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Theorem List for Metamath Proof Explorer - 23001-23100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdmdbr4ati 23001* Dual modular pair property in terms of atoms. (Contributed by NM, 15-Jan-2005.) (New usage is discouraged.)
HAtoms

Theoremdmdbr5ati 23002* Dual modular pair property in terms of atoms. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
HAtoms

Theoremdmdbr6ati 23003* Dual modular pair property in terms of atoms. The modular law takes the form of the shearing identity. (Contributed by NM, 18-Jan-2005.) (New usage is discouraged.)
HAtoms

Theoremdmdbr7ati 23004* Dual modular pair property in terms of atoms. (Contributed by NM, 18-Jan-2005.) (New usage is discouraged.)
HAtoms

Theoremmdoc1i 23005 Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremmdoc2i 23006 Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremdmdoc1i 23007 Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremdmdoc2i 23008 Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremmdcompli 23009 A condition equivalent to the modular pair property. Part of proof of Theorem 1.14 of [MaedaMaeda] p. 4. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremdmdcompli 23010 A condition equivalent to the dual modular pair property. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremmddmdin0i 23011* If dual modular implies modular whenever meet is zero, then dual modular implies modular for arbitrary lattice elements. This theorem is needed for the remark after Lemma 7 of [Holland] p. 1524 to hold. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremcdjreui 23012* A member of the sum of disjoint subspaces has a unique decomposition. Part of Lemma 5 of [Holland] p. 1520. (Contributed by NM, 20-May-2005.) (New usage is discouraged.)

Theoremcdj1i 23013* Two ways to express " and are completely disjoint subspaces." (1) => (2) in Lemma 5 of [Holland] p. 1520. (Contributed by NM, 21-May-2005.) (New usage is discouraged.)

Theoremcdj3lem1 23014* A property of " and are completely disjoint subspaces." Part of Lemma 5 of [Holland] p. 1520. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)

Theoremcdj3lem2 23015* Lemma for cdj3i 23021. Value of the first-component function . (Contributed by NM, 23-May-2005.) (New usage is discouraged.)

Theoremcdj3lem2a 23016* Lemma for cdj3i 23021. Closure of the first-component function . (Contributed by NM, 25-May-2005.) (New usage is discouraged.)

Theoremcdj3lem2b 23017* Lemma for cdj3i 23021. The first-component function is bounded if the subspaces are completely disjoint. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)

Theoremcdj3lem3 23018* Lemma for cdj3i 23021. Value of the second-component function . (Contributed by NM, 23-May-2005.) (New usage is discouraged.)

Theoremcdj3lem3a 23019* Lemma for cdj3i 23021. Closure of the second-component function . (Contributed by NM, 26-May-2005.) (New usage is discouraged.)

Theoremcdj3lem3b 23020* Lemma for cdj3i 23021. The second-component function is bounded if the subspaces are completely disjoint. (Contributed by NM, 31-May-2005.) (New usage is discouraged.)

Theoremcdj3i 23021* Two ways to express " and are completely disjoint subspaces." (1) <=> (3) in Lemma 5 of [Holland] p. 1520. (Contributed by NM, 1-Jun-2005.) (New usage is discouraged.)

PART 18  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)

18.1  Mathboxes for user contributions

18.1.1  Mathbox guidelines

Theoremmathbox 23022 (This theorem is a dummy placeholder for these guidelines. The name of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.)

A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of set.mm.

For contributors:

By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of set.mm.

Mathboxes are provided to help keep your work synchronized with changes in set.mm, but they shouldn't be depended on as a permanent archive. If you want to preserve your original contribution, it is your responsibility to keep your own copy of it along with the version of set.mm that works with it.

Guidelines:

1. If at all possible, please use only 0-ary class constants for new definitions, for example as in df-div 9424.

2. Try to follow the style of the rest of set.mm. Each \$p and \$a statement must be immediately preceded with the comment that will be shown on its web page description. The metamath program command "write source set.mm /rewrap" will take care of wrapping comment lines and indentation conventions. All mathbox content will be on public display and should hopefully reflect the overall quality of the web site.

3. Before submitting a revised mathbox, please make sure it verifies against the current set.mm.

4. Mathboxes should be independent i.e. the proofs should verify with all other mathboxes removed. If you need a theorem from another mathbox, that is fine (and encouraged), but let me know so I can move the theorem to the main section. One way avoid undesired accidental use of other mathbox theorems is to develop your mathbox using a modified set.mm that has mathboxes removed.

Notes:

1. I may decide to move some theorems to the main part of set.mm for general use. In that case, an author acknowledgment will be added to the description of the theorem.

2. I may make changes to mathboxes to maintain the overall quality of set.mm. Normally I will let you know if a change might impact what you are working on.

3. If you use theorems from another user's mathbox, I don't provide assurance that they are based on correct or consistent \$a statements. (If you find such a problem, please let me know so it can be corrected.) (Contributed by NM, 20-Feb-2007.) (New usage is discouraged.)

18.2  Mathbox for Stefan Allan

Theoremfoo3 23023 A theorem about the universal class. (Contributed by Stefan Allan, 9-Dec-2008.)

Theoremxfree 23024 A partial converse to 19.9t 1782. (Contributed by Stefan Allan, 21-Dec-2008.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremxfree2 23025 A partial converse to 19.9t 1782. (Contributed by Stefan Allan, 21-Dec-2008.)

TheoremaddltmulALT 23026 A proof readability experiment for addltmul 9947. (Contributed by Stefan Allan, 30-Oct-2010.) (Proof modification is discouraged.)

18.3  Mathbox for Thierry Arnoux

Theoremmptcnv 23027* The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017.)

Theoremreximddv 23028* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Theoreminfi 23029 If is finite, is finite. (Contributed by Thierry Arnoux, 17-Apr-2017.)

Theoremfzspl 23030 Split the last element of a finite set of sequential integers. (more generic than fzsuc 10835) (Contributed by Thierry Arnoux, 7-Nov-2016.)

Theoremfzsplit3 23031 Split a finite interval of integers into two parts. (Contributed by Thierry Arnoux, 2-May-2017.)

Theorembcm1n 23032 The proportion of one binomial coefficient to another with decreased by 1. (Contributed by Thierry Arnoux, 9-Nov-2016.)

Theoremltesubnnd 23033 Subtracting an integer number from another number decreases it. See ltsubrpd 10418 (Contributed by Thierry Arnoux, 18-Apr-2017.)

Theoremifeqeqx 23034* An equality theorem tailored for ballotlemsf1o 23072 (Contributed by Thierry Arnoux, 14-Apr-2017.)

Theoremfdmrn 23035 A different way to write is a function. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Theoremnvof1o 23036 An involution is a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Theoremf1o3d 23037* Describe an implicit one-to-one onto function. (Contributed by Thierry Arnoux, 23-Apr-2017.)

Theoremrinvf1o 23038 Sufficient conditions for the restriction of an involution to be a bijection (Contributed by Thierry Arnoux, 7-Dec-2016.)

Theoremelabreximd 23039* Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)

Theoremabrexss 23040* A necessary condition for an image set to be a subset. (Contributed by Thierry Arnoux, 6-Feb-2017.)

Theoremdfimafnf 23041* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Thierry Arnoux, 24-Apr-2017.)

Theoremfunimass4f 23042 Membership relation for the values of a function whose image is a subclass. (Contributed by Thierry Arnoux, 24-Apr-2017.)

Theoremaddeq0 23043 Two complex which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 2-May-2017.)

18.3.1  Bertrand's Ballot Problem

Theoremballotlemoex 23044* is a set. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Theoremballotlem1 23045* The size of the universe is a binomial coefficient. (Contributed by Thierry Arnoux, 23-Nov-2016.)

Theoremballotlemelo 23046* Elementhood in . (Contributed by Thierry Arnoux, 17-Apr-2017.)

Theoremballotlem2 23047* The probability that the first vote picked in a count is a B (Contributed by Thierry Arnoux, 23-Nov-2016.)

Theoremballotlemfval 23048* The value of F. (Contributed by Thierry Arnoux, 23-Nov-2016.)

Theoremballotlemfelz 23049* has values in . (Contributed by Thierry Arnoux, 23-Nov-2016.)

Theoremballotlemfp1 23050* If the th ballot is for A, goes up 1. If the th ballot is for B, goes down 1. (Contributed by Thierry Arnoux, 24-Nov-2016.)

Theoremballotlemfc0 23051* takes value 0 between negative and positive values. (Contributed by Thierry Arnoux, 24-Nov-2016.)

Theoremballotlemfcc 23052* takes value 0 between positive and negative values. (Contributed by Thierry Arnoux, 2-Apr-2017.)

Theoremballotlemfmpn 23053* finishes counting at . (Contributed by Thierry Arnoux, 25-Nov-2016.)

Theoremballotlemfval0 23054* always starts counting at 0 . (Contributed by Thierry Arnoux, 25-Nov-2016.)

Theoremballotleme 23055* Elements of . (Contributed by Thierry Arnoux, 14-Dec-2016.)

Theoremballotlemodife 23056* Elements of . (Contributed by Thierry Arnoux, 7-Dec-2016.)

Theoremballotlem4 23057* If the first pick is a vote for B, A is not ahead throughout the count (Contributed by Thierry Arnoux, 25-Nov-2016.)

Theoremballotlem5 23058* If A is not ahead throughout, there is a where votes are tied. (Contributed by Thierry Arnoux, 1-Dec-2016.)

Theoremballotlemi 23059* Value of for a given counting . (Contributed by Thierry Arnoux, 1-Dec-2016.)

Theoremballotlemiex 23060* Properties of . (Contributed by Thierry Arnoux, 12-Dec-2016.)

Theoremballotlemi1 23061* The first tie cannot be reached at the first pick. (Contributed by Thierry Arnoux, 12-Mar-2017.)

Theoremballotlemii 23062* The first tie cannot be reached at the first pick. (Contributed by Thierry Arnoux, 4-Apr-2017.)

Theoremballotlemsup 23063* The set of zeroes of satisfies the conditions to have a supremum (Contributed by Thierry Arnoux, 1-Dec-2016.)

Theoremballotlemimin 23064* is the first tie. (Contributed by Thierry Arnoux, 1-Dec-2016.)

Theoremballotlemic 23065* If the first vote is for B, the vote on the first tie is for A. (Contributed by Thierry Arnoux, 1-Dec-2016.)

Theoremballotlem1c 23066* If the first vote is for A, the vote on the first tie is for B. (Contributed by Thierry Arnoux, 4-Apr-2017.)

Theoremballotlemsval 23067* Value of (Contributed by Thierry Arnoux, 12-Apr-2017.)

Theoremballotlemsv 23068* Value of evaluated at for a given counting . (Contributed by Thierry Arnoux, 12-Apr-2017.)

Theoremballotlemsgt1 23069* maps values less than to values greater than 1. (Contributed by Thierry Arnoux, 28-Apr-2017.)

Theoremballotlemsdom 23070* Domain of for a given counting . (Contributed by Thierry Arnoux, 12-Apr-2017.)

Theoremballotlemsel1i 23071* The range is invariant under . (Contributed by Thierry Arnoux, 28-Apr-2017.)

Theoremballotlemsf1o 23072* The defined is a bijection, and an involution. (Contributed by Thierry Arnoux, 14-Apr-2017.)

Theoremballotlemsi 23073* The image by of the first tie pick is the first pick. (Contributed by Thierry Arnoux, 14-Apr-2017.)

Theoremballotlemsima 23074* The image by of an interval before the first pick. (Contributed by Thierry Arnoux, 5-May-2017.)

Theoremballotlemieq 23075* If two countings share the same first tie, they also have the same swap function. (Contributed by Thierry Arnoux, 18-Apr-2017.)

Theoremballotlemrval 23076* Value of . (Contributed by Thierry Arnoux, 14-Apr-2017.)

Theoremballotlemscr 23077* The image of by (Contributed by Thierry Arnoux, 21-Apr-2017.)

Theoremballotlemrv 23078* Value of evaluated at . (Contributed by Thierry Arnoux, 17-Apr-2017.)

Theoremballotlemrv1 23079* Value of before the tie. (Contributed by Thierry Arnoux, 11-Apr-2017.)

Theoremballotlemrv2 23080* Value of after the tie. (Contributed by Thierry Arnoux, 11-Apr-2017.)

Theoremballotlemro 23081* Range of is included in . (Contributed by Thierry Arnoux, 17-Apr-2017.)

Theoremballotlemgval 23082* Expand the value of . (Contributed by Thierry Arnoux, 21-Apr-2017.)

Theoremballotlemgun 23083* A property of the defined operator (Contributed by Thierry Arnoux, 26-Apr-2017.)

Theoremballotlemfg 23084* Express the value of in terms of . (Contributed by Thierry Arnoux, 21-Apr-2017.)

Theoremballotlemfrc 23085* Express the value of in terms of the newly defined . (Contributed by Thierry Arnoux, 21-Apr-2017.)

Theoremballotlemfrci 23086* Reverse counting preserves a tie at the first tie. (Contributed by Thierry Arnoux, 21-Apr-2017.)

Theoremballotlemfrceq 23087* Value of for a reverse counting . (Contributed by Thierry Arnoux, 27-Apr-2017.)

Theoremballotlemfrcn0 23088* Value of for a reversed counting , before the first tie, cannot be zero . (Contributed by Thierry Arnoux, 25-Apr-2017.)

Theoremballotlemrc 23089* Range of . (Contributed by Thierry Arnoux, 19-Apr-2017.)

Theoremballotlemirc 23090* Applying does not change first ties. (Contributed by Thierry Arnoux, 19-Apr-2017.)

Theoremballotlemrinv0 23091* Lemma for ballotlemrinv 23092. (Contributed by Thierry Arnoux, 18-Apr-2017.)

Theoremballotlemrinv 23092* is its own inverse : it is an involution. (Contributed by Thierry Arnoux, 10-Apr-2017.)

Theoremballotlem1ri 23093* When the vote on the first tie is for A, the first vote is also for A on the reverse counting. (Contributed by Thierry Arnoux, 18-Apr-2017.)

Theoremballotlem7 23094* is a bijection between two subsets of : one where a vote for A is picked first, and one where a vote for B is picked first (Contributed by Thierry Arnoux, 12-Dec-2016.)

Theoremballotlem8 23095* There are as many countings with ties starting with a ballot for A as there are starting with a ballot for B. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Theoremballotth 23096* Bertrand's ballot problem : the probability that A is ahead throughout the counting. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Theorem3o1cs 23097 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)

Theorem3o2cs 23098 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)

Theorem3o3cs 23099 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)

18.3.2  Division in the extended real number system

Syntaxcxdiv 23100 Extend class notation to include division of extended reals.
/𝑒

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