &   𝐽 = (TopOpen‘ℂfld)       (𝜑𝐶(𝑇 D (𝐹𝐺))(𝐾 · 𝐿))
 
Theoremgg-plycn 35115 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 11186. (Revised by GG, 16-Mar-2025.)
(𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))
 
Theoremgg-psercn2 35116* Since by pserulm 25916 the series converges uniformly, it is also continuous by ulmcn 25893. (Contributed by Mario Carneiro, 3-Mar-2015.) Avoid ax-mulf 11186. (Revised by GG, 16-Mar-2025.)
𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   𝐹 = (𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   𝐻 = (𝑖 ∈ ℕ0 ↦ (𝑦𝑆 ↦ (seq0( + , (𝐺𝑦))‘𝑖)))    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑𝑀 < 𝑅)    &   (𝜑𝑆 ⊆ (abs “ (0[,]𝑀)))       (𝜑𝐹 ∈ (𝑆cn→ℂ))
 
Theoremgg-rmulccn 35117* Multiplication by a real constant is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.) Avoid ax-mulf 11186. (Revised by GG, 16-Mar-2025.)
𝐽 = (topGen‘ran (,))    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝑥 ∈ ℝ ↦ (𝑥 · 𝐶)) ∈ (𝐽 Cn 𝐽))
 
21.11.1.1  Miscellaneous
 
Theoremgg-cnfldex 35118 The field of complex numbers is a set. Alternative proof of cnfldex 20932. This version direcly uses df-cnfld 20930, which is discouraged, however it saves all complex numbers axioms and ax-pow 5362. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 16-Mar-2025.)
fld ∈ V
 
Theoremgg-cmvth 35119* Cauchy's Mean Value Theorem. If 𝐹, 𝐺 are real continuous functions on [𝐴, 𝐵] differentiable on (𝐴, 𝐵), then there is some 𝑥 ∈ (𝐴, 𝐵) such that 𝐹' (𝑥) / 𝐺' (𝑥) = (𝐹(𝐴) − 𝐹(𝐵)) / (𝐺(𝐴) − 𝐺(𝐵)). (We express the condition without division, so that we need no nonzero constraints.) (Contributed by Mario Carneiro, 29-Dec-2016.) Use mpomulcn 35100 instead of mulcn 24365 as direct dependency. (Revised by GG, 16-Mar-2025.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))       (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)(((𝐹𝐵) − (𝐹𝐴)) · ((ℝ D 𝐺)‘𝑥)) = (((𝐺𝐵) − (𝐺𝐴)) · ((ℝ D 𝐹)‘𝑥)))
 
Theoremgg-dvfsumle 35120* Compare a finite sum to an integral (the integral here is given as a function with a known derivative). (Contributed by Mario Carneiro, 14-May-2016.) Use mpomulcn 35100 instead of mulcn 24365 as direct dependency. (Revised by GG, 16-Mar-2025.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))    &   ((𝜑𝑥 ∈ (𝑀(,)𝑁)) → 𝐵𝑉)    &   (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))    &   (𝑥 = 𝑀𝐴 = 𝐶)    &   (𝑥 = 𝑁𝐴 = 𝐷)    &   ((𝜑𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)    &   ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝑋𝐵)       (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ (𝐷𝐶))
 
Theoremgg-dvfsumlem2 35121* Lemma for dvfsumrlim 25530. (Contributed by Mario Carneiro, 17-May-2016.) Use mpomulcn 35100 instead of mulcn 24365 as direct dependency. (Revised by GG, 16-Mar-2025.)
𝑆 = (𝑇(,)+∞)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝑀 ≤ (𝐷 + 1))    &   (𝜑𝑇 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐵𝑉)    &   ((𝜑𝑥𝑍) → 𝐵 ∈ ℝ)    &   (𝜑 → (ℝ D (𝑥𝑆𝐴)) = (𝑥𝑆𝐵))    &   (𝑥 = 𝑘𝐵 = 𝐶)    &   (𝜑𝑈 ∈ ℝ*)    &   ((𝜑 ∧ (𝑥𝑆𝑘𝑆) ∧ (𝐷𝑥𝑥𝑘𝑘𝑈)) → 𝐶𝐵)    &   𝐻 = (𝑥𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶𝐴)))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)    &   (𝜑𝐷𝑋)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝑈)    &   (𝜑𝑌 ≤ ((⌊‘𝑋) + 1))       (𝜑 → ((𝐻𝑌) ≤ (𝐻𝑋) ∧ ((𝐻𝑋) − 𝑋 / 𝑥𝐵) ≤ ((𝐻𝑌) − 𝑌 / 𝑥𝐵)))
 
Theoremgg-cxpcn 35122* Domain of continuity of the complex power function. (Contributed by Mario Carneiro, 1-May-2016.) Use mpomulcn 35100 instead of mulcn 24365 as direct dependency. (Revised by GG, 16-Mar-2025.)
𝐷 = (ℂ ∖ (-∞(,]0))    &   𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t 𝐷)       (𝑥𝐷, 𝑦 ∈ ℂ ↦ (𝑥𝑐𝑦)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)
 
21.12  Mathbox for Jeff Hankins
 
21.12.1  Miscellany
 
Theorema1i14 35123 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
(𝜓 → (𝜒𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theorema1i24 35124 Add two antecedents to a wff. Deduction associated with a1i13 27. (Contributed by Jeff Hankins, 5-Aug-2009.)
(𝜑 → (𝜒𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp5d 35125 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(((𝜑𝜓) ∧ 𝜒) → ((𝜃𝜏) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp5g 35126 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((𝜑𝜓) → (((𝜒𝜃) ∧ 𝜏) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp5k 35127 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (((𝜓 ∧ (𝜒𝜃)) ∧ 𝜏) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp56 35128 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((((𝜑𝜓) ∧ 𝜒) ∧ (𝜃𝜏)) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp58 35129 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(((𝜑𝜓) ∧ ((𝜒𝜃) ∧ 𝜏)) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp510 35130 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((𝜑 ∧ (((𝜓𝜒) ∧ 𝜃) ∧ 𝜏)) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp511 35131 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((𝜑 ∧ ((𝜓 ∧ (𝜒𝜃)) ∧ 𝜏)) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp512 35132 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((𝜑 ∧ ((𝜓𝜒) ∧ (𝜃𝜏))) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theorem3com12d 35133 Commutation in consequent. Swap 1st and 2nd. (Contributed by Jeff Hankins, 17-Nov-2009.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑 → (𝜒𝜓𝜃))
 
Theoremimp5p 35134 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜑 → (𝜓 → ((𝜒𝜃𝜏) → 𝜂)))
 
Theoremimp5q 35135 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       ((𝜑𝜓) → ((𝜒𝜃𝜏) → 𝜂))
 
Theoremecase13d 35136 Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → ¬ 𝜃)    &   (𝜑 → (𝜒𝜓𝜃))       (𝜑𝜓)
 
Theoremsubtr 35137 Transitivity of implicit substitution. (Contributed by Jeff Hankins, 13-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝑌    &   𝑥𝑍    &   (𝑥 = 𝐴𝑋 = 𝑌)    &   (𝑥 = 𝐵𝑋 = 𝑍)       ((𝐴𝐶𝐵𝐷) → (𝐴 = 𝐵𝑌 = 𝑍))
 
Theoremsubtr2 35138 Transitivity of implicit substitution into a wff. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝜓    &   𝑥𝜒    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       ((𝐴𝐶𝐵𝐷) → (𝐴 = 𝐵 → (𝜓𝜒)))
 
Theoremtrer 35139* A relation intersected with its converse is an equivalence relation if the relation is transitive. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)
(∀𝑎𝑏𝑐((𝑎 𝑏𝑏 𝑐) → 𝑎 𝑐) → ( ) Er dom ( ))
 
Theoremelicc3 35140 An equivalent membership condition for closed intervals. (Contributed by Jeff Hankins, 14-Jul-2009.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶𝐶 < 𝐵) ∨ 𝐶 = 𝐵))))
 
Theoremfinminlem 35141* A useful lemma about finite sets. If a property holds for a finite set, it holds for a minimal set. (Contributed by Jeff Hankins, 4-Dec-2009.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥 ∈ Fin 𝜑 → ∃𝑥(𝜑 ∧ ∀𝑦((𝑦𝑥𝜓) → 𝑥 = 𝑦)))
 
Theoremgtinf 35142* Any number greater than an infimum is greater than some element of the set. (Contributed by Jeff Hankins, 29-Sep-2013.) (Revised by AV, 10-Oct-2021.)
(((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑆 𝑥𝑦) ∧ (𝐴 ∈ ℝ ∧ inf(𝑆, ℝ, < ) < 𝐴)) → ∃𝑧𝑆 𝑧 < 𝐴)
 
Theoremopnrebl 35143* A set is open in the standard topology of the reals precisely when every point can be enclosed in an open ball. (Contributed by Jeff Hankins, 23-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)
(𝐴 ∈ (topGen‘ran (,)) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥𝐴𝑦 ∈ ℝ+ ((𝑥𝑦)(,)(𝑥 + 𝑦)) ⊆ 𝐴))
 
Theoremopnrebl2 35144* A set is open in the standard topology of the reals precisely when every point can be enclosed in an arbitrarily small ball. (Contributed by Jeff Hankins, 22-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)
(𝐴 ∈ (topGen‘ran (,)) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝑧𝑦 ∧ ((𝑥𝑧)(,)(𝑥 + 𝑧)) ⊆ 𝐴)))
 
Theoremnn0prpwlem 35145* Lemma for nn0prpw 35146. Use strong induction to show that every positive integer has unique prime power divisors. (Contributed by Jeff Hankins, 28-Sep-2013.)
(𝐴 ∈ ℕ → ∀𝑘 ∈ ℕ (𝑘 < 𝐴 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝𝑛) ∥ 𝑘 ↔ (𝑝𝑛) ∥ 𝐴)))
 
Theoremnn0prpw 35146* Two nonnegative integers are the same if and only if they are divisible by the same prime powers. (Contributed by Jeff Hankins, 29-Sep-2013.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 = 𝐵 ↔ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝𝑛) ∥ 𝐴 ↔ (𝑝𝑛) ∥ 𝐵)))
 
21.12.2  Basic topological facts
 
Theoremtopbnd 35147 Two equivalent expressions for the boundary of a topology. (Contributed by Jeff Hankins, 23-Sep-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))
 
Theoremopnbnd 35148 A set is open iff it is disjoint from its boundary. (Contributed by Jeff Hankins, 23-Sep-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴𝐽 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅))
 
Theoremcldbnd 35149 A set is closed iff it contains its boundary. (Contributed by Jeff Hankins, 1-Oct-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴))
 
Theoremntruni 35150* A union of interiors is a subset of the interior of the union. The reverse inclusion may not hold. (Contributed by Jeff Hankins, 31-Aug-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → 𝑜𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂))
 
Theoremclsun 35151 A pairwise union of closures is the closure of the union. (Contributed by Jeff Hankins, 31-Aug-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
 
Theoremclsint2 35152* The closure of an intersection is a subset of the intersection of the closures. (Contributed by Jeff Hankins, 31-Aug-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ((cls‘𝐽)‘ 𝐶) ⊆ 𝑐𝐶 ((cls‘𝐽)‘𝑐))
 
Theoremopnregcld 35153* A set is regularly closed iff it is the closure of some open set. (Contributed by Jeff Hankins, 27-Sep-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))
 
Theoremcldregopn 35154* A set if regularly open iff it is the interior of some closed set. (Contributed by Jeff Hankins, 27-Sep-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)))
 
Theoremneiin 35155 Two neighborhoods intersect to form a neighborhood of the intersection. (Contributed by Jeff Hankins, 31-Aug-2009.)
((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)))
 
Theoremhmeoclda 35156 Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.) (Revised by Mario Carneiro, 3-Jun-2014.)
(((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ (𝐽Homeo𝐾)) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝐹𝑆) ∈ (Clsd‘𝐾))
 
Theoremhmeocldb 35157 Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.)
(((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ (𝐽Homeo𝐾)) ∧ 𝑆 ∈ (Clsd‘𝐾)) → (𝐹𝑆) ∈ (Clsd‘𝐽))
 
21.12.3  Topology of the real numbers
 
TheoremivthALT 35158* An alternate proof of the Intermediate Value Theorem ivth 24953 using topology. (Contributed by Jeff Hankins, 17-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ∃𝑥 ∈ (𝐴(,)𝐵)(𝐹𝑥) = 𝑈)
 
21.12.4  Refinements
 
Syntaxcfne 35159 Extend class definition to include the "finer than" relation.
class Fne
 
Definitiondf-fne 35160* Define the fineness relation for covers. (Contributed by Jeff Hankins, 28-Sep-2009.)
Fne = {⟨𝑥, 𝑦⟩ ∣ ( 𝑥 = 𝑦 ∧ ∀𝑧𝑥 𝑧 (𝑦 ∩ 𝒫 𝑧))}
 
Theoremfnerel 35161 Fineness is a relation. (Contributed by Jeff Hankins, 28-Sep-2009.)
Rel Fne
 
Theoremisfne 35162* The predicate "𝐵 is finer than 𝐴". This property is, in a sense, the opposite of refinement, as refinement requires every element to be a subset of an element of the original and fineness requires that every element of the original have a subset in the finer cover containing every point. I do not know of a literature reference for this. (Contributed by Jeff Hankins, 28-Sep-2009.)
𝑋 = 𝐴    &   𝑌 = 𝐵       (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥))))
 
Theoremisfne4 35163 The predicate "𝐵 is finer than 𝐴 " in terms of the topology generation function. (Contributed by Mario Carneiro, 11-Sep-2015.)
𝑋 = 𝐴    &   𝑌 = 𝐵       (𝐴Fne𝐵 ↔ (𝑋 = 𝑌𝐴 ⊆ (topGen‘𝐵)))
 
Theoremisfne4b 35164 A condition for a topology to be finer than another. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
𝑋 = 𝐴    &   𝑌 = 𝐵       (𝐵𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))
 
Theoremisfne2 35165* The predicate "𝐵 is finer than 𝐴". (Contributed by Jeff Hankins, 28-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
𝑋 = 𝐴    &   𝑌 = 𝐵       (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
 
Theoremisfne3 35166* The predicate "𝐵 is finer than 𝐴". (Contributed by Jeff Hankins, 11-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
𝑋 = 𝐴    &   𝑌 = 𝐵       (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦(𝑦𝐵𝑥 = 𝑦))))
 
Theoremfnebas 35167 A finer cover covers the same set as the original. (Contributed by Jeff Hankins, 28-Sep-2009.)
𝑋 = 𝐴    &   𝑌 = 𝐵       (𝐴Fne𝐵𝑋 = 𝑌)
 
Theoremfnetg 35168 A finer cover generates a topology finer than the original set. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝐴Fne𝐵𝐴 ⊆ (topGen‘𝐵))
 
Theoremfnessex 35169* If 𝐵 is finer than 𝐴 and 𝑆 is an element of 𝐴, every point in 𝑆 is an element of a subset of 𝑆 which is in 𝐵. (Contributed by Jeff Hankins, 28-Sep-2009.)
((𝐴Fne𝐵𝑆𝐴𝑃𝑆) → ∃𝑥𝐵 (𝑃𝑥𝑥𝑆))
 
Theoremfneuni 35170* If 𝐵 is finer than 𝐴, every element of 𝐴 is a union of elements of 𝐵. (Contributed by Jeff Hankins, 11-Oct-2009.)
((𝐴Fne𝐵𝑆𝐴) → ∃𝑥(𝑥𝐵𝑆 = 𝑥))
 
Theoremfneint 35171* If a cover is finer than another, every point can be approached more closely by intersections. (Contributed by Jeff Hankins, 11-Oct-2009.)
(𝐴Fne𝐵 {𝑥𝐵𝑃𝑥} ⊆ {𝑥𝐴𝑃𝑥})
 
Theoremfness 35172 A cover is finer than its subcovers. (Contributed by Jeff Hankins, 11-Oct-2009.)
𝑋 = 𝐴    &   𝑌 = 𝐵       ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → 𝐴Fne𝐵)
 
Theoremfneref 35173 Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.)
(𝐴𝑉𝐴Fne𝐴)
 
Theoremfnetr 35174 Transitivity of the fineness relation. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
((𝐴Fne𝐵𝐵Fne𝐶) → 𝐴Fne𝐶)
 
Theoremfneval 35175 Two covers are finer than each other iff they are both bases for the same topology. (Contributed by Mario Carneiro, 11-Sep-2015.)
= (Fne ∩ Fne)       ((𝐴𝑉𝐵𝑊) → (𝐴 𝐵 ↔ (topGen‘𝐴) = (topGen‘𝐵)))
 
Theoremfneer 35176 Fineness intersected with its converse is an equivalence relation. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
= (Fne ∩ Fne)        Er V
 
Theoremtopfne 35177 Fineness for covers corresponds precisely with fineness for topologies. (Contributed by Jeff Hankins, 29-Sep-2009.)
𝑋 = 𝐽    &   𝑌 = 𝐾       ((𝐾 ∈ Top ∧ 𝑋 = 𝑌) → (𝐽𝐾𝐽Fne𝐾))
 
Theoremtopfneec 35178 A cover is equivalent to a topology iff it is a base for that topology. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
= (Fne ∩ Fne)       (𝐽 ∈ Top → (𝐴 ∈ [𝐽] ↔ (topGen‘𝐴) = 𝐽))
 
Theoremtopfneec2 35179 A topology is precisely identified with its equivalence class. (Contributed by Jeff Hankins, 12-Oct-2009.)
= (Fne ∩ Fne)       ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ([𝐽] = [𝐾] 𝐽 = 𝐾))
 
Theoremfnessref 35180* A cover is finer iff it has a subcover which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
𝑋 = 𝐴    &   𝑌 = 𝐵       (𝑋 = 𝑌 → (𝐴Fne𝐵 ↔ ∃𝑐(𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
 
Theoremrefssfne 35181* A cover is a refinement iff it is a subcover of something which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
𝑋 = 𝐴    &   𝑌 = 𝐵       (𝑋 = 𝑌 → (𝐵Ref𝐴 ↔ ∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
 
21.12.5  Neighborhood bases determine topologies
 
Theoremneibastop1 35182* A collection of neighborhood bases determines a topology. Part of Theorem 4.5 of Stephen Willard's General Topology. (Contributed by Jeff Hankins, 8-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅}))    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥) ∧ 𝑤 ∈ (𝐹𝑥))) → ((𝐹𝑥) ∩ 𝒫 (𝑣𝑤)) ≠ ∅)    &   𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑜 ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅}       (𝜑𝐽 ∈ (TopOn‘𝑋))
 
Theoremneibastop2lem 35183* Lemma for neibastop2 35184. (Contributed by Jeff Hankins, 12-Sep-2009.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅}))    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥) ∧ 𝑤 ∈ (𝐹𝑥))) → ((𝐹𝑥) ∩ 𝒫 (𝑣𝑤)) ≠ ∅)    &   𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑜 ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅}    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥))) → 𝑥𝑣)    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥))) → ∃𝑡 ∈ (𝐹𝑥)∀𝑦𝑡 ((𝐹𝑦) ∩ 𝒫 𝑣) ≠ ∅)    &   (𝜑𝑃𝑋)    &   (𝜑𝑁𝑋)    &   (𝜑𝑈 ∈ (𝐹𝑃))    &   (𝜑𝑈𝑁)    &   𝐺 = (rec((𝑎 ∈ V ↦ 𝑧𝑎 𝑥𝑋 ((𝐹𝑥) ∩ 𝒫 𝑧)), {𝑈}) ↾ ω)    &   𝑆 = {𝑦𝑋 ∣ ∃𝑓 ran 𝐺((𝐹𝑦) ∩ 𝒫 𝑓) ≠ ∅}       (𝜑 → ∃𝑢𝐽 (𝑃𝑢𝑢𝑁))
 
Theoremneibastop2 35184* In the topology generated by a neighborhood base, a set is a neighborhood of a point iff it contains a subset in the base. (Contributed by Jeff Hankins, 9-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅}))    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥) ∧ 𝑤 ∈ (𝐹𝑥))) → ((𝐹𝑥) ∩ 𝒫 (𝑣𝑤)) ≠ ∅)    &   𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑜 ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅}    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥))) → 𝑥𝑣)    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥))) → ∃𝑡 ∈ (𝐹𝑥)∀𝑦𝑡 ((𝐹𝑦) ∩ 𝒫 𝑣) ≠ ∅)       ((𝜑𝑃𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑁𝑋 ∧ ((𝐹𝑃) ∩ 𝒫 𝑁) ≠ ∅)))
 
Theoremneibastop3 35185* The topology generated by a neighborhood base is unique. (Contributed by Jeff Hankins, 16-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅}))    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥) ∧ 𝑤 ∈ (𝐹𝑥))) → ((𝐹𝑥) ∩ 𝒫 (𝑣𝑤)) ≠ ∅)    &   𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑜 ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅}    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥))) → 𝑥𝑣)    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥))) → ∃𝑡 ∈ (𝐹𝑥)∀𝑦𝑡 ((𝐹𝑦) ∩ 𝒫 𝑣) ≠ ∅)       (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})
 
21.12.6  Lattice structure of topologies
 
Theoremtopmtcl 35186 The meet of a collection of topologies on 𝑋 is again a topology on 𝑋. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) ∈ (TopOn‘𝑋))
 
Theoremtopmeet 35187* Two equivalent formulations of the meet of a collection of topologies. (Contributed by Jeff Hankins, 4-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) = {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗})
 
Theoremtopjoin 35188* Two equivalent formulations of the join of a collection of topologies. (Contributed by Jeff Hankins, 6-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) = {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘})
 
Theoremfnemeet1 35189* The meet of a collection of equivalence classes of covers with respect to fineness. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → (𝒫 𝑋 𝑡𝑆 (topGen‘𝑡))Fne𝐴)
 
Theoremfnemeet2 35190* The meet of equivalence classes under the fineness relation-part two. (Contributed by Jeff Hankins, 6-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (𝑇Fne(𝒫 𝑋 𝑡𝑆 (topGen‘𝑡)) ↔ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑇Fne𝑥)))
 
Theoremfnejoin1 35191* Join of equivalence classes under the fineness relation-part one. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴Fneif(𝑆 = ∅, {𝑋}, 𝑆))
 
Theoremfnejoin2 35192* Join of equivalence classes under the fineness relation-part two. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 ↔ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)))
 
21.12.7  Filter bases
 
Theoremfgmin 35193 Minimality property of a generated filter: every filter that contains 𝐵 contains its generated filter. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)
((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐵𝐹 ↔ (𝑋filGen𝐵) ⊆ 𝐹))
 
Theoremneifg 35194* The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 23328. (Contributed by Jeff Hankins, 3-Sep-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → (𝑋filGen{𝑥𝐽𝑆𝑥}) = ((nei‘𝐽)‘𝑆))
 
21.12.8  Directed sets, nets
 
Theoremtailfval 35195* The tail function for a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
𝑋 = dom 𝐷       (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
 
Theoremtailval 35196 The tail of an element in a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
𝑋 = dom 𝐷       ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → ((tail‘𝐷)‘𝐴) = (𝐷 “ {𝐴}))
 
Theoremeltail 35197 An element of a tail. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
𝑋 = dom 𝐷       ((𝐷 ∈ DirRel ∧ 𝐴𝑋𝐵𝐶) → (𝐵 ∈ ((tail‘𝐷)‘𝐴) ↔ 𝐴𝐷𝐵))
 
Theoremtailf 35198 The tail function of a directed set sends its elements to its subsets. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
𝑋 = dom 𝐷       (𝐷 ∈ DirRel → (tail‘𝐷):𝑋⟶𝒫 𝑋)
 
Theoremtailini 35199 A tail contains its initial element. (Contributed by Jeff Hankins, 25-Nov-2009.)
𝑋 = dom 𝐷       ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → 𝐴 ∈ ((tail‘𝐷)‘𝐴))
 
Theoremtailfb 35200 The collection of tails of a directed set is a filter base. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
𝑋 = dom 𝐷       ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ran (tail‘𝐷) ∈ (fBas‘𝑋))
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