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Theorem List for Metamath Proof Explorer - 24301-24400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrrhval 24301 Value of the canonical homormorphism from the real numbers to a complete space. (Contributed by Thierry Arnoux, 2-Nov-2017.)
RRHom CnExtQQHom

Theoremrrhcn 24302 If the topology of is Hausdorff, and is a complete uniform space, then the canonical homomorphism from the real numbers to is continuous. (Contributed by Thierry Arnoux, 17-Jan-2018.)
NrmRing       Mod NrmMod       chr               CUnifSp              UnifSt metUnif       RRHom

Theoremrrhf 24303 If the topology of is Hausdorff, Cauchy sequences have at most one limit, i.e. the canonical homomorphism of into is a function. (Contributed by Thierry Arnoux, 2-Nov-2017.)
NrmRing       Mod NrmMod       chr               CUnifSp              UnifSt metUnif       RRHom

19.3.10.6  Embedding into ` RR* `

Syntaxcxrh 24304 Map the extended real numbers into a complete lattice.
RR*Hom

Definitiondf-xrh 24305* Define an embedding from the extended real number into a complete lattice. (Contributed by Thierry Arnoux, 19-Feb-2018.)
RR*Hom RRHom RRHom RRHom

Theoremxrhval 24306* The value of the embedding from the extended real numbers into a complete lattice. (Contributed by Thierry Arnoux, 19-Feb-2018.)
RRHom                     RR*Hom RRHom

19.3.10.7  Canonical embeddings into ` RR `

Theoremzrhre 24307 The RHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
RHomflds

Theoremqqhre 24308 The QQHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
QQHomflds

Theoremrrhre 24309 The RRHom homomorphism for the real numbers structure is the identity. (Contributed by Thierry Arnoux, 22-Oct-2017.)
RRHomflds

19.3.11  Real and complex functions

19.3.11.1  Logarithm laws generalized to an arbitrary base - logb

Define "log using an arbitrary base" function and then prove some of its properties. This builds on previous work by Stefan O'Rear. Note that logb is generalized to an arbitrary base and arbitrary parameter in , but it doesn't accept infinities as arguments, unlike .

Metamath doesn't care what letters are used to represent classes. Usually classes begin with the letter "A", but here we use "B" and "X" to more clearly distinguish between "base" and "other parameter of log".

There are different ways this could be defined in Metamath. The approach used here is intentionally similar to existing 2-parameter Metamath functions. The way defined here supports two notations, logb and logb where is the base and is the other parameter. An alternative would be to support the notational form logb; that looks a little more like traditional notation, but is different than other 2-parameter functions. It's not obvious to me which is better, so here we try out one way as an experiment. Feedback and help welcome.

Syntaxclogb 24310 Extend class notation to include the logarithm generalized to an arbitrary base.
logb

Definitiondf-logb 24311* Define the logb operator. This is the logarithm generalized to an arbitrary base. It can be used as logb for "log base B of X". In the most common traditional notation, base B is a subscript of "log". You could also use logb, which looks like a less-common notation that some use where the base is a preceding superscript. Note: This definition doesn't prevent bases of 1 or 0; proofs may need to forbid them. (Contributed by David A. Wheeler, 21-Jan-2017.)
logb

Theoremlogbval 24312 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the other operand here. Proof is similar to modval 11193. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.)
logb

Theoremlogb2aval 24313 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used in the 2-argument form logb (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.)
logb

Theoremeldifpr 24314 Membership in a set with two elements removed. Similar to eldifsn 3884 and eldiftp 24315. (Contributed by Mario Carneiro, 18-Jul-2017.)

Theoremeldiftp 24315 Membership in a set with three elements removed. Similar to eldifsn 3884 and eldifpr 24314. (Contributed by David A. Wheeler, 22-Jul-2017.)

Theoremlogeq0im1 24316 if then (Contributed by David A. Wheeler, 22-Jul-2017.)

Theoremlogccne0OLD 24317 log isn't 0 if argument isn't 0 or 1. Unlike logne0 20436, this handles complex numbers. (Contributed by David A. Wheeler, 17-Jul-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremlogccne0 24318 log isn't 0 if argument isn't 0 or 1. Unlike logne0 20436, this handles complex numbers. (Contributed by David A. Wheeler, 17-Jul-2017.)

Theoremlogbcl 24319 General logarithm closure. (Contributed by David A. Wheeler, 17-Jul-2017.)
logb

Theoremlogbid1 24320 General logarithm when base and arg match (Contributed by David A. Wheeler, 22-Jul-2017.)
logb

Theoremrnlogblem 24321 Useful lemma for working with integer logarithm bases (with is a common case, e.g. base 2, base 3 or base 10) (Contributed by Thierry Arnoux, 26-Sep-2017.)

Theoremrnlogbval 24322 Value of the general logarithm with integer base. (Contributed by Thierry Arnoux, 27-Sep-2017.)
logb

Theoremrnlogbcl 24323 Closure of the general logarithm with integer base on positive reals. See logbcl 24319. (Contributed by Thierry Arnoux, 27-Sep-2017.)
logb

Theoremrelogbcl 24324 Closure of the general logarithm with integer base on positive reals. (Contributed by Thierry Arnoux, 27-Sep-2017.)
logb

Theoremlogb1 24325 The natural logarithm of in base . See log1 20419 (Contributed by Thierry Arnoux, 27-Sep-2017.)
logb

Theoremnnlogbexp 24326 Identity law for general logarithm with integer base. (Contributed by Thierry Arnoux, 27-Sep-2017.)
logb

Theoremlogbrec 24327 Logarithm of a reciprocal changes sign. See logrec 20600 (Contributed by Thierry Arnoux, 27-Sep-2017.)
logb logb

Theoremlogblt 24328 The general logarithm function is monotone. See logltb 20433 (Contributed by Thierry Arnoux, 27-Sep-2017.)
logb logb

Theoremlog2le1 24329 is less than . This is just a weaker form of log2ub 20728 when no tight upper bound is required. (Contributed by Thierry Arnoux, 27-Sep-2017.)

19.3.11.2  Indicator Functions

Syntaxcind 24330 Extend class notation with the indicator function generator.
𝟭

Definitiondf-ind 24331* Define the indicator function generator. (Contributed by Thierry Arnoux, 20-Jan-2017.)
𝟭

Theoremindv 24332* Value of the indicator function generator with domain . (Contributed by Thierry Arnoux, 23-Aug-2017.)
𝟭

Theoremindval 24333* Value of the indicator function generator for a set and a domain . (Contributed by Thierry Arnoux, 2-Feb-2017.)
𝟭

Theoremindval2 24334 Alternate value of the indicator function generator. (Contributed by Thierry Arnoux, 2-Feb-2017.)
𝟭

Theoremindf 24335 An indicator function as a function with domain and codomain. (Contributed by Thierry Arnoux, 13-Aug-2017.)
𝟭

Theoremindfval 24336 Value of the indicator function. (Contributed by Thierry Arnoux, 13-Aug-2017.)
𝟭

Theorempr01ssre 24337 The range of the indicator function is a subset of . (Contributed by Thierry Arnoux, 14-Aug-2017.)

Theoremind1 24338 Value of the indicator function where it is . (Contributed by Thierry Arnoux, 14-Aug-2017.)
𝟭

Theoremind0 24339 Value of the indicator function where it is . (Contributed by Thierry Arnoux, 14-Aug-2017.)
𝟭

Theoremind1a 24340 Value of the indicator function where it is . (Contributed by Thierry Arnoux, 22-Aug-2017.)
𝟭

Theoremindpi1 24341 Preimage of the singleton by the indicator function. See i1f1lem 19520. (Contributed by Thierry Arnoux, 21-Aug-2017.)
𝟭

Theoremindsum 24342* Finite sum of a product with the indicator function / cross-product with the indicator function. (Contributed by Thierry Arnoux, 14-Aug-2017.)
𝟭

Theoremindf1o 24343 The bijection between a power set and the set of indicator functions. (Contributed by Thierry Arnoux, 14-Aug-2017.)
𝟭

Theoremindpreima 24344 A function with range as an indicator of the preimage of (Contributed by Thierry Arnoux, 23-Aug-2017.)
𝟭

Theoremindf1ofs 24345* The bijection between finite subsets and the indicator functions with finite support. (Contributed by Thierry Arnoux, 22-Aug-2017.)
𝟭

19.3.11.3  Extended sum

Syntaxcesum 24346 Extend class notation to include infinite summations.
Σ*

Definitiondf-esum 24347 Define a short-hand for the possibly infinite sum over the extended non-negative reals. Σ* is relying on the properties of the tsums, developped by Mario Carneiro. (Contributed by Thierry Arnoux, 21-Sep-2016.)
Σ* s tsums

Theoremesumex 24348 An extended sum is a set by definition. (Contributed by Thierry Arnoux, 5-Sep-2017.)
Σ*

Theoremesumcl 24349* Closure for extended sum in the extended positive reals. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Σ*

Theoremesumeq12dvaf 24350 Equality deduction for extended sum. (Contributed by Thierry Arnoux, 26-Mar-2017.)
Σ* Σ*

Theoremesumeq12dva 24351* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.) (Revised by Thierry Arnoux, 29-Jun-2017.)
Σ* Σ*

Theoremesumeq12d 24352* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
Σ* Σ*

Theoremesumeq1 24353* Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
Σ* Σ*

Theoremesumeq1d 24354 Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
Σ* Σ*

Theoremesumeq2 24355* Equality theorem for extended sum. (Contributed by Thierry Arnoux, 24-Dec-2016.)
Σ* Σ*

Theoremesumeq2d 24356 Equality deduction for extended sum. (Contributed by Thierry Arnoux, 21-Sep-2016.)
Σ* Σ*

Theoremesumeq2dv 24357* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Σ* Σ*

Theoremesumeq2sdv 24358* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Σ* Σ*

Theoremnfesum1 24359 Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
Σ*

Theoremcbvesum 24360* Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)
Σ* Σ*

Theoremcbvesumv 24361* Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)
Σ* Σ*

Theoremesumid 24362 Identify the extended sum as any limit points of the infinite sum. (Contributed by Thierry Arnoux, 9-May-2017.)
s tsums        Σ*

Theoremesumval 24363* Develop the value of the extended sum. (Contributed by Thierry Arnoux, 4-Jan-2017.)
s g        Σ*

Theoremesumel 24364* The extended sum is a limit point of the corresponding infinite group sum. (Contributed by Thierry Arnoux, 24-Mar-2017.)
Σ* s tsums

Theoremesumnul 24365 Extended sum over the empty set. (Contributed by Thierry Arnoux, 19-Feb-2017.)
Σ*

Theoremesum0 24366* Extended sum of zero. (Contributed by Thierry Arnoux, 3-Mar-2017.)
Σ*

Theoremesumf1o 24367* Re-index an extended sum using a bijection. (Contributed by Thierry Arnoux, 6-Apr-2017.)
Σ* Σ*

Theoremesumc 24368* Convert from the collection form to the class-variable form of a sum. (Contributed by Thierry Arnoux, 10-May-2017.)
Σ* Σ*

Theoremesumsplit 24369 Split an extended sum into two parts. (Contributed by Thierry Arnoux, 9-May-2017.)
Σ* Σ* Σ*

Σ* Σ* Σ*

Theoremesumle 24371* If all of the terms of an extended sums compare, so do the sums. (Contributed by Thierry Arnoux, 8-Jun-2017.)
Σ* Σ*

Theoremesummono 24372* Extended sum is monotonic. (Contributed by Thierry Arnoux, 19-Oct-2017.)
Σ* Σ*

Theoremgsumesum 24373* Relate a group sum on s to a finite extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
s g Σ*

Theoremesumlub 24374* The extended sum is the lowest upper bound for the partial sums. (Contributed by Thierry Arnoux, 19-Oct-2017.)
Σ*        Σ*

Σ* Σ* Σ*

Theoremesumlef 24376* If all of the terms of an extended sums compare, so do the sums. (Contributed by Thierry Arnoux, 8-Jun-2017.)
Σ* Σ*

Theoremesumcst 24377* The extended sum of a constant. (Contributed by Thierry Arnoux, 3-Mar-2017.) (Revised by Thierry Arnoux, 5-Jul-2017.)
Σ*

Theoremesumsn 24378* The extended sum of a singleton is the term. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Σ*

Theoremesumpr 24379* Extended sum over a pair. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Σ*

Theoremesumpr2 24380* Extended sum over a pair, with a relaxed condition compared to esumpr 24379. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Σ*

Theoremesumfzf 24381* Formulating a partial extended sum over integers using the recursive sequence builder. (Contributed by Thierry Arnoux, 18-Oct-2017.)
Σ*

Theoremesumfsup 24382 Formulating an extended sum over integers using the recursive sequence builder. (Contributed by Thierry Arnoux, 18-Oct-2017.)
Σ*

Theoremesumfsupre 24383 Formulating an extended sum over integers using the recursive sequence builder. This version is limited to real valued functions. (Contributed by Thierry Arnoux, 19-Oct-2017.)
Σ*

Theoremesumss 24384 Change the index set to a subset by adding zeroes. (Contributed by Thierry Arnoux, 19-Jun-2017.)
Σ* Σ*

Theoremesumpinfval 24385* The value of the extended sum of non-negative terms, with at least one infinite term. (Contributed by Thierry Arnoux, 19-Jun-2017.)
Σ*

Theoremesumpfinvallem 24386 Lemma for esumpfinval 24387 (Contributed by Thierry Arnoux, 28-Jun-2017.)
fld g s g

Theoremesumpfinval 24387* The value of the extended sum of a finite set of non-negative finite terms (Contributed by Thierry Arnoux, 28-Jun-2017.)
Σ*

Theoremesumpfinvalf 24388 Same as esumpfinval 24387, minus distinct variable restrictions. (Contributed by Thierry Arnoux, 28-Aug-2017.)
Σ*

Theoremesumpinfsum 24389* The value of the extended sum of infinitely many terms greater than one. (Contributed by Thierry Arnoux, 29-Jun-2017.)
Σ*

Theoremesumpcvgval 24390* The value of the extended sum when the corresponding series sum is convergent. (Contributed by Thierry Arnoux, 31-Jul-2017.)
Σ*

Theoremesumpmono 24391* The partial sums in an extended sum form a monotonic sequence. (Contributed by Thierry Arnoux, 31-Aug-2017.)
Σ* Σ*

Theoremesumcocn 24392* Lemma for esummulc2 24394 and co. Composing with a continuous function preserves extended sums (Contributed by Thierry Arnoux, 29-Jun-2017.)
ordTop t                                           Σ* Σ*

Theoremesummulc1 24393* An extended sum multiplied by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
Σ* Σ*

Theoremesummulc2 24394* An extended sum multiplied by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
Σ* Σ*

Theoremesumdivc 24395* An extended sum divided by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
Σ* /𝑒 Σ* /𝑒

Theoremhashf2 24396 Lemma for hasheuni 24397 (Contributed by Thierry Arnoux, 19-Nov-2016.)

Theoremhasheuni 24397* The cardinality of a disjoint union, not necessarily finite. cf. hashuni 12545. (Contributed by Thierry Arnoux, 19-Nov-2016.) (Revised by Thierry Arnoux, 2-Jan-2017.) (Revised by Thierry Arnoux, 20-Jun-2017.)
Disj Σ*

Theoremesumcvg 24398* The sequence of partial sums of an extended sum converges to the whole sum. cf. fsumcvg2 12462. (Contributed by Thierry Arnoux, 5-Sep-2017.)
s        Σ*                      Σ*

Theoremesumcvg2 24399* Simpler version of esumcvg 24398. (Contributed by Thierry Arnoux, 5-Sep-2017.)
s                             Σ* Σ*

19.3.12  Mixed Function/Constant operation

Syntaxcofc 24400 Extend class notation to include mapping of an operation to an operation for a function and a constant.
𝑓/𝑐

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