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Theorem List for Metamath Proof Explorer - 22801-22900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremchirredlem2 22801* Lemma for chirredi 22804. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
HAtoms HAtoms

Theoremchirredlem3 22802* Lemma for chirredi 22804. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

Theoremchirredlem4 22803* Lemma for chirredi 22804. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

Theoremchirredi 22804* The Hilbert lattice is irreducible: any element that commutes with all elements must be zero or one. Theorem 14.8.4 of [BeltramettiCassinelli] p. 166. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)

Theoremchirred 22805* The Hilbert lattice is irreducible: any element that commutes with all elements must be zero or one. Theorem 14.8.4 of [BeltramettiCassinelli] p. 166. (Contributed by NM, 16-Jun-2006.) (New usage is discouraged.)

15.9.55  Atoms (cont.)

Theorematcvat3i 22806 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 22807* 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 22808 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 22809 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 22810 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 22811 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 22812 Absorption of an incomparable atom. (Contributed by NM, 18-Jul-2004.) (New usage is discouraged.)
HAtoms

15.9.56  Modular symmetry

Theoremmdsymlem1 22813* Lemma for mdsymi 22821. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)

Theoremmdsymlem2 22814* Lemma for mdsymi 22821. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

Theoremmdsymlem3 22815* Lemma for mdsymi 22821. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

Theoremmdsymlem4 22816* Lemma for mdsymi 22821. 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 22817* Lemma for mdsymi 22821. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
HAtoms HAtoms HAtoms

Theoremmdsymlem6 22818* Lemma for mdsymi 22821. 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 22819* Lemma for mdsymi 22821. 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 22820* Lemma for mdsymi 22821. Lemma 4(ii) of [Maeda] p. 168. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.)

Theoremmdsymi 22821 M-symmetry of the Hilbert lattice. Lemma 5 of [Maeda] p. 168. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.)

Theoremmdsym 22822 M-symmetry of the Hilbert lattice. Lemma 5 of [Maeda] p. 168. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)

Theoremdmdsym 22823 Dual M-symmetry of the Hilbert lattice. (Contributed by NM, 25-Jul-2007.) (New usage is discouraged.)

Theorematdmd2 22824 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 22825 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 22826 Commuting subspaces form a modular pair. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)

Theoremcmdmdi 22827 Commuting subspaces form a dual modular pair. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)

Theoremsumdmdlem 22828 Lemma for sumdmdi 22830. The span of vector not in the subspace sum is "trimmed off." (Contributed by NM, 18-Dec-2004.) (New usage is discouraged.)

Theoremsumdmdlem2 22829* Lemma for sumdmdi 22830. (Contributed by NM, 23-Dec-2004.) (New usage is discouraged.)
HAtoms

Theoremsumdmdi 22830 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 22831* Dual modular pair property in terms of atoms. (Contributed by NM, 15-Jan-2005.) (New usage is discouraged.)
HAtoms

Theoremdmdbr5ati 22832* Dual modular pair property in terms of atoms. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
HAtoms

Theoremdmdbr6ati 22833* 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 22834* Dual modular pair property in terms of atoms. (Contributed by NM, 18-Jan-2005.) (New usage is discouraged.)
HAtoms

Theoremmdoc1i 22835 Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremmdoc2i 22836 Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremdmdoc1i 22837 Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremdmdoc2i 22838 Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremmdcompli 22839 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 22840 A condition equivalent to the dual modular pair property. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremmddmdin0i 22841* 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 22842* 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 22843* 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 22844* 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 22845* Lemma for cdj3i 22851. Value of the first-component function . (Contributed by NM, 23-May-2005.) (New usage is discouraged.)

Theoremcdj3lem2a 22846* Lemma for cdj3i 22851. Closure of the first-component function . (Contributed by NM, 25-May-2005.) (New usage is discouraged.)

Theoremcdj3lem2b 22847* Lemma for cdj3i 22851. The first-component function is bounded if the subspaces are completely disjoint. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)

Theoremcdj3lem3 22848* Lemma for cdj3i 22851. Value of the second-component function . (Contributed by NM, 23-May-2005.) (New usage is discouraged.)

Theoremcdj3lem3a 22849* Lemma for cdj3i 22851. Closure of the second-component function . (Contributed by NM, 26-May-2005.) (New usage is discouraged.)

Theoremcdj3lem3b 22850* Lemma for cdj3i 22851. The second-component function is bounded if the subspaces are completely disjoint. (Contributed by NM, 31-May-2005.) (New usage is discouraged.)

Theoremcdj3i 22851* 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 16  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)

16.1  Mathboxes for user contributions

16.1.1  Mathbox guidelines

Theoremmathbox 22852 (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 9304.

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.)

16.2  Mathbox for Stefan Allan

Theoremfoo3 22853 A theorem about the universal class. (Contributed by Stefan Allan, 9-Dec-2008.)

Theoremxfree 22854 A partial converse to 19.9t 1761. (Contributed by Stefan Allan, 21-Dec-2008.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremxfree2 22855 A partial converse to 19.9t 1761. (Contributed by Stefan Allan, 21-Dec-2008.)

TheoremaddltmulALT 22856 A proof readability experiment for addltmul 9826. (Contributed by Stefan Allan, 30-Oct-2010.) (Proof modification is discouraged.)

16.3  Mathbox for Mario Carneiro

16.3.1  Miscellaneous stuff

Theoremquartfull 22857 The quartic equation, written out in full. This actually makes a fairly good Metamath stress test. Note that the length of this formula could be shortened significantly if the intermediate expressions were expanded and simplified, but it's not like this theorem will be used anyway. (Contributed by Mario Carneiro, 6-May-2015.)
; ; ; ;; ; ; ; ;; ; ; ;;        ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;;        ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;; ; ; ;; ; ; ; ;; ; ; ; ;; ; ; ;;

16.3.2  Zeta function

Syntaxczeta 22858 The Riemann zeta function.

Definitiondf-zeta 22859* Define the Riemann zeta function. This definition uses a series expansion of the alternating zeta function ~? zetaalt that is convergent everywhere except , but going from the alternating zeta function to the regular zeta function requires dividing by , which has zeroes other than . To extract the correct value of the zeta function at these points, we extend the divided alternating zeta function by continuity. (Contributed by Mario Carneiro, 18-Jul-2014.)

Theoremzetacvg 22860* The zeta series is convergent. (Contributed by Mario Carneiro, 18-Jul-2014.)

16.3.3  Gamma function

Syntaxclgam 22861 Logarithm of the Gamma function.

Syntaxcgam 22862 The Gamma function.

Definitiondf-lgam 22863* Define the log-Gamma function. We can work with this form of the gamma function a bit easier than the equivalent expression for the gamma function itself, and moreover this function is not actually equal to because the branch cuts are placed differently (we do have , though). This definition is attributed to Euler, and unlike the usual integral definition is defined on the entire complex plane except the nonpositive integers , where the function has simple poles. (Contributed by Mario Carneiro, 12-Jul-2014.)

Definitiondf-gam 22864 Define the Gamma function. See df-lgam 22863 for more information about the reason for this definition in terms of the log-gamma function. (Contributed by Mario Carneiro, 12-Jul-2014.)

Theoremeldmgm 22865 Elementhood in the set of non-nonpositive integers. (Contributed by Mario Carneiro, 12-Jul-2014.)

Theoremdmgmaddn0 22866 If is not a nonpositive integer, then is nonzero for any nonnegative integer . (Contributed by Mario Carneiro, 12-Jul-2014.)

Theoremdmgmseqn0 22867* If is not a nonpositive integer, then is nonzero for any . (Contributed by Mario Carneiro, 12-Jul-2014.)

16.3.4  Derangements and the Subfactorial

Theoremderanglem 22868* Lemma for derangements. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremderangval 22869* Define the derangement function, which counts the number of bijections from a set to itself such that no element is mapped to itself. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremderangf 22870* The derangement number is a function from finite sets to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremderang0 22871* The derangement number of the empty set. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremderangsn 22872* The derangement number of a singleton. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremderangenlem 22873* One half of derangen 22874. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremderangen 22874* The derangement number is a cardinal invariant, i.e. it only depends on the size of a set and not on its contents. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremsubfacval 22875* The subfactorial is defined as the number of derangements (see derangval 22869) of the set . (Contributed by Mario Carneiro, 21-Jan-2015.)

Theoremderangen2 22876* Write the derangement number in terms of the subfactorial. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremsubfacf 22877* The subfactorial is a function from nonnegative integers to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremsubfaclefac 22878* The subfactorial is less than the factorial. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremsubfac0 22879* The subfactorial at zero. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremsubfac1 22880* The subfactorial at one. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremsubfacp1lem1 22881* Lemma for subfacp1 22888. The set together with partitions the set . (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremsubfacp1lem2a 22882* Lemma for subfacp1 22888. Properties of a bijection on augmented with the two-element flip to get a bijection on . (Contributed by Mario Carneiro, 23-Jan-2015.)