HomeHome Intuitionistic Logic Explorer
Theorem List (p. 136 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 - 13501-13600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlspsnel5a 13501 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.)
𝑆 = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    &   (πœ‘ β†’ 𝑋 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ (π‘β€˜{𝑋}) βŠ† π‘ˆ)
 
Theoremlspprid1 13502 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ 𝑋 ∈ (π‘β€˜{𝑋, π‘Œ}))
 
Theoremlspprid2 13503 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ π‘Œ ∈ (π‘β€˜{𝑋, π‘Œ}))
 
Theoremlspprvacl 13504 The sum of two vectors belongs to their span. (Contributed by NM, 20-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ∈ (π‘β€˜{𝑋, π‘Œ}))
 
Theoremlssats2 13505* A way to express atomisticity (a subspace is the union of its atoms). (Contributed by NM, 3-Feb-2015.)
𝑆 = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    β‡’   (πœ‘ β†’ π‘ˆ = βˆͺ π‘₯ ∈ π‘ˆ (π‘β€˜{π‘₯}))
 
Theoremlspsneli 13506 A scalar product with a vector belongs to the span of its singleton. (Contributed by NM, 2-Jul-2014.)
𝑉 = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝐴 ∈ 𝐾)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝐴 Β· 𝑋) ∈ (π‘β€˜{𝑋}))
 
Theoremlspsn 13507* Span of the singleton of a vector. (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   π‘‰ = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{𝑋}) = {𝑣 ∣ βˆƒπ‘˜ ∈ 𝐾 𝑣 = (π‘˜ Β· 𝑋)})
 
Theoremlspsnel 13508* Member of span of the singleton of a vector. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   π‘‰ = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘ˆ ∈ (π‘β€˜{𝑋}) ↔ βˆƒπ‘˜ ∈ 𝐾 π‘ˆ = (π‘˜ Β· 𝑋)))
 
Theoremlspsnvsi 13509 Span of a scalar product of a singleton. (Contributed by NM, 23-Apr-2014.) (Proof shortened by Mario Carneiro, 4-Sep-2014.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   π‘‰ = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{(𝑅 Β· 𝑋)}) βŠ† (π‘β€˜{𝑋}))
 
Theoremlspsnss2 13510* Comparable spans of singletons must have proportional vectors. (Contributed by NM, 7-Jun-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜π‘†)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ ((π‘β€˜{𝑋}) βŠ† (π‘β€˜{π‘Œ}) ↔ βˆƒπ‘˜ ∈ 𝐾 𝑋 = (π‘˜ Β· π‘Œ)))
 
Theoremlspsnneg 13511 Negation does not change the span of a singleton. (Contributed by NM, 24-Apr-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘€ = (invgβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{(π‘€β€˜π‘‹)}) = (π‘β€˜{𝑋}))
 
Theoremlspsnsub 13512 Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ (π‘β€˜{(𝑋 βˆ’ π‘Œ)}) = (π‘β€˜{(π‘Œ βˆ’ 𝑋)}))
 
Theoremlspsn0 13513 Span of the singleton of the zero vector. (Contributed by NM, 15-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ (π‘β€˜{ 0 }) = { 0 })
 
Theoremlsp0 13514 Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.)
0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ (π‘β€˜βˆ…) = { 0 })
 
Theoremlspuni0 13515 Union of the span of the empty set. (Contributed by NM, 14-Mar-2015.)
0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ βˆͺ (π‘β€˜βˆ…) = 0 )
 
Theoremlspun0 13516 The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 βŠ† 𝑉)    β‡’   (πœ‘ β†’ (π‘β€˜(𝑋 βˆͺ { 0 })) = (π‘β€˜π‘‹))
 
Theoremlspsneq0 13517 Span of the singleton is the zero subspace iff the vector is zero. (Contributed by NM, 27-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ ((π‘β€˜{𝑋}) = { 0 } ↔ 𝑋 = 0 ))
 
Theoremlspsneq0b 13518 Equal singleton spans imply both arguments are zero or both are nonzero. (Contributed by NM, 21-Mar-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ (𝑋 = 0 ↔ π‘Œ = 0 ))
 
Theoremlmodindp1 13519 Two independent (non-colinear) vectors have nonzero sum. (Contributed by NM, 22-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) β‰  0 )
 
Theoremlsslsp 13520 Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) TODO: Shouldn't we swap π‘€β€˜πΊ and π‘β€˜πΊ since we are computing a property of π‘β€˜πΊ? (Like we say sin 0 = 0 and not 0 = sin 0.) - NM 15-Mar-2015.
𝑋 = (π‘Š β†Ύs π‘ˆ)    &   π‘€ = (LSpanβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘‹)    &   πΏ = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝐿 ∧ 𝐺 βŠ† π‘ˆ) β†’ (π‘€β€˜πΊ) = (π‘β€˜πΊ))
 
Theoremlss0v 13521 The zero vector in a submodule equals the zero vector in the including module. (Contributed by NM, 15-Mar-2015.)
𝑋 = (π‘Š β†Ύs π‘ˆ)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (0gβ€˜π‘‹)    &   πΏ = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝐿) β†’ 𝑍 = 0 )
 
Theoremlsspropdg 13522* If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   (πœ‘ β†’ 𝐡 βŠ† π‘Š)    &   ((πœ‘ ∧ (π‘₯ ∈ π‘Š ∧ 𝑦 ∈ π‘Š)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) ∈ π‘Š)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))    &   (πœ‘ β†’ 𝑃 = (Baseβ€˜(Scalarβ€˜πΎ)))    &   (πœ‘ β†’ 𝑃 = (Baseβ€˜(Scalarβ€˜πΏ)))    &   (πœ‘ β†’ 𝐾 ∈ 𝑋)    &   (πœ‘ β†’ 𝐿 ∈ π‘Œ)    β‡’   (πœ‘ β†’ (LSubSpβ€˜πΎ) = (LSubSpβ€˜πΏ))
 
Theoremlsppropd 13523* If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 24-Apr-2024.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   (πœ‘ β†’ 𝐡 βŠ† π‘Š)    &   ((πœ‘ ∧ (π‘₯ ∈ π‘Š ∧ 𝑦 ∈ π‘Š)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) ∈ π‘Š)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))    &   (πœ‘ β†’ 𝑃 = (Baseβ€˜(Scalarβ€˜πΎ)))    &   (πœ‘ β†’ 𝑃 = (Baseβ€˜(Scalarβ€˜πΏ)))    &   (πœ‘ β†’ 𝐾 ∈ 𝑋)    &   (πœ‘ β†’ 𝐿 ∈ π‘Œ)    β‡’   (πœ‘ β†’ (LSpanβ€˜πΎ) = (LSpanβ€˜πΏ))
 
7.6  The complex numbers as an algebraic extensible structure
 
7.6.1  Definition and basic properties
 
Syntaxcpsmet 13524 Extend class notation with the class of all pseudometric spaces.
class PsMet
 
Syntaxcxmet 13525 Extend class notation with the class of all extended metric spaces.
class ∞Met
 
Syntaxcmet 13526 Extend class notation with the class of all metrics.
class Met
 
Syntaxcbl 13527 Extend class notation with the metric space ball function.
class ball
 
Syntaxcfbas 13528 Extend class definition to include the class of filter bases.
class fBas
 
Syntaxcfg 13529 Extend class definition to include the filter generating function.
class filGen
 
Syntaxcmopn 13530 Extend class notation with a function mapping each metric space to the family of its open sets.
class MetOpen
 
Syntaxcmetu 13531 Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
class metUnif
 
Definitiondf-psmet 13532* Define the set of all pseudometrics on a given base set. In a pseudo metric, two distinct points may have a distance zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet = (π‘₯ ∈ V ↦ {𝑑 ∈ (ℝ* β†‘π‘š (π‘₯ Γ— π‘₯)) ∣ βˆ€π‘¦ ∈ π‘₯ ((𝑦𝑑𝑦) = 0 ∧ βˆ€π‘§ ∈ π‘₯ βˆ€π‘€ ∈ π‘₯ (𝑦𝑑𝑧) ≀ ((𝑀𝑑𝑦) +𝑒 (𝑀𝑑𝑧)))})
 
Definitiondf-xmet 13533* Define the set of all extended metrics on a given base set. The definition is similar to df-met 13534, but we also allow the metric to take on the value +∞. (Contributed by Mario Carneiro, 20-Aug-2015.)
∞Met = (π‘₯ ∈ V ↦ {𝑑 ∈ (ℝ* β†‘π‘š (π‘₯ Γ— π‘₯)) ∣ βˆ€π‘¦ ∈ π‘₯ βˆ€π‘§ ∈ π‘₯ (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ βˆ€π‘€ ∈ π‘₯ (𝑦𝑑𝑧) ≀ ((𝑀𝑑𝑦) +𝑒 (𝑀𝑑𝑧)))})
 
Definitiondf-met 13534* Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric. However, we will often also call the metric itself a "metric space".) Equivalent to Definition 14-1.1 of [Gleason] p. 223. (Contributed by NM, 25-Aug-2006.)
Met = (π‘₯ ∈ V ↦ {𝑑 ∈ (ℝ β†‘π‘š (π‘₯ Γ— π‘₯)) ∣ βˆ€π‘¦ ∈ π‘₯ βˆ€π‘§ ∈ π‘₯ (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ βˆ€π‘€ ∈ π‘₯ (𝑦𝑑𝑧) ≀ ((𝑀𝑑𝑦) + (𝑀𝑑𝑧)))})
 
Definitiondf-bl 13535* Define the metric space ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.)
ball = (𝑑 ∈ V ↦ (π‘₯ ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (π‘₯𝑑𝑦) < 𝑧}))
 
Definitiondf-mopn 13536 Define a function whose value is the family of open sets of a metric space. (Contributed by NM, 1-Sep-2006.)
MetOpen = (𝑑 ∈ βˆͺ ran ∞Met ↦ (topGenβ€˜ran (ballβ€˜π‘‘)))
 
Definitiondf-fbas 13537* Define the class of all filter bases. Note that a filter base on one set is also a filter base for any superset, so there is not a unique base set that can be recovered. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
fBas = (𝑀 ∈ V ↦ {π‘₯ ∈ 𝒫 𝒫 𝑀 ∣ (π‘₯ β‰  βˆ… ∧ βˆ… βˆ‰ π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ βˆ€π‘§ ∈ π‘₯ (π‘₯ ∩ 𝒫 (𝑦 ∩ 𝑧)) β‰  βˆ…)})
 
Definitiondf-fg 13538* Define the filter generating function. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
filGen = (𝑀 ∈ V, π‘₯ ∈ (fBasβ€˜π‘€) ↦ {𝑦 ∈ 𝒫 𝑀 ∣ (π‘₯ ∩ 𝒫 𝑦) β‰  βˆ…})
 
Definitiondf-metu 13539* Define the function mapping metrics to the uniform structure generated by that metric. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
metUnif = (𝑑 ∈ βˆͺ ran PsMet ↦ ((dom dom 𝑑 Γ— dom dom 𝑑)filGenran (π‘Ž ∈ ℝ+ ↦ (◑𝑑 β€œ (0[,)π‘Ž)))))
 
Syntaxccnfld 13540 Extend class notation with the field of complex numbers.
class β„‚fld
 
Definitiondf-icnfld 13541 The field of complex numbers. Other number fields and rings can be constructed by applying the β†Ύs restriction operator.

The contract of this set is defined entirely by cnfldex 13543, cnfldadd 13545, cnfldmul 13546, cnfldcj 13547, and cnfldbas 13544.

We may add additional members to this in the future.

At least for now, this structure does not include a topology, order, a distance function, or function mapping metrics.

(Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Thierry Arnoux, 15-Dec-2017.) (New usage is discouraged.)

β„‚fld = ({⟨(Baseβ€˜ndx), β„‚βŸ©, ⟨(+gβ€˜ndx), + ⟩, ⟨(.rβ€˜ndx), Β· ⟩} βˆͺ {⟨(*π‘Ÿβ€˜ndx), βˆ—βŸ©})
 
Theoremcnfldstr 13542 The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
β„‚fld Struct ⟨1, 13⟩
 
Theoremcnfldex 13543 The field of complex numbers is a set. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
β„‚fld ∈ V
 
Theoremcnfldbas 13544 The base set 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.)
β„‚ = (Baseβ€˜β„‚fld)
 
Theoremcnfldadd 13545 The addition 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.)
+ = (+gβ€˜β„‚fld)
 
Theoremcnfldmul 13546 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.)
Β· = (.rβ€˜β„‚fld)
 
Theoremcnfldcj 13547 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.)
βˆ— = (*π‘Ÿβ€˜β„‚fld)
 
Theoremcncrng 13548 The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.)
β„‚fld ∈ CRing
 
Theoremcnring 13549 The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
β„‚fld ∈ Ring
 
Theoremcnfld0 13550 Zero is the zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
0 = (0gβ€˜β„‚fld)
 
Theoremcnfld1 13551 One is the unity element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
1 = (1rβ€˜β„‚fld)
 
Theoremcnfldneg 13552 The additive inverse in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
(𝑋 ∈ β„‚ β†’ ((invgβ€˜β„‚fld)β€˜π‘‹) = -𝑋)
 
Theoremcnfldplusf 13553 The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.)
+ = (+π‘“β€˜β„‚fld)
 
Theoremcnfldsub 13554 The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.)
βˆ’ = (-gβ€˜β„‚fld)
 
Theoremcnfldmulg 13555 The group multiple function in the field of complex numbers. (Contributed by Mario Carneiro, 14-Jun-2015.)
((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„‚) β†’ (𝐴(.gβ€˜β„‚fld)𝐡) = (𝐴 Β· 𝐡))
 
Theoremcnfldexp 13556 The exponentiation operator in the field of complex numbers (for nonnegative exponents). (Contributed by Mario Carneiro, 15-Jun-2015.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„•0) β†’ (𝐡(.gβ€˜(mulGrpβ€˜β„‚fld))𝐴) = (𝐴↑𝐡))
 
Theoremcnsubmlem 13557* Lemma for nn0subm 13562 and friends. (Contributed by Mario Carneiro, 18-Jun-2015.)
(π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ β„‚)    &   ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (π‘₯ + 𝑦) ∈ 𝐴)    &   0 ∈ 𝐴    β‡’   π΄ ∈ (SubMndβ€˜β„‚fld)
 
Theoremcnsubglem 13558* Lemma for cnsubrglem 13559 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.)
(π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ β„‚)    &   ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (π‘₯ + 𝑦) ∈ 𝐴)    &   (π‘₯ ∈ 𝐴 β†’ -π‘₯ ∈ 𝐴)    &   π΅ ∈ 𝐴    β‡’   π΄ ∈ (SubGrpβ€˜β„‚fld)
 
Theoremcnsubrglem 13559* Lemma for zsubrg 13560 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.)
(π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ β„‚)    &   ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (π‘₯ + 𝑦) ∈ 𝐴)    &   (π‘₯ ∈ 𝐴 β†’ -π‘₯ ∈ 𝐴)    &   1 ∈ 𝐴    &   ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (π‘₯ Β· 𝑦) ∈ 𝐴)    β‡’   π΄ ∈ (SubRingβ€˜β„‚fld)
 
Theoremzsubrg 13560 The integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
β„€ ∈ (SubRingβ€˜β„‚fld)
 
Theoremgzsubrg 13561 The gaussian integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
β„€[i] ∈ (SubRingβ€˜β„‚fld)
 
Theoremnn0subm 13562 The nonnegative integers form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 18-Jun-2015.)
β„•0 ∈ (SubMndβ€˜β„‚fld)
 
Theoremrege0subm 13563 The nonnegative reals form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 20-Jun-2015.)
(0[,)+∞) ∈ (SubMndβ€˜β„‚fld)
 
Theoremzsssubrg 13564 The integers are a subset of any subring of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝑅 ∈ (SubRingβ€˜β„‚fld) β†’ β„€ βŠ† 𝑅)
 
7.6.2  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 (β„‚fld β†Ύs β„€), the field of complex numbers restricted to the integers. In zringring 13568 it is shown that this restriction is a ring, and zringbas 13571 shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as Definition df-zring 13566 of the ring of integers.

Remark: Instead of using the symbol "ZZrng" analogous to β„‚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) 13566).

 
Syntaxczring 13565 Extend class notation with the (unital) ring of integers.
class β„€ring
 
Definitiondf-zring 13566 The (unital) ring of integers. (Contributed by Alexander van der Vekens, 9-Jun-2019.)
β„€ring = (β„‚fld β†Ύs β„€)
 
Theoremzringcrng 13567 The ring of integers is a commutative ring. (Contributed by AV, 13-Jun-2019.)
β„€ring ∈ CRing
 
Theoremzringring 13568 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.)
β„€ring ∈ Ring
 
Theoremzringabl 13569 The ring of integers is an (additive) abelian group. (Contributed by AV, 13-Jun-2019.)
β„€ring ∈ Abel
 
Theoremzringgrp 13570 The ring of integers is an (additive) group. (Contributed by AV, 10-Jun-2019.)
β„€ring ∈ Grp
 
Theoremzringbas 13571 The integers are the base of the ring of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.)
β„€ = (Baseβ€˜β„€ring)
 
Theoremzringplusg 13572 The addition operation of the ring of integers. (Contributed by Thierry Arnoux, 8-Nov-2017.) (Revised by AV, 9-Jun-2019.)
+ = (+gβ€˜β„€ring)
 
Theoremzringmulg 13573 The multiplication (group power) operation of the group of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.)
((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ (𝐴(.gβ€˜β„€ring)𝐡) = (𝐴 Β· 𝐡))
 
Theoremzringmulr 13574 The multiplication operation of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
Β· = (.rβ€˜β„€ring)
 
Theoremzring0 13575 The zero element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
0 = (0gβ€˜β„€ring)
 
Theoremzring1 13576 The unity element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
1 = (1rβ€˜β„€ring)
 
Theoremzringnzr 13577 The ring of integers is a nonzero ring. (Contributed by AV, 18-Apr-2020.)
β„€ring ∈ NzRing
 
Theoremdvdsrzring 13578 Ring divisibility in the ring of integers corresponds to ordinary divisibility in β„€. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.)
βˆ₯ = (βˆ₯rβ€˜β„€ring)
 
Theoremzringinvg 13579 The additive inverse of an element of the ring of integers. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
(𝐴 ∈ β„€ β†’ -𝐴 = ((invgβ€˜β„€ring)β€˜π΄))
 
Theoremzringsubgval 13580 Subtraction in the ring of integers. (Contributed by AV, 3-Aug-2019.)
βˆ’ = (-gβ€˜β„€ring)    β‡’   ((𝑋 ∈ β„€ ∧ π‘Œ ∈ β„€) β†’ (𝑋 βˆ’ π‘Œ) = (𝑋 βˆ’ π‘Œ))
 
Theoremzringmpg 13581 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.)
((mulGrpβ€˜β„‚fld) β†Ύs β„€) = (mulGrpβ€˜β„€ring)
 
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 13582 Syntax for the class of topologies.
class Top
 
Definitiondf-top 13583* Define the class of topologies. It is a proper class. See istopg 13584 and istopfin 13585 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 = {π‘₯ ∣ (βˆ€π‘¦ ∈ 𝒫 π‘₯βˆͺ 𝑦 ∈ π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ βˆ€π‘§ ∈ π‘₯ (𝑦 ∩ 𝑧) ∈ π‘₯)}
 
Theoremistopg 13584* Express the predicate "𝐽 is a topology". See istopfin 13585 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.)

(𝐽 ∈ 𝐴 β†’ (𝐽 ∈ Top ↔ (βˆ€π‘₯(π‘₯ βŠ† 𝐽 β†’ βˆͺ π‘₯ ∈ 𝐽) ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ 𝐽 (π‘₯ ∩ 𝑦) ∈ 𝐽)))
 
Theoremistopfin 13585* Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 13584. 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 β†’ (βˆ€π‘₯(π‘₯ βŠ† 𝐽 β†’ βˆͺ π‘₯ ∈ 𝐽) ∧ βˆ€π‘₯((π‘₯ βŠ† 𝐽 ∧ π‘₯ β‰  βˆ… ∧ π‘₯ ∈ Fin) β†’ ∩ π‘₯ ∈ 𝐽)))
 
Theoremuniopn 13586 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 ∧ 𝐴 βŠ† 𝐽) β†’ βˆͺ 𝐴 ∈ 𝐽)
 
Theoremiunopn 13587* The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
((𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ 𝐽) β†’ βˆͺ π‘₯ ∈ 𝐴 𝐡 ∈ 𝐽)
 
Theoreminopn 13588 The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐡 ∈ 𝐽) β†’ (𝐴 ∩ 𝐡) ∈ 𝐽)
 
Theoremfiinopn 13589 The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
(𝐽 ∈ Top β†’ ((𝐴 βŠ† 𝐽 ∧ 𝐴 β‰  βˆ… ∧ 𝐴 ∈ Fin) β†’ ∩ 𝐴 ∈ 𝐽))
 
Theoremunopn 13590 The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐡 ∈ 𝐽) β†’ (𝐴 βˆͺ 𝐡) ∈ 𝐽)
 
Theorem0opn 13591 The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)
(𝐽 ∈ Top β†’ βˆ… ∈ 𝐽)
 
Theorem0ntop 13592 The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
Β¬ βˆ… ∈ Top
 
Theoremtopopn 13593 The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
𝑋 = βˆͺ 𝐽    β‡’   (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
 
Theoremeltopss 13594 A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) β†’ 𝐴 βŠ† 𝑋)
 
8.1.1.2  Topologies on sets
 
Syntaxctopon 13595 Syntax for the function of topologies on sets.
class TopOn
 
Definitiondf-topon 13596* 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 = βˆͺ 𝑗})
 
Theoremfuntopon 13597 The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
Fun TopOn
 
Theoremistopon 13598 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 ∈ (TopOnβ€˜π΅) ↔ (𝐽 ∈ Top ∧ 𝐡 = βˆͺ 𝐽))
 
Theoremtopontop 13599 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOnβ€˜π΅) β†’ 𝐽 ∈ Top)
 
Theoremtoponuni 13600 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOnβ€˜π΅) β†’ 𝐡 = βˆͺ 𝐽)
    < 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 14901-14917
  Copyright terms: Public domain < Previous  Next >