HomeHome Intuitionistic Logic Explorer
Theorem List (p. 137 of 150)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 13601-13700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnfldmul 13601 The multiplication operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
· = (.r‘ℂfld)
 
Theoremcnfldcj 13602 The conjugation operation of the field of complex numbers. (Contributed by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 17-Dec-2017.)
∗ = (*𝑟‘ℂfld)
 
Theoremcncrng 13603 The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.)
fld ∈ CRing
 
Theoremcnring 13604 The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
fld ∈ Ring
 
Theoremcnfld0 13605 Zero is the zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
0 = (0g‘ℂfld)
 
Theoremcnfld1 13606 One is the unity element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
1 = (1r‘ℂfld)
 
Theoremcnfldneg 13607 The additive inverse in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
(𝑋 ∈ ℂ → ((invg‘ℂfld)‘𝑋) = -𝑋)
 
Theoremcnfldplusf 13608 The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.)
+ = (+𝑓‘ℂfld)
 
Theoremcnfldsub 13609 The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.)
− = (-g‘ℂfld)
 
Theoremcnfldmulg 13610 The group multiple function in the field of complex numbers. (Contributed by Mario Carneiro, 14-Jun-2015.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℂ) → (𝐴(.g‘ℂfld)𝐵) = (𝐴 · 𝐵))
 
Theoremcnfldexp 13611 The exponentiation operator in the field of complex numbers (for nonnegative exponents). (Contributed by Mario Carneiro, 15-Jun-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ0) → (𝐵(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴𝐵))
 
Theoremcnsubmlem 13612* Lemma for nn0subm 13617 and friends. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑥𝐴𝑥 ∈ ℂ)    &   ((𝑥𝐴𝑦𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   0 ∈ 𝐴       𝐴 ∈ (SubMnd‘ℂfld)
 
Theoremcnsubglem 13613* Lemma for cnsubrglem 13614 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.)
(𝑥𝐴𝑥 ∈ ℂ)    &   ((𝑥𝐴𝑦𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   (𝑥𝐴 → -𝑥𝐴)    &   𝐵𝐴       𝐴 ∈ (SubGrp‘ℂfld)
 
Theoremcnsubrglem 13614* Lemma for zsubrg 13615 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.)
(𝑥𝐴𝑥 ∈ ℂ)    &   ((𝑥𝐴𝑦𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   (𝑥𝐴 → -𝑥𝐴)    &   1 ∈ 𝐴    &   ((𝑥𝐴𝑦𝐴) → (𝑥 · 𝑦) ∈ 𝐴)       𝐴 ∈ (SubRing‘ℂfld)
 
Theoremzsubrg 13615 The integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
ℤ ∈ (SubRing‘ℂfld)
 
Theoremgzsubrg 13616 The gaussian integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
ℤ[i] ∈ (SubRing‘ℂfld)
 
Theoremnn0subm 13617 The nonnegative integers form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 18-Jun-2015.)
0 ∈ (SubMnd‘ℂfld)
 
Theoremrege0subm 13618 The nonnegative reals form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 20-Jun-2015.)
(0[,)+∞) ∈ (SubMnd‘ℂfld)
 
Theoremzsssubrg 13619 The integers are a subset of any subring of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝑅 ∈ (SubRing‘ℂfld) → ℤ ⊆ 𝑅)
 
7.7.2  Ring of integers

According to Wikipedia ("Integer", 25-May-2019, https://en.wikipedia.org/wiki/Integer) "The integers form a unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring. This universal property, namely to be an initial object in the category of [unital] rings, characterizes the ring 𝑍." In set.mm, there was no explicit definition for the ring of integers until June 2019, but it was denoted by (ℂflds ℤ), the field of complex numbers restricted to the integers. In zringring 13623 it is shown that this restriction is a ring, and zringbas 13626 shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as Definition df-zring 13621 of the ring of integers.

Remark: Instead of using the symbol "ZZrng" analogous to fld used for the field of complex numbers, we have chosen the version with an "i" to indicate that the ring of integers is a unital ring, see also Wikipedia ("Rng (algebra)", 9-Jun-2019, https://en.wikipedia.org/wiki/Rng_(algebra) 13621).

 
Syntaxczring 13620 Extend class notation with the (unital) ring of integers.
class ring
 
Definitiondf-zring 13621 The (unital) ring of integers. (Contributed by Alexander van der Vekens, 9-Jun-2019.)
ring = (ℂflds ℤ)
 
Theoremzringcrng 13622 The ring of integers is a commutative ring. (Contributed by AV, 13-Jun-2019.)
ring ∈ CRing
 
Theoremzringring 13623 The ring of integers is a ring. (Contributed by AV, 20-May-2019.) (Revised by AV, 9-Jun-2019.) (Proof shortened by AV, 13-Jun-2019.)
ring ∈ Ring
 
Theoremzringabl 13624 The ring of integers is an (additive) abelian group. (Contributed by AV, 13-Jun-2019.)
ring ∈ Abel
 
Theoremzringgrp 13625 The ring of integers is an (additive) group. (Contributed by AV, 10-Jun-2019.)
ring ∈ Grp
 
Theoremzringbas 13626 The integers are the base of the ring of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.)
ℤ = (Base‘ℤring)
 
Theoremzringplusg 13627 The addition operation of the ring of integers. (Contributed by Thierry Arnoux, 8-Nov-2017.) (Revised by AV, 9-Jun-2019.)
+ = (+g‘ℤring)
 
Theoremzringmulg 13628 The multiplication (group power) operation of the group of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴(.g‘ℤring)𝐵) = (𝐴 · 𝐵))
 
Theoremzringmulr 13629 The multiplication operation of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
· = (.r‘ℤring)
 
Theoremzring0 13630 The zero element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
0 = (0g‘ℤring)
 
Theoremzring1 13631 The unity element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
1 = (1r‘ℤring)
 
Theoremzringnzr 13632 The ring of integers is a nonzero ring. (Contributed by AV, 18-Apr-2020.)
ring ∈ NzRing
 
Theoremdvdsrzring 13633 Ring divisibility in the ring of integers corresponds to ordinary divisibility in . (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.)
∥ = (∥r‘ℤring)
 
Theoremzringinvg 13634 The additive inverse of an element of the ring of integers. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
(𝐴 ∈ ℤ → -𝐴 = ((invg‘ℤring)‘𝐴))
 
Theoremzringsubgval 13635 Subtraction in the ring of integers. (Contributed by AV, 3-Aug-2019.)
= (-g‘ℤring)       ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑋𝑌) = (𝑋 𝑌))
 
Theoremzringmpg 13636 The multiplicative group of the ring of integers is the restriction of the multiplicative group of the complex numbers to the integers. (Contributed by AV, 15-Jun-2019.)
((mulGrp‘ℂfld) ↾s ℤ) = (mulGrp‘ℤring)
 
PART 8  BASIC TOPOLOGY
 
8.1  Topology
 
8.1.1  Topological spaces

A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union), and it may sometimes be more convenient to consider topologies without reference to the underlying set.

 
8.1.1.1  Topologies
 
Syntaxctop 13637 Syntax for the class of topologies.
class Top
 
Definitiondf-top 13638* Define the class of topologies. It is a proper class. See istopg 13639 and istopfin 13640 for the corresponding characterizations, using respectively binary intersections like in this definition and nonempty finite intersections.

The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see

Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241.

(Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.)

Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
 
Theoremistopg 13639* Express the predicate "𝐽 is a topology". See istopfin 13640 for another characterization using nonempty finite intersections instead of binary intersections.

Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use 𝑇 to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

(𝐽𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
 
Theoremistopfin 13640* Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 13639. It is not clear we can prove the converse without adding additional conditions. (Contributed by NM, 19-Jul-2006.) (Revised by Jim Kingdon, 14-Jan-2023.)
(𝐽 ∈ Top → (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥((𝑥𝐽𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → 𝑥𝐽)))
 
Theoremuniopn 13641 The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝐽)
 
Theoremiunopn 13642* The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → 𝑥𝐴 𝐵𝐽)
 
Theoreminopn 13643 The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
 
Theoremfiinopn 13644 The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
(𝐽 ∈ Top → ((𝐴𝐽𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → 𝐴𝐽))
 
Theoremunopn 13645 The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
 
Theorem0opn 13646 The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)
(𝐽 ∈ Top → ∅ ∈ 𝐽)
 
Theorem0ntop 13647 The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
¬ ∅ ∈ Top
 
Theoremtopopn 13648 The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
𝑋 = 𝐽       (𝐽 ∈ Top → 𝑋𝐽)
 
Theoremeltopss 13649 A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝑋)
 
8.1.1.2  Topologies on sets
 
Syntaxctopon 13650 Syntax for the function of topologies on sets.
class TopOn
 
Definitiondf-topon 13651* Define the function that associates with a set the set of topologies on it. (Contributed by Stefan O'Rear, 31-Jan-2015.)
TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = 𝑗})
 
Theoremfuntopon 13652 The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
Fun TopOn
 
Theoremistopon 13653 Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))
 
Theoremtopontop 13654 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
 
Theoremtoponuni 13655 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐵 = 𝐽)
 
Theoremtopontopi 13656 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐽 ∈ (TopOn‘𝐵)       𝐽 ∈ Top
 
Theoremtoponunii 13657 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐽 ∈ (TopOn‘𝐵)       𝐵 = 𝐽
 
Theoremtoptopon 13658 Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝑋 = 𝐽       (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
 
Theoremtoptopon2 13659 A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
(𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
 
Theoremtopontopon 13660 A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
(𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ (TopOn‘ 𝐽))
 
Theoremtoponrestid 13661 Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022.)
𝐴 ∈ (TopOn‘𝐵)       𝐴 = (𝐴t 𝐵)
 
Theoremtoponsspwpwg 13662 The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.)
(𝐴𝑉 → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
 
Theoremdmtopon 13663 The domain of TopOn is V. (Contributed by BJ, 29-Apr-2021.)
dom TopOn = V
 
Theoremfntopon 13664 The class TopOn is a function with domain V. (Contributed by BJ, 29-Apr-2021.)
TopOn Fn V
 
Theoremtoponmax 13665 The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐵𝐽)
 
Theoremtoponss 13666 A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴𝑋)
 
Theoremtoponcom 13667 If 𝐾 is a topology on the base set of topology 𝐽, then 𝐽 is a topology on the base of 𝐾. (Contributed by Mario Carneiro, 22-Aug-2015.)
((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘ 𝐽)) → 𝐽 ∈ (TopOn‘ 𝐾))
 
Theoremtoponcomb 13668 Biconditional form of toponcom 13667. (Contributed by BJ, 5-Dec-2021.)
((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘ 𝐾) ↔ 𝐾 ∈ (TopOn‘ 𝐽)))
 
Theoremtopgele 13669 The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
(𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽𝐽 ⊆ 𝒫 𝑋))
 
8.1.1.3  Topological spaces
 
Syntaxctps 13670 Syntax for the class of topological spaces.
class TopSp
 
Definitiondf-topsp 13671 Define the class of topological spaces (as extensible structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)
TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
 
Theoremistps 13672 Express the predicate "is a topological space". (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐴 = (Base‘𝐾)    &   𝐽 = (TopOpen‘𝐾)       (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴))
 
Theoremistps2 13673 Express the predicate "is a topological space". (Contributed by NM, 20-Oct-2012.)
𝐴 = (Base‘𝐾)    &   𝐽 = (TopOpen‘𝐾)       (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = 𝐽))
 
Theoremtpsuni 13674 The base set of a topological space. (Contributed by FL, 27-Jun-2014.)
𝐴 = (Base‘𝐾)    &   𝐽 = (TopOpen‘𝐾)       (𝐾 ∈ TopSp → 𝐴 = 𝐽)
 
Theoremtpstop 13675 The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.)
𝐽 = (TopOpen‘𝐾)       (𝐾 ∈ TopSp → 𝐽 ∈ Top)
 
Theoremtpspropd 13676 A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))       (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))
 
Theoremtopontopn 13677 Express the predicate "is a topological space". (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐴 = (Base‘𝐾)    &   𝐽 = (TopSet‘𝐾)       (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾))
 
Theoremtsettps 13678 If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐴 = (Base‘𝐾)    &   𝐽 = (TopSet‘𝐾)       (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)
 
Theoremistpsi 13679 Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)
(Base‘𝐾) = 𝐴    &   (TopOpen‘𝐾) = 𝐽    &   𝐴 = 𝐽    &   𝐽 ∈ Top       𝐾 ∈ TopSp
 
Theoremeltpsg 13680 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), 𝐽⟩}       (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)
 
Theoremeltpsi 13681 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by NM, 20-Oct-2012.) (Revised by Mario Carneiro, 13-Aug-2015.)
𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), 𝐽⟩}    &   𝐴 = 𝐽    &   𝐽 ∈ Top       𝐾 ∈ TopSp
 
8.1.2  Topological bases
 
Syntaxctb 13682 Syntax for the class of topological bases.
class TopBases
 
Definitiondf-bases 13683* Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 13685). Note that "bases" is the plural of "basis". (Contributed by NM, 17-Jul-2006.)
TopBases = {𝑥 ∣ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ⊆ (𝑥 ∩ 𝒫 (𝑦𝑧))}
 
Theoremisbasisg 13684* Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
(𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
 
Theoremisbasis2g 13685* Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
(𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
 
Theoremisbasis3g 13686* Express the predicate "the set 𝐵 is a basis for a topology". Definition of basis in [Munkres] p. 78. (Contributed by NM, 17-Jul-2006.)
(𝐵𝐶 → (𝐵 ∈ TopBases ↔ (∀𝑥𝐵 𝑥 𝐵 ∧ ∀𝑥 𝐵𝑦𝐵 𝑥𝑦 ∧ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))))
 
Theorembasis1 13687 Property of a basis. (Contributed by NM, 16-Jul-2006.)
((𝐵 ∈ TopBases ∧ 𝐶𝐵𝐷𝐵) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷)))
 
Theorembasis2 13688* Property of a basis. (Contributed by NM, 17-Jul-2006.)
(((𝐵 ∈ TopBases ∧ 𝐶𝐵) ∧ (𝐷𝐵𝐴 ∈ (𝐶𝐷))) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷)))
 
Theoremfiinbas 13689* If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)
 
Theorembaspartn 13690* A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝑃𝑉 ∧ ∀𝑥𝑃𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → 𝑃 ∈ TopBases)
 
Theoremtgval2 13691* Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 13704) that (topGen‘𝐵) is indeed a topology (on 𝐵, see unitg 13702). See also tgval 12717 and tgval3 13698. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))})
 
Theoremeltg 13692 Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
 
Theoremeltg2 13693* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴))))
 
Theoremeltg2b 13694* Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴)))
 
Theoremeltg4i 13695 An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐴 ∈ (topGen‘𝐵) → 𝐴 = (𝐵 ∩ 𝒫 𝐴))
 
Theoremeltg3i 13696 The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
((𝐵𝑉𝐴𝐵) → 𝐴 ∈ (topGen‘𝐵))
 
Theoremeltg3 13697* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Jim Kingdon, 4-Mar-2023.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥𝐵𝐴 = 𝑥)))
 
Theoremtgval3 13698* Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 12717 and tgval2 13691. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
(𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ ∃𝑦(𝑦𝐵𝑥 = 𝑦)})
 
Theoremtg1 13699 Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
(𝐴 ∈ (topGen‘𝐵) → 𝐴 𝐵)
 
Theoremtg2 13700* Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
((𝐴 ∈ (topGen‘𝐵) ∧ 𝐶𝐴) → ∃𝑥𝐵 (𝐶𝑥𝑥𝐴))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-14973
  Copyright terms: Public domain < Previous  Next >