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Theorem List for Metamath Proof Explorer - 19301-19400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcmscmet 19301 The induced metric on a complete normed group is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)
CMetSp

Theorembncmet 19302 The induced metric on Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
Ban

Theoremcmsms 19303 A complete metric space is a metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
CMetSp

Theoremcmspropd 19304 Property deduction for a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
CMetSp CMetSp

Theoremcmsss 19305 The restriction of a complete metric space is complete iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.)
s                      CMetSp CMetSp

Theoremlssbn 19306 A subspace of a Banach space is a Banach space iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.)
s                      Ban Ban

Theoremcmetcusp1OLD 19307 If the uniform set of a complete metric space is the uniform structure generated by its metric, then it is a complete uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.)
UnifSt       CMetSp metUnifOLD CUnifSp

Theoremcmetcusp1 19308 If the uniform set of a complete metric space is the uniform structure generated by its metric, then it is a complete uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.)
UnifSt       CMetSp metUnif CUnifSp

TheoremcmetcuspOLD 19309 The uniform space generated by a complete metric is a complete uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
toUnifSpmetUnifOLD CUnifSp

Theoremcmetcusp 19310 The uniform space generated by a complete metric is a complete uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
toUnifSpmetUnif CUnifSp

Theoremcncms 19311 The field of complex numbers is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
fld CMetSp

Theoremcnflduss 19312 The uniform structure of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
UnifStfld       metUnif

Theoremcnfldcusp 19313 The field of complex numbers is a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.)
fld CUnifSp

Theoremresscdrg 19314 The real numbers are a subset of any complete subfield in the complexes. (Contributed by Mario Carneiro, 15-Oct-2015.)
flds        SubRingfld CMetSp

Theoremcncdrg 19315 The only complete subfields of the complexes are and . (Contributed by Mario Carneiro, 15-Oct-2015.)
flds        SubRingfld CMetSp

Theoremsrabn 19316 The subring algebra over a complete normed ring is a Banach space iff the subring is a closed division ring. (Contributed by Mario Carneiro, 15-Oct-2015.)
subringAlg               NrmRing CMetSp SubRing Ban s

Theoremrlmbn 19317 The ring module over a complete normed division ring is a Banach space. (Contributed by Mario Carneiro, 15-Oct-2015.)
NrmRing CMetSp ringLMod Ban

Theoremishl 19318 The predicate "is a complex Hilbert space." A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
Ban

Theoremhlbn 19319 Every complex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
Ban

Theoremhlcph 19320 Every complex Hilbert space is a complex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremhlphl 19321 Every complex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)

Theoremhlcms 19322 Every complex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.)
CMetSp

Theoremhlprlem 19323 Lemma for hlpr 19325. (Contributed by Mario Carneiro, 15-Oct-2015.)
Scalar              SubRingfld flds flds CMetSp

Theoremhlress 19324 The scalar field of a complex Hilbert space contains . (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar

Theoremhlpr 19325 The scalar field of a complex Hilbert space is either or . (Contributed by Mario Carneiro, 15-Oct-2015.)
Scalar

Theoremishl2 19326 A Hilbert space is a complete complex pre-Hilbert space over or . (Contributed by Mario Carneiro, 15-Oct-2015.)
Scalar              CMetSp

11.5.6  Minimizing Vector Theorem

Theoremminveclem1 19327* Lemma for minvec 19339. The set of all distances from points of to are a nonempty set of nonnegative reals. (Contributed by Mario Carneiro, 8-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem4c 19328* Lemma for minvec 19339. The infimum of the distances to is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem2 19329* Lemma for minvec 19339. Any two points and in are close to each other if they are close to the infimum of distance to . (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem3a 19330* Lemma for minvec 19339. is a complete metric when restricted to . (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem3b 19331* Lemma for minvec 19339. The set of vectors within a fixed distance of the infimum forms a filter base. (Contributed by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem3 19332* Lemma for minvec 19339. The filter formed by taking elements successively closer to the infimum is Cauchy. (Contributed by Mario Carneiro, 8-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp                                                 CauFil

Theoremminveclem4a 19333* Lemma for minvec 19339. converges to a point in . (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem4b 19334* Lemma for minvec 19339. The convergent point of the Cauchy sequence is a member of the base space. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem4 19335* Lemma for minvec 19339. The convergent point of the Cauchy sequence attains the minimum distance, and so is closer to than any other point in . (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem5 19336* Lemma for minvec 19339. Discharge the assumptions in minveclem4 19335. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem6 19337* Lemma for minvec 19339. Any minimal point is less than away from . (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem7 19338* Lemma for minvec 19339. Since any two minimal points are distance zero away from each other, the minimal point is unique. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminvec 19339* Minimizing vector theorem, or the Hilbert projection theorem. There is exactly one vector in a complete subspace that minimizes the distance to an arbitrary vector in a parent inner product space. Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

11.5.7  Projection Theorem

Theorempjthlem1 19340* Lemma for pjth 19342. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 17-Oct-2015.)

Theorempjthlem2 19341 Lemma for pjth 19342. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.)

Theorempjth 19342 Projection Theorem: Any Hilbert space vector can be decomposed uniquely into a member of a closed subspace and a member of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.)

Theorempjth2 19343 Projection Theorem with abbreviations: A topologically closed subspace is a projection subspace. (Contributed by Mario Carneiro, 17-Oct-2015.)

Theoremcldcss 19344 Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.)

Theoremcldcss2 19345 Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.)

Theoremhlhil 19346 Corollary of the Projection Theorem: A complex Hilbert space is a Hilbert space (in the algebraic sense, meaning that all algebraically closed subspaces have a projection decomposition). (Contributed by Mario Carneiro, 17-Oct-2015.)

PART 12  BASIC REAL AND COMPLEX ANALYSIS

12.1  Continuity

12.1.1  Intermediate value theorem

Theorempmltpclem1 19347* Lemma for pmltpc 19349. (Contributed by Mario Carneiro, 1-Jul-2014.)

Theorempmltpclem2 19348* Lemma for pmltpc 19349. (Contributed by Mario Carneiro, 1-Jul-2014.)

Theorempmltpc 19349* Any function on the reals is either increasing, decreasing, or has a triple of points in a vee formation. (This theorem was created on demand by Mario Carneiro for the 6PCM conference in Bialystok, 1-Jul-2014.) (Contributed by Mario Carneiro, 1-Jul-2014.)

Theoremivthlem1 19350* Lemma for ivth 19353. The set of all values with less than is lower bounded by and upper bounded by . (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremivthlem2 19351* Lemma for ivth 19353. Show that the supremum of cannot be less than . If it was, continuity of implies that there are points just above the supremum that are also less than , a contradiction. (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremivthlem3 19352* Lemma for ivth 19353, the intermediate value theorem. Show that cannot be greater than , and so establish the existence of a root of the function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 17-Jun-2014.)

Theoremivth 19353* The intermediate value theorem, increasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Proof shortened by Mario Carneiro, 30-Apr-2014.)

Theoremivth2 19354* The intermediate value theorem, decreasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremivthle 19355* The intermediate value theorem with weak inequality, increasing case. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremivthle2 19356* The intermediate value theorem with weak inequality, decreasing case. (Contributed by Mario Carneiro, 12-May-2014.)

Theoremivthicc 19357* The interval between any two points of a continuous real function is contained in the range of the function. Equivalently, the range of a continuous real function is convex. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremevthicc 19358* Specialization of the Extreme Value Theorem to a closed interval of . (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremevthicc2 19359* Combine ivthicc 19357 with evthicc 19358 to exactly describe the image of a closed interval. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremcniccbdd 19360* A continuous function on a closed interval is bounded. (Contributed by Mario Carneiro, 7-Sep-2014.)

12.2  Integrals

12.2.1  Lebesgue measure

Syntaxcovol 19361 Extend class notation with the outer Lebesgue measure.

Syntaxcvol 19362 Extend class notation with the Lebesgue measure.

Definitiondf-ovol 19363* Define the outer Lebesgue measure for subsets of the reals. Here is a function from the natural numbers to pairs with , and the outer volume of the set is the infimum over all such functions such that the union of the open intervals covers of the sum of . (Contributed by Mario Carneiro, 16-Mar-2014.)

Definitiondf-vol 19364* Define the Lebesgue measure, which is just the outer measure with a peculiar domain of definition. The property of being Lebesgue-measurable can be expressed as . (Contributed by Mario Carneiro, 17-Mar-2014.)

Theoremovolfcl 19365 Closure for the interval endpoint function. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolfioo 19366* Unpack the interval covering property of the outer measure definition. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolficc 19367* Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolficcss 19368 Any (closed) interval covering is a subset of the reals. (Contributed by Mario Carneiro, 24-Mar-2015.)

Theoremovolfsval 19369 The value of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolfsf 19370 Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolsf 19371 Closure for the partial sums of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolval 19372* The value of the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremelovolm 19373* Elementhood in the set of approximations to the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremelovolmr 19374* Sufficient condition for elementhood in the set . (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolmge0 19375* The set is composed of nonnegative extended real numbers. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolcl 19376 The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovollb 19377 The outer volume is a lower bound on the sum of all interval coverings of . (Contributed by Mario Carneiro, 15-Jun-2014.)

Theoremovolgelb 19378* The outer volume is the greatest lower bound on the sum of all interval coverings of . (Contributed by Mario Carneiro, 15-Jun-2014.)

Theoremovolge0 19379 The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolf 19380 The domain and range of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovollecl 19381 If an outer volume is bounded above, then it is real. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremovolsslem 19382* Lemma for ovolss 19383. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolss 19383 The volume of a set is monotone with respect to set inclusion. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremovolsscl 19384 If a set is contained in another of bounded measure, it too is bounded. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremovolssnul 19385 A subset of a nullset is null. (Contributed by Mario Carneiro, 19-Mar-2014.)

Theoremovollb2lem 19386* Lemma for ovollb2 19387. (Contributed by Mario Carneiro, 24-Mar-2015.)

Theoremovollb2 19387 It is often more convenient to do calculations with *closed* coverings rather than open ones; here we show that it makes no difference (compare ovollb 19377). (Contributed by Mario Carneiro, 24-Mar-2015.)

Theoremovolctb 19388 The volume of a denumerable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)

Theoremovolq 19389 The rational numbers have 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)

Theoremovolctb2 19390 The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.)

Theoremovol0 19391 The empty set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 17-Mar-2014.)

Theoremovolfi 19392 A finite set has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 13-Aug-2014.)

Theoremovolsn 19393 A singleton has 0 outer Lebesgue measure. (Contributed by Mario Carneiro, 15-Aug-2014.)

Theoremovolunlem1a 19394* Lemma for ovolun 19397. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremovolunlem1 19395* Lemma for ovolun 19397. (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremovolunlem2 19396 Lemma for ovolun 19397. (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremovolun 19397 The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 19403, this does not require any choice principles.) (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremovolunnul 19398 Adding a nullset does not change the measure of a set. (Contributed by Mario Carneiro, 25-Mar-2015.)

Theoremovolfiniun 19399* The Lebesgue outer measure function is finitely sub-additive. Finite sum version. (Contributed by Mario Carneiro, 19-Jun-2014.)

Theoremovoliunlem1 19400* Lemma for ovoliun 19403. (Contributed by Mario Carneiro, 12-Jun-2014.)

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