HomeHome Metamath Proof Explorer
Theorem List (p. 230 of 314)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21444)
  Hilbert Space Explorer  Hilbert Space Explorer
(21445-22967)
  Users' Mathboxes  Users' Mathboxes
(22968-31305)
 

Theorem List for Metamath Proof Explorer - 22901-23000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcvp 22901 The Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  (
 ( A  i^i  B )  =  0H  <->  A  <oH  ( A 
 vH  B ) ) )
 
Theorematnssm0 22902 The meet of a Hilbert lattice element and an incomparable atom is the zero subspace. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( -.  B  C_  A  <->  ( A  i^i  B )  =  0H )
 )
 
Theorematnemeq0 22903 The meet of distinct atoms is the zero subspace. (Contributed by NM, 25-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e. HAtoms  /\  B  e. HAtoms )  ->  ( A  =/=  B  <->  ( A  i^i  B )  =  0H )
 )
 
Theorematssma 22904 The meet with an atom's superset is the atom. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e. HAtoms  /\  B  e.  CH )  ->  ( A  C_  B  <->  ( A  i^i  B )  e. HAtoms ) )
 
Theorematcv0eq 22905 Two atoms covering the zero subspace are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e. HAtoms  /\  B  e. HAtoms )  ->  ( 0H  <oH  ( A  vH  B )  <->  A  =  B )
 )
 
Theorematcv1 22906 Two atoms covering the zero subspace are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e. HAtoms  /\  C  e. HAtoms )  /\  A  <oH  ( B  vH  C ) )  ->  ( A  =  0H  <->  B  =  C ) )
 
Theorematexch 22907 The Hilbert lattice satisfies the atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem related to vector analysis was originally proved by Hermann Grassmann in 1862. Also Definition 3.4-3(b) in [MegPav2000] p. 2345 (PDF p. 8) (use atnemeq0 22903 to obtain atom inequality). (Contributed by NM, 27-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms ) 
 ->  ( ( B  C_  ( A  vH  C ) 
 /\  ( A  i^i  B )  =  0H )  ->  C  C_  ( A  vH  B ) ) )
 
Theorematomli 22908 An assertion holding in atomic orthomodular lattices that is equivalent to the exchange axiom. Proposition 3.2.17 of [PtakPulmannova] p. 66. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( B  e. HAtoms  ->  ( ( A  vH  B )  i^i  ( _|_ `  A ) )  e.  (HAtoms  u. 
 { 0H } )
 )
 
Theorematoml2i 22909 An assertion holding in atomic orthomodular lattices that is equivalent to the exchange axiom. Proposition P8(ii) of [BeltramettiCassinelli1] p. 400. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( B  e. HAtoms  /\ 
 -.  B  C_  A )  ->  ( ( A 
 vH  B )  i^i  ( _|_ `  A ) )  e. HAtoms )
 
Theorematordi 22910 An ordering law for a Hilbert lattice atom and a commuting subspace. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( B  e. HAtoms  /\  A  C_H  B ) 
 ->  ( B  C_  A  \/  B  C_  ( _|_ `  A ) ) )
 
Theorematcvatlem 22911 Lemma for atcvati 22912. (Contributed by NM, 27-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( ( B  e. HAtoms  /\  C  e. HAtoms )  /\  ( A  =/=  0H  /\  A  C.  ( B 
 vH  C ) ) )  ->  ( -.  B  C_  A  ->  A  e. HAtoms ) )
 
Theorematcvati 22912 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (Contributed by NM, 28-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( B  e. HAtoms  /\  C  e. HAtoms )  ->  (
 ( A  =/=  0H  /\  A  C.  ( B 
 vH  C ) ) 
 ->  A  e. HAtoms ) )
 
Theorematcvat2i 22913 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( B  e. HAtoms  /\  C  e. HAtoms )  ->  (
 ( -.  B  =  C  /\  A  <oH  ( B 
 vH  C ) ) 
 ->  A  e. HAtoms ) )
 
Theorematord 22914 An ordering law for a Hilbert lattice atom and a commuting subspace. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms  /\  A  C_H  B )  ->  ( B  C_  A  \/  B  C_  ( _|_ `  A )
 ) )
 
Theorematcvat2 22915 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (Contributed by NM, 29-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms ) 
 ->  ( ( -.  B  =  C  /\  A  <oH  ( B  vH  C ) )  ->  A  e. HAtoms ) )
 
15.9.54  Irreducibility
 
Theoremchirredlem1 22916* Lemma for chirredi 22920. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( ( p  e. HAtoms  /\  ( q  e. 
 CH  /\  q  C_  ( _|_ `  A ) ) )  /\  ( ( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q ) ) )  ->  ( p  i^i  ( _|_ `  r
 ) )  =  0H )
 
Theoremchirredlem2 22917* Lemma for chirredi 22920. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e.  CH  /\  q  C_  ( _|_ `  A )
 ) )  /\  (
 ( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
 ) ) )  ->  ( ( _|_ `  r
 )  i^i  ( p  vH  q ) )  =  q )
 
Theoremchirredlem3 22918* Lemma for chirredi 22920. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  ( x  e. 
 CH  ->  A  C_H  x )   =>    |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. HAtoms  /\  q  C_  ( _|_ `  A ) ) )  /\  ( r  e. HAtoms  /\  r  C_  ( p  vH  q ) ) )  ->  ( r  C_  A  ->  r  =  p ) )
 
Theoremchirredlem4 22919* Lemma for chirredi 22920. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  ( x  e. 
 CH  ->  A  C_H  x )   =>    |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. HAtoms  /\  q  C_  ( _|_ `  A ) ) )  /\  ( r  e. HAtoms  /\  r  C_  ( p  vH  q ) ) )  ->  ( r  =  p  \/  r  =  q ) )
 
Theoremchirredi 22920* 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.)
 |-  A  e.  CH   &    |-  ( x  e. 
 CH  ->  A  C_H  x )   =>    |-  ( A  =  0H  \/  A  =  ~H )
 
Theoremchirred 22921* 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.)
 |-  (
 ( A  e.  CH  /\ 
 A. x  e.  CH  A  C_H  x )  ->  ( A  =  0H  \/  A  =  ~H )
 )
 
15.9.55  Atoms (cont.)
 
Theorematcvat3i 22922 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.)
 |-  A  e.  CH   =>    |-  ( ( B  e. HAtoms  /\  C  e. HAtoms )  ->  (
 ( ( -.  B  =  C  /\  -.  C  C_  A )  /\  B  C_  ( A  vH  C ) )  ->  ( A  i^i  ( B  vH  C ) )  e. HAtoms ) )
 
Theorematcvat4i 22923* 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.)
 |-  A  e.  CH   =>    |-  ( ( B  e. HAtoms  /\  C  e. HAtoms )  ->  (
 ( A  =/=  0H  /\  B  C_  ( A  vH  C ) )  ->  E. x  e. HAtoms  ( x 
 C_  A  /\  B  C_  ( C  vH  x ) ) ) )
 
Theorematdmd 22924 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.)
 |-  (
 ( A  e. HAtoms  /\  B  e.  CH )  ->  A  MH* 
 B )
 
Theorematmd 22925 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.)
 |-  (
 ( A  e. HAtoms  /\  B  e.  CH )  ->  A  MH  B )
 
Theorematmd2 22926 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.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  A  MH  B )
 
Theorematabsi 22927 Absorption of an incomparable atom. Similar to Exercise 7.1 of [MaedaMaeda] p. 34. (Contributed by NM, 15-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( C  e. HAtoms  ->  ( -.  C  C_  ( A  vH  B )  ->  (
 ( A  vH  C )  i^i  B )  =  ( A  i^i  B ) ) )
 
Theorematabs2i 22928 Absorption of an incomparable atom. (Contributed by NM, 18-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( C  e. HAtoms  ->  ( -.  C  C_  ( A  vH  B )  ->  (
 ( A  vH  C )  i^i  ( A  vH  B ) )  =  A ) )
 
15.9.56  Modular symmetry
 
Theoremmdsymlem1 22929* Lemma for mdsymi 22937. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  =  ( A  vH  p )   =>    |-  ( ( ( p  e.  CH  /\  ( B  i^i  C )  C_  A )  /\  ( B 
 MH*  A  /\  p  C_  ( A  vH  B ) ) )  ->  p  C_  A )
 
Theoremmdsymlem2 22930* Lemma for mdsymi 22937. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  =  ( A  vH  p )   =>    |-  ( ( ( p  e. HAtoms  /\  ( B  i^i  C )  C_  A )  /\  ( B  MH*  A  /\  p  C_  ( A 
 vH  B ) ) )  ->  ( B  =/=  0H  ->  E. r  e. HAtoms  E. q  e. HAtoms  ( p  C_  ( q  vH  r )  /\  ( q 
 C_  A  /\  r  C_  B ) ) ) )
 
Theoremmdsymlem3 22931* Lemma for mdsymi 22937. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  =  ( A  vH  p )   =>    |-  ( ( ( ( p  e. HAtoms  /\  -.  ( B  i^i  C )  C_  A )  /\  p  C_  ( A  vH  B ) )  /\  A  =/=  0H )  ->  E. r  e. HAtoms  E. q  e. HAtoms  ( p  C_  ( q  vH  r )  /\  ( q 
 C_  A  /\  r  C_  B ) ) )
 
Theoremmdsymlem4 22932* Lemma for mdsymi 22937. This is the forward direction of Lemma 4(i) of [Maeda] p. 168. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  =  ( A  vH  p )   =>    |-  ( p  e. HAtoms  ->  ( ( B  MH*  A  /\  ( ( A  =/=  0H 
 /\  B  =/=  0H )  /\  p  C_  ( A  vH  B ) ) )  ->  E. q  e. HAtoms  E. r  e. HAtoms  ( p  C_  ( q  vH  r )  /\  ( q 
 C_  A  /\  r  C_  B ) ) ) )
 
Theoremmdsymlem5 22933* Lemma for mdsymi 22937. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  =  ( A  vH  p )   =>    |-  ( ( q  e. HAtoms  /\  r  e. HAtoms )  ->  ( -.  q  =  p 
 ->  ( ( p  C_  ( q  vH  r ) 
 /\  ( q  C_  A  /\  r  C_  B ) )  ->  ( ( ( c  e.  CH  /\  A  C_  c )  /\  p  e. HAtoms )  ->  ( p  C_  c  ->  p  C_  ( ( c  i^i  B )  vH  A ) ) ) ) ) )
 
Theoremmdsymlem6 22934* Lemma for mdsymi 22937. 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.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  =  ( A  vH  p )   =>    |-  ( A. p  e. HAtoms  ( p  C_  ( A 
 vH  B )  ->  E. q  e. HAtoms  E. r  e. HAtoms  ( p  C_  (
 q  vH  r )  /\  ( q  C_  A  /\  r  C_  B ) ) )  ->  B  MH* 
 A )
 
Theoremmdsymlem7 22935* Lemma for mdsymi 22937. 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.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  =  ( A  vH  p )   =>    |-  ( ( A  =/=  0H 
 /\  B  =/=  0H )  ->  ( B  MH*  A  <->  A. p  e. HAtoms  ( p 
 C_  ( A  vH  B )  ->  E. q  e. HAtoms  E. r  e. HAtoms  ( p  C_  ( q  vH  r )  /\  ( q 
 C_  A  /\  r  C_  B ) ) ) ) )
 
Theoremmdsymlem8 22936* Lemma for mdsymi 22937. Lemma 4(ii) of [Maeda] p. 168. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  =  ( A  vH  p )   =>    |-  ( ( A  =/=  0H 
 /\  B  =/=  0H )  ->  ( B  MH*  A  <->  A  MH*  B ) )
 
Theoremmdsymi 22937 M-symmetry of the Hilbert lattice. Lemma 5 of [Maeda] p. 168. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH  B  <->  B  MH  A )
 
Theoremmdsym 22938 M-symmetry of the Hilbert lattice. Lemma 5 of [Maeda] p. 168. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  B  MH  A ) )
 
Theoremdmdsym 22939 Dual M-symmetry of the Hilbert lattice. (Contributed by NM, 25-Jul-2007.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  B 
 MH*  A ) )
 
Theorematdmd2 22940 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.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  A  MH* 
 B )
 
Theoremsumdmdii 22941 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.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( A  +H  B )  =  ( A  vH  B )  ->  A  MH* 
 B )
 
Theoremcmmdi 22942 Commuting subspaces form a modular pair. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  ->  A  MH  B )
 
Theoremcmdmdi 22943 Commuting subspaces form a dual modular pair. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  ->  A  MH* 
 B )
 
Theoremsumdmdlem 22944 Lemma for sumdmdi 22946. The span of vector  C not in the subspace sum is "trimmed off." (Contributed by NM, 18-Dec-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( C  e.  ~H  /\ 
 -.  C  e.  ( A  +H  B ) ) 
 ->  ( ( B  +H  ( span `  { C }
 ) )  i^i  A )  =  ( B  i^i  A ) )
 
Theoremsumdmdlem2 22945* Lemma for sumdmdi 22946. (Contributed by NM, 23-Dec-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A. x  e. HAtoms  ( ( x  vH  B )  i^i  ( A  vH  B ) )  C_  ( ( ( x 
 vH  B )  i^i 
 A )  vH  B )  ->  ( A  +H  B )  =  ( A  vH  B ) )
 
Theoremsumdmdi 22946 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.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( A  +H  B )  =  ( A  vH  B )  <->  A  MH*  B )
 
Theoremdmdbr4ati 22947* Dual modular pair property in terms of atoms. (Contributed by NM, 15-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH*  B  <->  A. x  e. HAtoms  (
 ( x  vH  B )  i^i  ( A  vH  B ) )  C_  ( ( ( x 
 vH  B )  i^i 
 A )  vH  B ) )
 
Theoremdmdbr5ati 22948* Dual modular pair property in terms of atoms. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH*  B  <->  A. x  e. HAtoms  ( x  C_  ( A  vH  B )  ->  x  C_  ( ( ( x 
 vH  B )  i^i 
 A )  vH  B ) ) )
 
Theoremdmdbr6ati 22949* 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.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH*  B  <->  A. x  e. HAtoms  (
 ( A  vH  B )  i^i  x )  =  ( ( ( ( x  vH  B )  i^i  A )  vH  B )  i^i  x ) )
 
Theoremdmdbr7ati 22950* Dual modular pair property in terms of atoms. (Contributed by NM, 18-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH*  B  <->  A. x  e. HAtoms  (
 ( A  vH  B )  i^i  x )  C_  ( ( ( x 
 vH  B )  i^i 
 A )  vH  B ) )
 
Theoremmdoc1i 22951 Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  A  MH  ( _|_ `  A )
 
Theoremmdoc2i 22952 Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( _|_ `  A )  MH  A
 
Theoremdmdoc1i 22953 Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  A  MH*  ( _|_ `  A )
 
Theoremdmdoc2i 22954 Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( _|_ `  A )  MH*  A
 
Theoremmdcompli 22955 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.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH  B  <->  ( A  i^i  ( _|_ `  ( A  i^i  B ) ) )  MH  ( B  i^i  ( _|_ `  ( A  i^i  B ) ) ) )
 
Theoremdmdcompli 22956 A condition equivalent to the dual modular pair property. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH*  B  <->  ( A  i^i  ( _|_ `  ( A  i^i  B ) ) ) 
 MH*  ( B  i^i  ( _|_ `  ( A  i^i  B ) ) ) )
 
Theoremmddmdin0i 22957* 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.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  A. x  e.  CH  A. y  e. 
 CH  ( ( x 
 MH*  y  /\  ( x  i^i  y )  =  0H )  ->  x  MH  y )   =>    |-  ( A  MH*  B  ->  A  MH  B )
 
Theoremcdjreui 22958* 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.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  (
 ( C  e.  ( A  +H  B )  /\  ( A  i^i  B )  =  0H )  ->  E! x  e.  A  E. y  e.  B  C  =  ( x  +h  y ) )
 
Theoremcdj1i 22959* Two ways to express " A and  B are completely disjoint subspaces." (1) => (2) in Lemma 5 of [Holland] p. 1520. (Contributed by NM, 21-May-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( E. w  e.  RR  ( 0  <  w  /\  A. y  e.  A  A. v  e.  B  ( ( normh `  y )  +  ( normh `  v )
 )  <_  ( w  x.  ( normh `  ( y  +h  v ) ) ) )  ->  E. x  e.  RR  ( 0  < 
 x  /\  A. y  e.  A  A. z  e.  B  ( ( normh `  y )  =  1 
 ->  x  <_  ( normh `  ( y  -h  z
 ) ) ) ) )
 
Theoremcdj3lem1 22960* A property of " A and  B are completely disjoint subspaces." Part of Lemma 5 of [Holland] p. 1520. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( E. x  e.  RR  ( 0  <  x  /\  A. y  e.  A  A. z  e.  B  ( ( normh `  y )  +  ( normh `  z )
 )  <_  ( x  x.  ( normh `  ( y  +h  z ) ) ) )  ->  ( A  i^i  B )  =  0H )
 
Theoremcdj3lem2 22961* Lemma for cdj3i 22967. Value of the first-component function  S. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  S  =  ( x  e.  ( A  +H  B )  |->  (
 iota_ z  e.  A E. w  e.  B  x  =  ( z  +h  w ) ) )   =>    |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  ->  ( S `  ( C  +h  D ) )  =  C )
 
Theoremcdj3lem2a 22962* Lemma for cdj3i 22967. Closure of the first-component function  S. (Contributed by NM, 25-May-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  S  =  ( x  e.  ( A  +H  B )  |->  (
 iota_ z  e.  A E. w  e.  B  x  =  ( z  +h  w ) ) )   =>    |-  ( ( C  e.  ( A  +H  B ) 
 /\  ( A  i^i  B )  =  0H )  ->  ( S `  C )  e.  A )
 
Theoremcdj3lem2b 22963* Lemma for cdj3i 22967. The first-component function  S is bounded if the subspaces are completely disjoint. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  S  =  ( x  e.  ( A  +H  B )  |->  (
 iota_ z  e.  A E. w  e.  B  x  =  ( z  +h  w ) ) )   =>    |-  ( E. v  e.  RR  ( 0  <  v  /\  A. x  e.  A  A. y  e.  B  ( ( normh `  x )  +  ( normh `  y )
 )  <_  ( v  x.  ( normh `  ( x  +h  y ) ) ) )  ->  E. v  e.  RR  ( 0  < 
 v  /\  A. u  e.  ( A  +H  B ) ( normh `  ( S `  u ) ) 
 <_  ( v  x.  ( normh `  u ) ) ) )
 
Theoremcdj3lem3 22964* Lemma for cdj3i 22967. Value of the second-component function  T. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  T  =  ( x  e.  ( A  +H  B )  |->  (
 iota_ w  e.  B E. z  e.  A  x  =  ( z  +h  w ) ) )   =>    |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  ->  ( T `  ( C  +h  D ) )  =  D )
 
Theoremcdj3lem3a 22965* Lemma for cdj3i 22967. Closure of the second-component function  T. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  T  =  ( x  e.  ( A  +H  B )  |->  (
 iota_ w  e.  B E. z  e.  A  x  =  ( z  +h  w ) ) )   =>    |-  ( ( C  e.  ( A  +H  B ) 
 /\  ( A  i^i  B )  =  0H )  ->  ( T `  C )  e.  B )
 
Theoremcdj3lem3b 22966* Lemma for cdj3i 22967. The second-component function  T is bounded if the subspaces are completely disjoint. (Contributed by NM, 31-May-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  T  =  ( x  e.  ( A  +H  B )  |->  (
 iota_ w  e.  B E. z  e.  A  x  =  ( z  +h  w ) ) )   =>    |-  ( E. v  e.  RR  ( 0  <  v  /\  A. x  e.  A  A. y  e.  B  ( ( normh `  x )  +  ( normh `  y )
 )  <_  ( v  x.  ( normh `  ( x  +h  y ) ) ) )  ->  E. v  e.  RR  ( 0  < 
 v  /\  A. u  e.  ( A  +H  B ) ( normh `  ( T `  u ) ) 
 <_  ( v  x.  ( normh `  u ) ) ) )
 
Theoremcdj3i 22967* Two ways to express " A and  B are completely disjoint subspaces." (1) <=> (3) in Lemma 5 of [Holland] p. 1520. (Contributed by NM, 1-Jun-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  S  =  ( x  e.  ( A  +H  B )  |->  (
 iota_ z  e.  A E. w  e.  B  x  =  ( z  +h  w ) ) )   &    |-  T  =  ( x  e.  ( A  +H  B )  |->  ( iota_ w  e.  B E. z  e.  A  x  =  ( z  +h  w ) ) )   &    |-  ( ph  <->  E. v  e.  RR  ( 0  <  v  /\  A. u  e.  ( A  +H  B ) (
 normh `  ( S `  u ) )  <_  ( v  x.  ( normh `  u ) ) ) )   &    |-  ( ps  <->  E. v  e.  RR  ( 0  <  v  /\  A. u  e.  ( A  +H  B ) (
 normh `  ( T `  u ) )  <_  ( v  x.  ( normh `  u ) ) ) )   =>    |-  ( E. v  e. 
 RR  ( 0  < 
 v  /\  A. x  e.  A  A. y  e.  B  ( ( normh `  x )  +  ( normh `  y ) ) 
 <_  ( v  x.  ( normh `  ( x  +h  y ) ) ) )  <->  ( ( A  i^i  B )  =  0H  /\  ph  /\  ps ) )
 
PART 16  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
 
16.1  Mathboxes for user contributions
 
16.1.1  Mathbox guidelines
 
Theoremmathbox 22968 (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 9378.

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.)

 |-  x  =  x
 
16.2  Mathbox for Stefan Allan
 
Theoremfoo3 22969 A theorem about the universal class. (Contributed by Stefan Allan, 9-Dec-2008.)
 |-  ph   =>    |- 
 _V  =  { x  |  ph }
 
Theoremxfree 22970 A partial converse to 19.9t 1761. (Contributed by Stefan Allan, 21-Dec-2008.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  ( A. x ( ph  ->  A. x ph )  <->  A. x ( E. x ph  ->  ph ) )
 
Theoremxfree2 22971 A partial converse to 19.9t 1761. (Contributed by Stefan Allan, 21-Dec-2008.)
 |-  ( A. x ( ph  ->  A. x ph )  <->  A. x ( -.  ph  ->  A. x  -.  ph ) )
 
TheoremaddltmulALT 22972 A proof readability experiment for addltmul 9900. (Contributed by Stefan Allan, 30-Oct-2010.) (Proof modification is discouraged.)
 |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 2  <  A  /\  2  <  B ) )  ->  ( A  +  B )  < 
 ( A  x.  B ) )
 
16.3  Mathbox for Thierry Arnoux
 
Theoremmptcnv 22973* The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017.)
 |-  ( ph  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( y  e.  C  /\  x  =  D ) ) )   =>    |-  ( ph  ->  `' ( x  e.  A  |->  B )  =  ( y  e.  C  |->  D ) )
 
Theoremreximddv 22974* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 7-Dec-2016.)
 |-  (
 ( ph  /\  ( x  e.  A  /\  ps ) )  ->  ch )   &    |-  ( ph  ->  E. x  e.  A  ps )   =>    |-  ( ph  ->  E. x  e.  A  ch )
 
Theoreminfi 22975 If  A is finite,  ( A  i^i  B ) is finite. (Contributed by Thierry Arnoux, 17-Apr-2017.)
 |-  ( A  e.  Fin  ->  ( A  i^i  B )  e. 
 Fin )
 
Theoremfzspl 22976 Split the last element of a finite set of sequential integers. (more generic than fzsuc 10787) (Contributed by Thierry Arnoux, 7-Nov-2016.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  ( M
 ... N )  =  ( ( M ... ( N  -  1
 ) )  u.  { N } ) )
 
Theoremfzsplit3 22977 Split a finite interval of integers into two parts. (Contributed by Thierry Arnoux, 2-May-2017.)
 |-  ( K  e.  ( M ... N )  ->  ( M ... N )  =  ( ( M ... ( K  -  1
 ) )  u.  ( K ... N ) ) )
 
Theorembcm1n 22978 The proportion of one binomial coefficient to another with  N decreased by 1. (Contributed by Thierry Arnoux, 9-Nov-2016.)
 |-  (
 ( K  e.  (
 0 ... ( N  -  1 ) )  /\  N  e.  NN )  ->  ( ( ( N  -  1 )  _C  K )  /  ( N  _C  K ) )  =  ( ( N  -  K )  /  N ) )
 
Theoremltesubnnd 22979 Subtracting an integer number from another number decreases it. See ltsubrpd 10371 (Contributed by Thierry Arnoux, 18-Apr-2017.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  (
 ( M  +  1 )  -  N ) 
 <_  M )
 
Theoremifeqeqx 22980* An equality theorem tailored for ballotlemsf1o 23019 (Contributed by Thierry Arnoux, 14-Apr-2017.)
 |-  ( x  =  X  ->  A  =  C )   &    |-  ( x  =  Y  ->  B  =  a )   &    |-  ( x  =  X  ->  ( ch  <->  th ) )   &    |-  ( x  =  Y  ->  ( ch  <->  ps ) )   &    |-  ( ph  ->  a  =  C )   &    |-  ( ( ph  /\  ps )  ->  th )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  e.  W )   =>    |-  ( ( ph  /\  x  =  if ( ps ,  X ,  Y )
 )  ->  a  =  if ( ch ,  A ,  B ) )
 
Theoremfl_cmpfun 22981 Functionality of a class given by a "maps to" notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |- 
 Fun  F
 
Theoremfdmrn 22982 A different way to write  F is a function. (Contributed by Thierry Arnoux, 7-Dec-2016.)
 |-  ( Fun  F  <->  F : dom  F --> ran  F )
 
Theoremnvof1o 22983 An involution is a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)
 |-  (
 ( F  Fn  A  /\  `' F  =  F )  ->  F : A -1-1-onto-> A )
 
Theoremf1o3d 22984* Describe an implicit one-to-one onto function. (Contributed by Thierry Arnoux, 23-Apr-2017.)
 |-  ( ph  ->  F  =  ( x  e.  A  |->  C ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  B )   &    |-  ( ( ph  /\  y  e.  B )  ->  D  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  ( x  =  D  <->  y  =  C ) )   =>    |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  ( y  e.  B  |->  D ) ) )
 
Theoremrinvf1o 22985 Sufficient conditions for the restriction of an involution to be a bijection (Contributed by Thierry Arnoux, 7-Dec-2016.)
 |-  Fun  F   &    |-  `' F  =  F   &    |-  ( F " A )  C_  B   &    |-  ( F " B )  C_  A   &    |-  A  C_  dom  F   &    |-  B  C_ 
 dom  F   =>    |-  ( F  |`  A ) : A -1-1-onto-> B
 
Theoremelabreximd 22986* Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
 |-  F/ x ph   &    |-  F/ x ch   &    |-  ( A  =  B  ->  ( ch  <->  ps ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  C )  ->  ps )   =>    |-  (
 ( ph  /\  A  e.  { y  |  E. x  e.  C  y  =  B } )  ->  ch )
 
Theoremabrexss 22987* A necessary condition for an image set to be a subset. (Contributed by Thierry Arnoux, 6-Feb-2017.)
 |-  F/_ x C   =>    |-  ( A. x  e.  A  B  e.  C  ->  { y  |  E. x  e.  A  y  =  B }  C_  C )
 
Theoremdfimafnf 22988* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Thierry Arnoux, 24-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x F   =>    |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( F " A )  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
 
Theoremfunimass4f 22989 Membership relation for the values of a function whose image is a subclass. (Contributed by Thierry Arnoux, 24-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x F   =>    |-  ( ( Fun 
 F  /\  A  C_  dom  F )  ->  ( ( F
 " A )  C_  B 
 <-> 
 A. x  e.  A  ( F `  x )  e.  B ) )
 
Theoremaddeq0 22990 Two complex which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 2-May-2017.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  =  0  <->  A  =  -u B ) )
 
16.3.1  Bertrand's Ballot Problem
 
Theoremballotlemoex 22991*  O is a set. (Contributed by Thierry Arnoux, 7-Dec-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   =>    |-  O  e.  _V
 
Theoremballotlem1 22992* The size of the universe is a binomial coefficient. (Contributed by Thierry Arnoux, 23-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   =>    |-  ( # `  O )  =  ( ( M  +  N )  _C  M )
 
Theoremballotlemelo 22993* Elementhood in  O. (Contributed by Thierry Arnoux, 17-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   =>    |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N ) )  /\  ( # `  C )  =  M ) )
 
Theoremballotlem2 22994* The probability that the first vote picked in a count is a B (Contributed by Thierry Arnoux, 23-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   =>    |-  ( P `  { c  e.  O  |  -.  1  e.  c } )  =  ( N  /  ( M  +  N ) )
 
Theoremballotlemfval 22995* The value of F. (Contributed by Thierry Arnoux, 23-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  ( ph  ->  C  e.  O )   &    |-  ( ph  ->  J  e.  ZZ )   =>    |-  ( ph  ->  (
 ( F `  C ) `  J )  =  ( ( # `  (
 ( 1 ... J )  i^i  C ) )  -  ( # `  (
 ( 1 ... J )  \  C ) ) ) )
 
Theoremballotlemfelz 22996*  ( F `  C ) has values in  ZZ. (Contributed by Thierry Arnoux, 23-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  ( ph  ->  C  e.  O )   &    |-  ( ph  ->  J  e.  ZZ )   =>    |-  ( ph  ->  (
 ( F `  C ) `  J )  e. 
 ZZ )
 
Theoremballotlemfp1 22997* If the  J th ballot is for A,  ( F `  C ) goes up 1. If the  J th ballot is for B,  ( F `  C ) goes down 1. (Contributed by Thierry Arnoux, 24-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  ( ph  ->  C  e.  O )   &    |-  ( ph  ->  J  e.  NN )   =>    |-  ( ph  ->  (
 ( -.  J  e.  C  ->  ( ( F `
  C ) `  J )  =  (
 ( ( F `  C ) `  ( J  -  1 ) )  -  1 ) ) 
 /\  ( J  e.  C  ->  ( ( F `
  C ) `  J )  =  (
 ( ( F `  C ) `  ( J  -  1 ) )  +  1 ) ) ) )
 
Theoremballotlemfc0 22998*  F takes value 0 between negative and positive values. (Contributed by Thierry Arnoux, 24-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  ( ph  ->  C  e.  O )   &    |-  ( ph  ->  J  e.  NN )   &    |-  ( ph  ->  E. i  e.  ( 1 ... J ) ( ( F `
  C ) `  i )  <_  0 )   &    |-  ( ph  ->  0  <  ( ( F `  C ) `  J ) )   =>    |-  ( ph  ->  E. k  e.  ( 1 ... J ) ( ( F `
  C ) `  k )  =  0
 )
 
Theoremballotlemfcc 22999*  F takes value 0 between positive and negative values. (Contributed by Thierry Arnoux, 2-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  ( ph  ->  C  e.  O )   &    |-  ( ph  ->  J  e.  NN )   &    |-  ( ph  ->  E. i  e.  ( 1 ... J ) 0  <_  (
 ( F `  C ) `  i ) )   &    |-  ( ph  ->  ( ( F `  C ) `  J )  <  0 )   =>    |-  ( ph  ->  E. k  e.  ( 1 ... J ) ( ( F `
  C ) `  k )  =  0
 )
 
Theoremballotlemfmpn 23000*  ( F `  C ) finishes counting at  ( M  -  N ). (Contributed by Thierry Arnoux, 25-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   =>    |-  ( C  e.  O  ->  ( ( F `  C ) `  ( M  +  N )
 )  =  ( M  -  N ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31305
  Copyright terms: Public domain < Previous  Next >