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Theorem List for Metamath Proof Explorer - 23901-24000   *Has distinct variable group(s)
TypeLabelDescription
Statement

18.8.6  Atoms (cont.)

Theorematcvat3i 23901 A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

Theorematcvat4i 23902* A condition implying existence of an atom with the properties shown. Lemma 3.2.20 of [PtakPulmannova] p. 68. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

Theorematdmd 23903 Two Hilbert lattice elements have the dual modular pair property if the first is an atom. Theorem 7.6(c) of [MaedaMaeda] p. 31. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
HAtoms

Theorematmd 23904 Two Hilbert lattice elements have the modular pair property if the first is an atom. Theorem 7.6(b) of [MaedaMaeda] p. 31. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
HAtoms

Theorematmd2 23905 Two Hilbert lattice elements have the dual modular pair property if the second is an atom. Part of Exercise 6 of [Kalmbach] p. 103. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
HAtoms

Theorematabsi 23906 Absorption of an incomparable atom. Similar to Exercise 7.1 of [MaedaMaeda] p. 34. (Contributed by NM, 15-Jul-2004.) (New usage is discouraged.)
HAtoms

Theorematabs2i 23907 Absorption of an incomparable atom. (Contributed by NM, 18-Jul-2004.) (New usage is discouraged.)
HAtoms

18.8.7  Modular symmetry

Theoremmdsymlem1 23908* Lemma for mdsymi 23916. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)

Theoremmdsymlem2 23909* Lemma for mdsymi 23916. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

Theoremmdsymlem3 23910* Lemma for mdsymi 23916. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

Theoremmdsymlem4 23911* Lemma for mdsymi 23916. This is the forward direction of Lemma 4(i) of [Maeda] p. 168. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

Theoremmdsymlem5 23912* Lemma for mdsymi 23916. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

Theoremmdsymlem6 23913* Lemma for mdsymi 23916. This is the converse direction of Lemma 4(i) of [Maeda] p. 168, and is based on the proof of Theorem 1(d) to (e) of [Maeda] p. 167. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

Theoremmdsymlem7 23914* Lemma for mdsymi 23916. Lemma 4(i) of [Maeda] p. 168. Note that Maeda's 1965 definition of dual modular pair has reversed arguments compared to the later (1970) definition given in Remark 29.6 of [MaedaMaeda] p. 130, which is the one that we use. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

Theoremmdsymlem8 23915* Lemma for mdsymi 23916. Lemma 4(ii) of [Maeda] p. 168. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.)

Theoremmdsymi 23916 M-symmetry of the Hilbert lattice. Lemma 5 of [Maeda] p. 168. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.)

Theoremmdsym 23917 M-symmetry of the Hilbert lattice. Lemma 5 of [Maeda] p. 168. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)

Theoremdmdsym 23918 Dual M-symmetry of the Hilbert lattice. (Contributed by NM, 25-Jul-2007.) (New usage is discouraged.)

Theorematdmd2 23919 Two Hilbert lattice elements have the dual modular pair property if the second is an atom. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
HAtoms

Theoremsumdmdii 23920 If the subspace sum of two Hilbert lattice elements is closed, then the elements are a dual modular pair. Remark in [MaedaMaeda] p. 139. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)

Theoremcmmdi 23921 Commuting subspaces form a modular pair. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)

Theoremcmdmdi 23922 Commuting subspaces form a dual modular pair. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)

Theoremsumdmdlem 23923 Lemma for sumdmdi 23925. The span of vector not in the subspace sum is "trimmed off." (Contributed by NM, 18-Dec-2004.) (New usage is discouraged.)

Theoremsumdmdlem2 23924* Lemma for sumdmdi 23925. (Contributed by NM, 23-Dec-2004.) (New usage is discouraged.)
HAtoms

Theoremsumdmdi 23925 The subspace sum of two Hilbert lattice elements is closed iff the elements are a dual modular pair. Theorem 2 of [Holland] p. 1519. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)

Theoremdmdbr4ati 23926* Dual modular pair property in terms of atoms. (Contributed by NM, 15-Jan-2005.) (New usage is discouraged.)
HAtoms

Theoremdmdbr5ati 23927* Dual modular pair property in terms of atoms. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
HAtoms

Theoremdmdbr6ati 23928* 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 23929* Dual modular pair property in terms of atoms. (Contributed by NM, 18-Jan-2005.) (New usage is discouraged.)
HAtoms

Theoremmdoc1i 23930 Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremmdoc2i 23931 Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremdmdoc1i 23932 Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremdmdoc2i 23933 Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremmdcompli 23934 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 23935 A condition equivalent to the dual modular pair property. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremmddmdin0i 23936* 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 23937* 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 23938* 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 23939* 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 23940* Lemma for cdj3i 23946. Value of the first-component function . (Contributed by NM, 23-May-2005.) (New usage is discouraged.)

Theoremcdj3lem2a 23941* Lemma for cdj3i 23946. Closure of the first-component function . (Contributed by NM, 25-May-2005.) (New usage is discouraged.)

Theoremcdj3lem2b 23942* Lemma for cdj3i 23946. The first-component function is bounded if the subspaces are completely disjoint. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)

Theoremcdj3lem3 23943* Lemma for cdj3i 23946. Value of the second-component function . (Contributed by NM, 23-May-2005.) (New usage is discouraged.)

Theoremcdj3lem3a 23944* Lemma for cdj3i 23946. Closure of the second-component function . (Contributed by NM, 26-May-2005.) (New usage is discouraged.)

Theoremcdj3lem3b 23945* Lemma for cdj3i 23946. The second-component function is bounded if the subspaces are completely disjoint. (Contributed by NM, 31-May-2005.) (New usage is discouraged.)

Theoremcdj3i 23946* 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 19  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)

19.1  Mathboxes for user contributions

19.1.1  Mathbox guidelines

Theoremmathbox 23947 (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 9680.

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 website.

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.)

19.2  Mathbox for Stefan Allan

Theoremfoo3 23948 A theorem about the universal class. (Contributed by Stefan Allan, 9-Dec-2008.)

Theoremxfree 23949 A partial converse to 19.9t 1794. (Contributed by Stefan Allan, 21-Dec-2008.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremxfree2 23950 A partial converse to 19.9t 1794. (Contributed by Stefan Allan, 21-Dec-2008.)

TheoremaddltmulALT 23951 A proof readability experiment for addltmul 10205. (Contributed by Stefan Allan, 30-Oct-2010.) (Proof modification is discouraged.)

19.3  Mathbox for Thierry Arnoux

19.3.1  Propositional Calculus - misc additions

Theorembian1d 23952 Adding a superfluous conjunct in a biconditionnal. (Contributed by Thierry Arnoux, 26-Feb-2017.)

Theoremor3di 23953 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)

Theoremor3dir 23954 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)

Theorem3o1cs 23955 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)

Theorem3o2cs 23956 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)

Theorem3o3cs 23957 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)

Theoremadantl3r 23958 Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)

Theoremadantl4r 23959 Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)

Theoremadantl5r 23960 Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)

Theoremadantl6r 23961 Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)

19.3.2  Predicate Calculus

19.3.2.1  Predicate Calculus - misc additions

Theoremabeq2f 23962 Equality of a class variable and a class abstraction. In this version, the fact that is a non-free variable in is explicitely stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.)

Theoremeqvincg 23963* A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.)

19.3.2.2  Restricted quantification - misc additions

Theoremreximddv 23964* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Theoremraleqbid 23965 Equality deduction for restricted universal quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)

Theoremrexeqbid 23966 Equality deduction for restricted existential quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)

Theoremralcom4f 23967* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)

Theoremrexcom4f 23968* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)

Theoremrabid2f 23969 An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)

Theorem19.9d2rf 23970 A deduction version of one direction of 19.9 1798 with two variables (Contributed by Thierry Arnoux, 20-Mar-2017.)

Theorem19.9d2r 23971* A deduction version of one direction of 19.9 1798 with two variables (Contributed by Thierry Arnoux, 30-Jan-2017.)

Theoremr19.41vv 23972* Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. Version with two quantifiers (Contributed by Thierry Arnoux, 25-Jan-2017.)

19.3.2.3  Substitution (without distinct variables) - misc additions

Theoremclelsb3f 23973 Substitution applied to an atomic wff (class version of elsb3 2181). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)

19.3.2.4  Existential "at most one" - misc additions

Theoremmo5f 23974* Alternate definition of "at most one." (Contributed by Thierry Arnoux, 1-Mar-2017.)

Theoremnmo 23975* Negation of "at most one". (Contributed by Thierry Arnoux, 26-Feb-2017.)

Theoremmoimd 23976* "At most one" is preserved through implication (notice wff reversal). (Contributed by Thierry Arnoux, 25-Feb-2017.)

19.3.2.5  Existential uniqueness - misc additions

Theorem2reuswap2 23977* A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.)

Theoremreuxfr3d 23978* Transfer existential uniqueness from a variable to another variable contained in expression . Cf. reuxfr2d 4748 (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)

Theoremreuxfr4d 23979* Transfer existential uniqueness from a variable to another variable contained in expression . Cf. reuxfrd 4750 (Contributed by Thierry Arnoux, 7-Apr-2017.)

Theoremrexunirn 23980* Restricted existential quantification over the union of the range of a function. Cf. rexrn 5874 and eluni2 4021. (Contributed by Thierry Arnoux, 19-Sep-2017.)

19.3.2.6  Restricted "at most one" - misc additions

TheoremrmoxfrdOLD 23981* Transfer "at most one" restricted quantification from a variable to another variable contained in expression . (Contributed by Thierry Arnoux, 7-Apr-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremrmoxfrd 23982* Transfer "at most one" restricted quantification from a variable to another variable contained in expression . (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)

Theoremssrmo 23983 "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)

Theoremrmo3f 23984* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)

Theoremrmo4fOLD 23985* Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremrmo4f 23986* Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)

19.3.3  General Set Theory

19.3.3.1  Class abstractions (a.k.a. class builders)

Theoremceqsexv2d 23987* Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.)

Theoremrabbidva2 23988* Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)

TheoremrabexgfGS 23989 Separation Scheme in terms of a restricted class abstraction. To be removed in profit of Glauco's equivalent version. (Contributed by Thierry Arnoux, 11-May-2017.)

19.3.3.2  Image Sets

Theoremabrexdomjm 23990* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremabrexdom2jm 23991* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremabrexexd 23992* Existence of a class abstraction of existentially restricted sets. (Contributed by Thierry Arnoux, 10-May-2017.)

Theoremelabreximd 23993* Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)

Theoremelabreximdv 23994* Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)

Theoremabrexss 23995* A necessary condition for an image set to be a subset. (Contributed by Thierry Arnoux, 6-Feb-2017.)

19.3.3.3  Set relations and operations - misc additions

Theoremeqri 23996 Infer equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 7-Oct-2017.)

Theoremrabss3d 23997* Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.)

Theoreminin 23998 Intersection with an intersection (Contributed by Thierry Arnoux, 27-Dec-2016.)

Theoremdifneqnul 23999 If the difference of two sets is not empty, then the sets are not equal. (Contributed by Thierry Arnoux, 28-Feb-2017.)

Theoremdifeq 24000 Rewriting an equation with set difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.)

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