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Most recent proofs    These are the 10 (GIF, Unicode) or 1000 (GIF, Unicode) most recent proofs in the set.mm database for the Metamath Proof Explorer (and the Hilbert Space Explorer). The set.mm database is maintained on GitHub with master (stable) and develop (development) versions. This page was created from develop commit 285ac8b, also available here: set.mm (30MB) or set.mm.bz2 (compressed, 8MB).

Other links    Email: Norm Megill.    Mailing list: Metamath Google Group Updated 19-May-2017 .    Syndication: RSS feed (courtesy of Dan Getz).    Related wikis: Wikiproofs (JHilbert) (Recent Changes); Ghilbert site; Ghilbert Google Group.

Recent news items    (7-May-2017) Thierry Arnoux added a new proof to the 100 theorem list, Betrand's Ballot Problem ballotth, bringing the Metamath total to 63.

(20-Apr-2017) Glauco Siliprandi added a new proof in the supplementary list on the 100 theorem list, Stone-Weierstrass Theorem stowei.

(13-Apr-2017) See equidK for a discussion of the recent theorems in my mathbox. -NM

(24-Mar-2017) Alan Sare updated his completeusersproof program.

(28-Feb-2017) David Moews added a new proof to the 100 theorem list, Product of Segments of Chords chordthm, bringing the Metamath total to 62.

(18-Feb-2017) Filip Cernatescu announced Milpgame 0.1 (MILP: Math is like a puzzle!).

(1-Jan-2017) Saveliy Skresanov added a new proof to the 100 theorem list, Isosceles triangle theorem isosctr, bringing the Metamath total to 61.

(1-Jan-2017) Mario Carneiro added 2 new proofs to the 100 theorem list, L'Hôpital's Rule lhop and Taylor's Theorem taylth, bringing the Metamath total to 60.

(28-Dec-2016) David A. Wheeler is putting together a page on Metamath (specifically set.mm) conventions. Comments are welcome on the Google Group thread.

(24-Dec-2016) Mario Carneiro introduced the abbreviation "F/ x ph" (symbols: turned F, x, phi) in df-nf to represent the "effectively not free" idiom "A. x ( ph -> A. x ph )". Theorem nf2 shows a version without nested quantifiers.

(22-Dec-2016) Naip Moro has developed a Metamath database for G. Spencer-Brown's Laws of Form. You can follow the Google Group discussion here.

(20-Dec-2016) In metamath program version 0.137, 'verify markup *' now checks that ax-XXX $a matches axXXX $p when the latter exists, per the discussion at https://groups.google.com/d/msg/metamath/Vtz3CKGmXnI/Fxq3j1I_EQAJ.

(24-Nov-2016) Mingl Yuan has kindly provided a mirror site in Beijing, China. He has also provided an rsync server; type "rsync cn.metamath.org::" in a bash shell to check its status (it should return "metamath metamath").

(14-Aug-2016) All HTML pages on this site should now be mobile-friendly and pass the Mobile-Friendly Test. If you find one that does not, let me know.

(14-Aug-2016) Daniel Whalen wrote a paper describing the use of using deep learning to prove 14% of test theorems taken from set.mm: Holophrasm: a neural Automated Theorem Prover for higher-order logic. The associated program is called Holophrasm.

(14-Aug-2016) David A. Wheeler created a video called Metamath Proof Explorer: A Modern Principia Mathematica

(12-Aug-2016) A Gitter chat room has been created for Metamath.

(9-Aug-2016) Mario Carneiro wrote a Metamath proof verifier in the Scala language as part of the ongoing Metamath -> MMT import project

(9-Aug-2016) David A. Wheeler created a GitHub project called metamath-test (last execution run) to check that different verifiers both pass good databases and detect errors in defective ones.

(4-Aug-2016) Mario gave two presentations at CICM 2016.

(17-Jul-2016) Thierry Arnoux has written EMetamath, a Metamath plugin for the Eclipse IDE.

(16-Jul-2016) Mario recovered Chris Capel's collapsible proof demo.

(13-Jul-2016) FL sent me an updated version of PDF (LaTeX source) developed with Lamport's pf2 package. See the 23-Apr-2012 entry below.

(12-Jul-2016) David A. Wheeler produced a new video for mmj2 called "Creating functions in Metamath". It shows a more efficient approach than his previous recent video "Creating functions in Metamath" (old) but it can be of interest to see both approaches.

(10-Jul-2016) Metamath program version 0.132 changes the command 'show restricted' to 'show discouraged' and adds a new command, 'set discouragement'. See the mmnotes.txt entry of 11-May-2016 (updated 10-Jul-2016).

(12-Jun-2016) Dan Getz has written Metamath.jl, a Metamath proof verifier written in the Julia language.    Older news...

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Last updated on 20-May-2017 at 12:40 AM ET.
Recent Additions to the Metamath Proof Explorer   Notes (last updated 14-May-2017 )
DateLabelDescription
Theorem
 
18-May-2017fsnunfv 5619 Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.)
 |-  ( ( X  e.  V  /\  Y  e.  W  /\  -.  X  e.  dom  F )  ->  ( ( F  u.  { <. X ,  Y >. } ) `  X )  =  Y )
 
5-May-2017ballotlemsima 23000 The image by  S of an interval before the first pick. (Contributed by Thierry Arnoux, 5-May-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
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( I `  C ) ) ) 
 ->  ( ( S `  C ) " (
 1 ... J ) )  =  ( ( ( S `  C ) `
  J ) ... ( I `  C ) ) )
 
2-May-2017addeq0 22969 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 ) )
 
2-May-2017fzsplit3 22956 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 ) ) )
 
2-May-2017ax12b 1834 Two equivalent ways of expressing ax-12 1633. See the comment for ax-12 1633. (Contributed by NM, 2-May-2017.)
 |-  ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 ) 
 <->  ( -.  x  =  y  ->  ( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z ) ) ) )
 
1-May-2017lvecdim 15837 The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 15824 and lbsacsbs 15836 to show that being a basis for a vector space is equivalent to being a basis for the associated algebraic closure system, and then using acsexdimd 14213. (Contributed by David Moews, 1-May-2017.)
 |-  J  =  (LBasis `  W )   =>    |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  S  ~~  T )
 
1-May-2017lbsacsbs 15836 Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 15834. (Contributed by David Moews, 1-May-2017.)
 |-  A  =  ( LSubSp `  W )   &    |-  N  =  (mrCls `  A )   &    |-  X  =  (
 Base `  W )   &    |-  I  =  (mrInd `  A )   &    |-  J  =  (LBasis `  W )   =>    |-  ( W  e.  LVec  ->  ( S  e.  J  <->  ( S  e.  I  /\  ( N `  S )  =  X ) ) )
 
1-May-2017lssacsex 15824 In a vector space, subspaces form an algebraic closure system whose closure operator has the exchange property. Strengthening of lssacs 15651 by lspsolv 15823. (Contributed by David Moews, 1-May-2017.)
 |-  A  =  ( LSubSp `  W )   &    |-  N  =  (mrCls `  A )   &    |-  X  =  (
 Base `  W )   =>    |-  ( W  e.  LVec 
 ->  ( A  e.  (ACS `  X )  /\  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y } ) )  \  ( N `  s ) ) y  e.  ( N `  ( s  u. 
 { z } )
 ) ) )
 
1-May-2017acsexdimd 14213 In an algebraic closure system whose closure operator has the exchange property, if two independent sets have equal closure, they are equinumerous. See mreexfidimd 13479 for the finite case and acsinfdimd 14212 for the infinite case. This is a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  e.  I
 )   &    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )   =>    |-  ( ph  ->  S  ~~  T )
 
1-May-2017acsinfdimd 14212 In an algebraic closure system, if two independent sets have equal closure and one is infinite, then they are equinumerous. This is proven by using acsdomd 14211 twice with acsinfd 14210. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  e.  I
 )   &    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )   &    |-  ( ph  ->  -.  S  e.  Fin )   =>    |-  ( ph  ->  S  ~~  T )
 
1-May-2017acsdomd 14211 In an algebraic closure system, if 
S and  T have the same closure and  S is infinite independent, then  T dominates  S. This follows from applying acsinfd 14210 and then applying unirnfdomd 8122 to the map given in acsmap2d 14209. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  C_  X )   &    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )   &    |-  ( ph  ->  -.  S  e.  Fin )   =>    |-  ( ph  ->  S  ~<_  T )
 
1-May-2017acsinfd 14210 In an algebraic closure system, if 
S and  T have the same closure and  S is infinite independent, then  T is infinite. This follows from applying unirnffid 7080 to the map given in acsmap2d 14209. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  C_  X )   &    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )   &    |-  ( ph  ->  -.  S  e.  Fin )   =>    |-  ( ph  ->  -.  T  e.  Fin )
 
1-May-2017acsmap2d 14209 In an algebraic closure system, if 
S and  T have the same closure and  S is independent, then there is a map  f from  T into the set of finite subsets of  S such that  S equals the union of  ran  f. This is proven by taking the map  f from acsmapd 14208 and observing that, since  S and  T have the same closure, the closure of  U. ran  f must contain  S. Since  S is independent, by mrissmrcd 13469,  U. ran  f must equal  S. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  C_  X )   &    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )   =>    |-  ( ph  ->  E. f
 ( f : T --> ( ~P S  i^i  Fin )  /\  S  =  U. ran  f ) )
 
1-May-2017acsmapd 14208 In an algebraic closure system, if 
T is contained in the closure of  S, there is a map  f from  T into the set of finite subsets of  S such that the closure of  U. ran  f contains  T. This is proven by applying acsficl2d 14206 to each element of  T. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  S 
 C_  X )   &    |-  ( ph  ->  T  C_  ( N `  S ) )   =>    |-  ( ph  ->  E. f
 ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f
 ) ) )
 
1-May-2017acsfiindd 14207 In an algebraic closure system, a set is independent if and only if all its finite subsets are independent. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( S  e.  I  <->  ( ~P S  i^i  Fin )  C_  I
 ) )
 
1-May-2017acsficl2d 14206 In an algebraic closure system, an element is in the closure of a set if and only if it is in the closure of a finite subset. Alternate form of acsficl 14201. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( Y  e.  ( N `
  S )  <->  E. x  e.  ( ~P S  i^i  Fin ) Y  e.  ( N `  x ) ) )
 
1-May-2017acsficld 14205 In an algebraic closure system, the closure of a set is the union of the closures of its finite subsets. Deduction form of acsficl 14201. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( N `  S )  =  U. ( N
 " ( ~P S  i^i  Fin ) ) )
 
1-May-2017acsmred 13485 An algebraic closure system is also a Moore system. Deduction form of acsmre 13481. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   =>    |-  ( ph  ->  A  e.  (Moore `  X )
 )
 
1-May-2017mreexfidimd 13479 In a Moore system whose closure operator has the exchange property, if two independent sets have equal closure and one is finite, then they are equinumerous. Proven by using mreexdomd 13478 twice. This implies a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  e.  I
 )   &    |-  ( ph  ->  S  e.  Fin )   &    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )   =>    |-  ( ph  ->  S 
 ~~  T )
 
1-May-2017mreexdomd 13478 In a Moore system whose closure operator has the exchange property, if  S is independent and contained in the closure of  T, and either  S or  T is finite, then  T dominates  S. This is an immediate consequence of mreexexd 13477. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  S 
 C_  ( N `  T ) )   &    |-  ( ph  ->  T  C_  X )   &    |-  ( ph  ->  ( S  e.  Fin  \/  T  e.  Fin ) )   &    |-  ( ph  ->  S  e.  I
 )   =>    |-  ( ph  ->  S  ~<_  T )
 
1-May-2017mreexexd 13477 Exchange-type theorem. In a Moore system whose closure operator has the exchange property, if  F and  G are disjoint from  H,  ( F  u.  H ) is independent,  F is contained in the closure of  ( G  u.  H ), and either  F or  G is finite, then there is a subset  q of  G equinumerous to  F such that  ( q  u.  H ) is independent. This implies the case of Proposition 4.2.1 in [FaureFrolicher] p. 86 where either  ( A  \  B ) or  ( B  \  A ) is finite. The theorem is proven by induction using mreexexlem3d 13475 for the base case and mreexexlem4d 13476 for the induction step. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  F 
 C_  ( X  \  H ) )   &    |-  ( ph  ->  G  C_  ( X  \  H ) )   &    |-  ( ph  ->  F  C_  ( N `  ( G  u.  H ) ) )   &    |-  ( ph  ->  ( F  u.  H )  e.  I
 )   &    |-  ( ph  ->  ( F  e.  Fin  \/  G  e.  Fin ) )   =>    |-  ( ph  ->  E. q  e.  ~P  G ( F  ~~  q  /\  ( q  u.  H )  e.  I )
 )
 
1-May-2017mreexexlem4d 13476 Induction step of the induction in mreexexd 13477. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  F 
 C_  ( X  \  H ) )   &    |-  ( ph  ->  G  C_  ( X  \  H ) )   &    |-  ( ph  ->  F  C_  ( N `  ( G  u.  H ) ) )   &    |-  ( ph  ->  ( F  u.  H )  e.  I
 )   &    |-  ( ph  ->  L  e.  om )   &    |-  ( ph  ->  A. h A. f  e. 
 ~P  ( X  \  h ) A. g  e.  ~P  ( X  \  h ) ( ( ( f  ~~  L  \/  g  ~~  L ) 
 /\  f  C_  ( N `  ( g  u.  h ) )  /\  ( f  u.  h )  e.  I )  ->  E. j  e.  ~P  g ( f  ~~  j  /\  ( j  u.  h )  e.  I
 ) ) )   &    |-  ( ph  ->  ( F  ~~  suc 
 L  \/  G  ~~  suc 
 L ) )   =>    |-  ( ph  ->  E. j  e.  ~P  G ( F  ~~  j  /\  ( j  u.  H )  e.  I )
 )
 
1-May-2017mreexexlem3d 13475 Base case of the induction in mreexexd 13477. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  F 
 C_  ( X  \  H ) )   &    |-  ( ph  ->  G  C_  ( X  \  H ) )   &    |-  ( ph  ->  F  C_  ( N `  ( G  u.  H ) ) )   &    |-  ( ph  ->  ( F  u.  H )  e.  I
 )   &    |-  ( ph  ->  ( F  =  (/)  \/  G  =  (/) ) )   =>    |-  ( ph  ->  E. i  e.  ~P  G ( F  ~~  i  /\  ( i  u.  H )  e.  I )
 )
 
1-May-2017mreexexlem2d 13474 Used in mreexexlem4d 13476 to prove the induction step in mreexexd 13477. See the proof of Proposition 4.2.1 in [FaureFrolicher] p. 86 to 87. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  F 
 C_  ( X  \  H ) )   &    |-  ( ph  ->  G  C_  ( X  \  H ) )   &    |-  ( ph  ->  F  C_  ( N `  ( G  u.  H ) ) )   &    |-  ( ph  ->  ( F  u.  H )  e.  I
 )   &    |-  ( ph  ->  Y  e.  F )   =>    |-  ( ph  ->  E. g  e.  G  ( -.  g  e.  ( F  \  { Y } )  /\  (
 ( F  \  { Y } )  u.  ( H  u.  { g }
 ) )  e.  I
 ) )
 
1-May-2017mreexexlemd 13473 This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 13477. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  X  e.  J )   &    |-  ( ph  ->  F 
 C_  ( X  \  H ) )   &    |-  ( ph  ->  G  C_  ( X  \  H ) )   &    |-  ( ph  ->  F  C_  ( N `  ( G  u.  H ) ) )   &    |-  ( ph  ->  ( F  u.  H )  e.  I
 )   &    |-  ( ph  ->  ( F  ~~  K  \/  G  ~~  K ) )   &    |-  ( ph  ->  A. t A. u  e.  ~P  ( X  \  t ) A. v  e.  ~P  ( X  \  t ) ( ( ( u  ~~  K  \/  v  ~~  K ) 
 /\  u  C_  ( N `  ( v  u.  t ) )  /\  ( u  u.  t
 )  e.  I ) 
 ->  E. i  e.  ~P  v ( u  ~~  i  /\  ( i  u.  t )  e.  I
 ) ) )   =>    |-  ( ph  ->  E. j  e.  ~P  G ( F  ~~  j  /\  ( j  u.  H )  e.  I )
 )
 
1-May-2017mreexmrid 13472 In a Moore system whose closure operator has the exchange property, if a set is independent and an element is not in its closure, then adding the element to the set gives another independent set. Lemma 4.1.5 in [FaureFrolicher] p. 84. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  -.  Y  e.  ( N `  S ) )   =>    |-  ( ph  ->  ( S  u.  { Y }
 )  e.  I )
 
1-May-2017mreexd 13471 In a Moore system, the closure operator is said to have the exchange property if, for all elements  y and  z of the base set and subsets  S of the base set such that  z is in the closure of  ( S  u.  { y } ) but not in the closure of  S,  y is in the closure of  ( S  u.  { z } ) (Definition 3.1.9 in [FaureFrolicher] p. 57 to 58.) This theorem allows us to construct substitution instances of this definition. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  S 
 C_  X )   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  Z  e.  ( N `  ( S  u.  { Y }
 ) ) )   &    |-  ( ph  ->  -.  Z  e.  ( N `  S ) )   =>    |-  ( ph  ->  Y  e.  ( N `  ( S  u.  { Z }
 ) ) )
 
1-May-2017mrissmrid 13470 In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  C_  S )   =>    |-  ( ph  ->  T  e.  I )
 
1-May-2017mrissmrcd 13469 In a Moore system, if an independent set is between a set and its closure, the two sets are equal (since the two sets must have equal closures by mressmrcd 13456, and so are equal by mrieqv2d 13468.) (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S 
 C_  ( N `  T ) )   &    |-  ( ph  ->  T  C_  S )   &    |-  ( ph  ->  S  e.  I )   =>    |-  ( ph  ->  S  =  T )
 
1-May-2017mrieqv2d 13468 In a Moore system, a set is independent if and only if all its proper subsets have closure properly contained in the closure of the set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( S  e.  I  <->  A. s ( s 
 C.  S  ->  ( N `  s )  C.  ( N `  S ) ) ) )
 
1-May-2017mrieqvd 13467 In a Moore system, a set is independent if and only if, for all elements of the set, the closure of the set with the element removed is unequal to the closure of the original set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  ( N `  ( S 
 \  { x }
 ) )  =/=  ( N `  S ) ) )
 
1-May-2017ismri2dad 13466 Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A  e.  (Moore `  X ) )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  Y  e.  S )   =>    |-  ( ph  ->  -.  Y  e.  ( N `  ( S  \  { Y }
 ) ) )
 
1-May-2017mrissd 13465 An independent set of a Moore system is a subset of the base set. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A  e.  (Moore `  X ) )   &    |-  ( ph  ->  S  e.  I )   =>    |-  ( ph  ->  S 
 C_  X )
 
1-May-2017mriss 13464 An independent set of a Moore system is a subset of the base set. (Contributed by David Moews, 1-May-2017.)
 |-  I  =  (mrInd `  A )   =>    |-  ( ( A  e.  (Moore `  X )  /\  S  e.  I )  ->  S  C_  X )
 
1-May-2017ismri2dd 13463 Definition of independence of a subset of the base set in a Moore system. One-way deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A  e.  (Moore `  X ) )   &    |-  ( ph  ->  S 
 C_  X )   &    |-  ( ph  ->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) )   =>    |-  ( ph  ->  S  e.  I )
 
1-May-2017ismri2d 13462 Criterion for a subset of the base set in a Moore system to be independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A  e.  (Moore `  X ) )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) ) )
 
1-May-2017ismri2 13461 Criterion for a subset of the base set in a Moore system to be independent. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   =>    |-  ( ( A  e.  (Moore `  X )  /\  S  C_  X )  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) ) )
 
1-May-2017ismri 13460 Criterion for a set to be an independent set of a Moore system. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   =>    |-  ( A  e.  (Moore `  X )  ->  ( S  e.  I  <->  ( S  C_  X  /\  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
 ) ) ) ) )
 
1-May-2017mrisval 13459 Value of the set of independent sets of a Moore system. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   =>    |-  ( A  e.  (Moore `  X )  ->  I  =  { s  e.  ~P X  |  A. x  e.  s  -.  x  e.  ( N `  (
 s  \  { x } ) ) }
 )
 
1-May-2017mrieqvlemd 13458 In a Moore system, if  Y is a member of  S,  ( S 
\  { Y }
) and  S have the same closure if and only if  Y is in the closure of  ( S  \  { Y } ). Used in the proof of mrieqvd 13467 and mrieqv2d 13468. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  S 
 C_  X )   &    |-  ( ph  ->  Y  e.  S )   =>    |-  ( ph  ->  ( Y  e.  ( N `  ( S  \  { Y } ) )  <->  ( N `  ( S  \  { Y } ) )  =  ( N `  S ) ) )
 
1-May-2017mressmrcd 13456 In a Moore system, if a set is between another set and its closure, the two sets have the same closure. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  S 
 C_  ( N `  T ) )   &    |-  ( ph  ->  T  C_  S )   =>    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )
 
1-May-2017mrcidmd 13455 Moore closure is idempotent. Deduction form of mrcidm 13448. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  U 
 C_  X )   =>    |-  ( ph  ->  ( N `  ( N `
  U ) )  =  ( N `  U ) )
 
1-May-2017mrcssidd 13454 A set is contained in its Moore closure. Deduction form of mrcssid 13446. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  U 
 C_  X )   =>    |-  ( ph  ->  U 
 C_  ( N `  U ) )
 
1-May-2017mrcssd 13453 Moore closure preserves subset ordering. Deduction form of mrcss 13445. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  U 
 C_  V )   &    |-  ( ph  ->  V  C_  X )   =>    |-  ( ph  ->  ( N `  U )  C_  ( N `  V ) )
 
1-May-2017mrcssvd 13452 The Moore closure of a set is a subset of the base. Deduction form of mrcssv 13443. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   =>    |-  ( ph  ->  ( N `  B )  C_  X )
 
1-May-2017df-mri 13417 In a Moore system, a set is independent if no element of the set is in the closure of the set with the element removed (Section 0.6 in [Gratzer] p. 27; Definition 4.1.1 in [FaureFrolicher] p. 83.) mrInd is a class function which takes a Moore system to its set of independent sets. (Contributed by David Moews, 1-May-2017.)
 |- mrInd  =  ( c  e.  U. ran Moore 
 |->  { s  e.  ~P U. c  |  A. x  e.  s  -.  x  e.  ( (mrCls `  c
 ) `  ( s  \  { x } )
 ) } )
 
1-May-2017df-mrc 13416 Define the Moore closure of a generating set, which is the smallest closed set containing all generating elements. Definition of Moore closure in [Schechter] p. 79. This generalizes topological closure (mrccls 16743) and linear span (mrclsp 15673).

A Moore closure operation  N is (1) extensive, i.e.,  x  C_  ( N `  x ) for all subsets  x of the base set (mrcssid 13446), (2) isotone, i.e.,  x  C_  y implies that  ( N `
 x )  C_  ( N `  y ) for all subsets  x and  y of the base set (mrcss 13445), and (3) idempotent, i.e.,  ( N `  ( N `  x )
)  =  ( N `
 x ) for all subsets  x of the base set (mrcidm 13448.) Operators satisfying these three properties are in bijective correspondence with Moore collections, so these properties may be used to give an alternate characterization of a Moore collection by providing a closure operation  N on the set of subsets of a given base set which satisfies (1), (2), and (3); the closed sets can be recovered as those sets which equal their closures (Section 4.5 in [Schechter] p. 82.) (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by David Moews, 1-May-2017.)

 |- mrCls  =  ( c  e.  U. ran Moore 
 |->  ( x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) )
 
1-May-2017df-mre 13415 Define a Moore collection, which is a family of subsets of a base set which preserve arbitrary intersection. Elements of a Moore collection are termed closed; Moore collections generalize the notion of closedness from topologies (cldmre 16742) and vector spaces (lssmre 15650) to the most general setting in which such concepts make sense. Definition of Moore collection of sets in [Schechter] p. 78. A Moore collection may also be called a closure system (Section 0.6 in [Gratzer] p. 23.) The name Moore collection is after Eliakim Hastings Moore, who discussed these systems in Part I of [Moore] p. 53 to 76.

See ismre 13419, mresspw 13421, mre1cl 13423 and mreintcl 13424 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 13429); as such the disjoint union of all Moore collections is sometimes considered as  U. ran Moore, justified by mreunirn 13430. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by David Moews, 1-May-2017.)

 |- Moore  =  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }
 )
 
1-May-2017unirnfdomd 8122 The union of the range of a function from a infinite set into the class of finite sets is dominated by its domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  F : T --> Fin )   &    |-  ( ph  ->  -.  T  e.  Fin )   &    |-  ( ph  ->  T  e.  V )   =>    |-  ( ph  ->  U. ran  F  ~<_  T )
 
1-May-2017infinf 8121 Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  e.  B  ->  ( -.  A  e.  Fin  <->  om  ~<_  A ) )
 
1-May-2017cardidd 8104 Any set is equinumerous to its cardinal number. Deduction form of cardid 8102. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  ( card `  A )  ~~  A )
 
1-May-2017cardidg 8103 Any set is equinumerous to its cardinal number. Closed theorem form of cardid 8102. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  e.  B  ->  ( card `  A )  ~~  A )
 
1-May-2017unirnffid 7080 The union of the range of a function from a finite set into the class of finite sets is finite. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  F : T --> Fin )   &    |-  ( ph  ->  T  e.  Fin )   =>    |-  ( ph  ->  U.
 ran  F  e.  Fin )
 
1-May-2017ensymd 6845 Symmetry of equinumerosity. Deduction form of ensym 6843. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  ~~  B )   =>    |-  ( ph  ->  B  ~~  A )
 
1-May-2017elfvexd 5455 If a function value is nonempty, its argument is a set. Deduction form of elfvex 5454. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ( B `  C ) )   =>    |-  ( ph  ->  C  e.  _V )
 
1-May-2017ordelinel 4428 The intersection of two ordinal classes is an element of a third if and only if either one of them is. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( ( A  i^i  B )  e.  C  <->  ( A  e.  C  \/  B  e.  C ) ) )
 
1-May-2017ordtri2or3 4427 A consequence of total ordering for ordinal classes. Similar to ordtri2or2 4426. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  ( A  i^i  B )  \/  B  =  ( A  i^i  B ) ) )
 
1-May-2017ssexd 4101 A subclass of a set is a set. Deduction form of ssexg 4100. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  B  e.  C )   &    |-  ( ph  ->  A 
 C_  B )   =>    |-  ( ph  ->  A  e.  _V )
 
1-May-2017unissd 3792 Subclass relationship for subclass union. Deduction form of uniss 3789. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  U. A  C_ 
 U. B )
 
1-May-2017unissi 3791 Subclass relationship for subclass union. Inference form of uniss 3789. (Contributed by David Moews, 1-May-2017.)
 |-  A  C_  B   =>    |- 
 U. A  C_  U. B
 
1-May-2017difsnpss 3699  ( B  \  { A } ) is a proper subclass of  B if and only if  A is a member of  B. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  e.  B  <->  ( B  \  { A } )  C.  B )
 
1-May-2017difsneq 3698  ( B  \  { A } ) equals  B if and only if 
A is not a member of  B. Generalization of difsn 3696. (Contributed by David Moews, 1-May-2017.)
 |-  ( -.  A  e.  B 
 <->  ( B  \  { A } )  =  B )
 
1-May-2017neldifsnd 3693  A is not in  ( B 
\  { A }
). Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  -.  A  e.  ( B  \  { A } ) )
 
1-May-2017neldifsn 3692  A is not in  ( B 
\  { A }
). (Contributed by David Moews, 1-May-2017.)
 |- 
 -.  A  e.  ( B  \  { A }
 )
 
1-May-2017elpwid 3575 An element of a power class is a subclass. Deduction form of elpwi 3574. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ~P B )   =>    |-  ( ph  ->  A 
 C_  B )
 
1-May-2017ssnelpssd 3460 Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3459. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C  e.  B )   &    |-  ( ph  ->  -.  C  e.  A )   =>    |-  ( ph  ->  A  C.  B )
 
1-May-2017dfss5 3316 Another definition of subclasshood. Similar to df-ss 3108, dfss 3109, and dfss1 3315. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  C_  B  <->  A  =  ( B  i^i  A ) )
 
1-May-2017unssbd 3295 If  ( A  u.  B ) is contained in  C, so is  B. One-way deduction form of unss 3291. Partial converse of unssd 3293. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  ( A  u.  B )  C_  C )   =>    |-  ( ph  ->  B  C_  C )
 
1-May-2017unssad 3294 If  ( A  u.  B ) is contained in  C, so is  A. One-way deduction form of unss 3291. Partial converse of unssd 3293. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  ( A  u.  B )  C_  C )   =>    |-  ( ph  ->  A  C_  C )
 
1-May-2017ssdif2d 3257 If  A is contained in  B and  C is contained in  D, then  ( A  \  D ) is contained in  ( B  \  C ). Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C 
 C_  D )   =>    |-  ( ph  ->  ( A  \  D ) 
 C_  ( B  \  C ) )
 
1-May-2017ssdifssd 3256 If  A is contained in  B, then  ( A 
\  C ) is also contained in  B. Deduction form of ssdifss 3249. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( A  \  C )  C_  B )
 
1-May-2017sscond 3255 If  A is contained in  B, then  ( C 
\  B ) is contained in  ( C  \  A ). Deduction form of sscon 3252. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( C  \  B )  C_  ( C  \  A ) )
 
1-May-2017ssdifd 3254 If  A is contained in  B, then  ( A 
\  C ) is contained in  ( B  \  C ). Deduction form of ssdif 3253. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( A  \  C )  C_  ( B  \  C ) )
 
1-May-2017difss2d 3248 If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 3247. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  ( B  \  C ) )   =>    |-  ( ph  ->  A  C_  B )
 
1-May-2017difss2 3247 If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  C_  ( B  \  C )  ->  A  C_  B )
 
1-May-2017difssd 3246 A difference of two classes is contained in the minuend. Deduction form of difss 3245. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  ( A  \  B )  C_  A )
 
1-May-2017psssstrd 3227 Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 3224. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C.  B )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C.  C )
 
1-May-2017sspsstrd 3226 Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3223. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B 
 C.  C )   =>    |-  ( ph  ->  A 
 C.  C )
 
1-May-2017psstrd 3225 Proper subclass inclusion is transitive. Deduction form of psstr 3222. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C.  B )   &    |-  ( ph  ->  B 
 C.  C )   =>    |-  ( ph  ->  A 
 C.  C )
 
1-May-2017pssned 3216 Proper subclasses are unequal. Deduction form of pssne 3214. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C.  B )   =>    |-  ( ph  ->  A  =/=  B )
 
1-May-2017ssneldd 3125 If an element is not in a class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  -.  C  e.  B )   =>    |-  ( ph  ->  -.  C  e.  A )
 
1-May-2017ssneld 3124 If a class is not in another class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( -.  C  e.  B  ->  -.  C  e.  A ) )
 
1-May-2017eldifbd 3107 If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3104. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ( B  \  C ) )   =>    |-  ( ph  ->  -.  A  e.  C )
 
1-May-2017eldifad 3106 If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3104. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ( B  \  C ) )   =>    |-  ( ph  ->  A  e.  B )
 
1-May-2017eldifd 3105 If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3104. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  -.  A  e.  C )   =>    |-  ( ph  ->  A  e.  ( B  \  C ) )
 
1-May-2017cbvrexv2 3090 Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)
 |-  ( x  =  y 
 ->  ( ps  <->  ch ) )   &    |-  ( x  =  y  ->  A  =  B )   =>    |-  ( E. x  e.  A  ps  <->  E. y  e.  B  ch )
 
1-May-2017cbvralv2 3089 Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)
 |-  ( x  =  y 
 ->  ( ps  <->  ch ) )   &    |-  ( x  =  y  ->  A  =  B )   =>    |-  ( A. x  e.  A  ps  <->  A. y  e.  B  ch )
 
1-May-2017cla4dv 2817 Rule of specialization, using implicit substitution. Analogous to rcla4dv 2838. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  ->  ch ) )
 
1-May-2017cbvrexdva 2723 Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. y  e.  A  ch ) )
 
1-May-2017cbvraldva 2722 Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. y  e.  A  ch ) )
 
1-May-2017cbvrexdva2 2721 Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   &    |-  (
 ( ph  /\  x  =  y )  ->  A  =  B )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. y  e.  B  ch ) )
 
1-May-2017cbvraldva2 2720 Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   &    |-  (
 ( ph  /\  x  =  y )  ->  A  =  B )   =>    |-  ( ph  ->  ( A. x  e.  A  ps 
 <-> 
 A. y  e.  B  ch ) )
 
1-May-2017rexeqbii 2545 Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
 |-  A  =  B   &    |-  ( ps 
 <->  ch )   =>    |-  ( E. x  e.  A  ps  <->  E. x  e.  B  ch )
 
1-May-2017raleqbii 2544 Equality deduction for restricted universal quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
 |-  A  =  B   &    |-  ( ps 
 <->  ch )   =>    |-  ( A. x  e.  A  ps  <->  A. x  e.  B  ch )
 
1-May-2017neleqtrrd 2352 If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  -.  C  e.  B )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  -.  C  e.  A )
 
1-May-2017neleqtrd 2351 If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  -.  C  e.  A )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  -.  C  e.  B )
 
1-May-2017eqneltrrd 2350 If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  -.  A  e.  C )   =>    |-  ( ph  ->  -.  B  e.  C )
 
1-May-2017eqneltrd 2349 If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  -.  B  e.  C )   =>    |-  ( ph  ->  -.  A  e.  C )
 
1-May-2017cbvexdva 2055 Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. y ch )
 )
 
1-May-2017cbvaldva 2054 Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( ph  /\  x  =  y )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. y ch )
 )
 
28-Apr-2017ballotlemsel1i 22997 The range  ( 1 ... ( I `  C
) ) is invariant under  ( S `  C ). (Contributed by Thierry Arnoux, 28-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
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( I `  C ) ) ) 
 ->  ( ( S `  C ) `  J )  e.  ( 1 ... ( I `  C ) ) )
 
28-Apr-2017ballotlemsgt1 22995  S maps values less than  ( I `  C ) to values greater than 1. (Contributed by Thierry Arnoux, 28-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
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
 1  <  ( ( S `  C ) `  J ) )
 
27-Apr-2017ballotlemfrceq 23013 Value of  F for a reverse counting  ( R `  C ). (Contributed by Thierry Arnoux, 27-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
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   &    |-  .^  =  ( u  e. 
 Fin ,  v  e.  Fin  |->  ( ( # `  (
 v  i^i  u )
 )  -  ( # `  ( v  \  u ) ) ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( I `  C ) ) ) 
 ->  ( ( F `  C ) `  (
 ( ( S `  C ) `  J )  -  1 ) )  =  -u ( ( F `
  ( R `  C ) ) `  J ) )
 
26-Apr-2017ballotlemgun 23009 A property of the defined  .^ operator (Contributed by Thierry Arnoux, 26-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
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   &    |-  .^  =  ( u  e. 
 Fin ,  v  e.  Fin  |->  ( ( # `  (
 v  i^i  u )
 )  -  ( # `  ( v  \  u ) ) ) )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ph  ->  V  e.  Fin )   &    |-  ( ph  ->  W  e.  Fin )   &    |-  ( ph  ->  ( V  i^i  W )  =  (/) )   =>    |-  ( ph  ->  ( U  .^  ( V  u.  W ) )  =  ( ( U  .^  V )  +  ( U  .^  W ) ) )
 
25-Apr-2017ballotlemfrcn0 23014 Value of  F for a reversed counting  ( R `  C ), before the first tie, cannot be zero . (Contributed by Thierry Arnoux, 25-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
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  ->  ( ( F `  ( R `  C ) ) `  J )  =/=  0 )
 
24-Apr-2017funimass4f 22968 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 ) )
 
24-Apr-2017dfimafnf 22967 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 ) } )
 
23-Apr-2017ax9dgenK 28137 Degenerate case of ax-9 1684. Uses only Tarski's FOL axiom schemes (see description for equidK 28115). (Contributed by NM, 23-Apr-2017.)
 |-  -.  A. x  -.  x  =  x
 
23-Apr-2017f1o3d 22963 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 ) ) )
 
21-Apr-2017ballotlemfrci 23012 Reverse counting preserves a tie at the first tie. (Contributed by Thierry Arnoux, 21-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
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   &    |-  .^  =  ( u  e. 
 Fin ,  v  e.  Fin  |->  ( ( # `  (
 v  i^i  u )
 )  -  ( # `  ( v  \  u ) ) ) )   =>    |-  ( C  e.  ( O  \  E )  ->  ( ( F `  ( R `  C ) ) `  ( I `
  C ) )  =  0 )
 
21-Apr-2017ballotlemfrc 23011 Express the value of  ( F `  ( R `  C )
) in terms of the newly defined  .^. (Contributed by Thierry Arnoux, 21-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
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   &    |-  .^  =  ( u  e. 
 Fin ,  v  e.  Fin  |->  ( ( # `  (
 v  i^i  u )
 )  -  ( # `  ( v  \  u ) ) ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 1 ... ( I `  C ) ) ) 
 ->  ( ( F `  ( R `  C ) ) `  J )  =  ( C  .^  ( ( ( S `
  C ) `  J ) ... ( I `  C ) ) ) )
 
21-Apr-2017ballotlemfg 23010 Express the value of  ( F `  C
) in terms of  .^. (Contributed by Thierry Arnoux, 21-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
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   &    |-  .^  =  ( u  e. 
 Fin ,  v  e.  Fin  |->  ( ( # `  (
 v  i^i  u )
 )  -  ( # `  ( v  \  u ) ) ) )   =>    |-  ( ( C  e.  ( O  \  E ) 
 /\  J  e.  (
 0 ... ( M  +  N ) ) ) 
 ->  ( ( F `  C ) `  J )  =  ( C  .^  ( 1 ... J ) ) )
 
21-Apr-2017ballotlemgval 23008 Expand the value of  .^. (Contributed by Thierry Arnoux, 21-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
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   &    |-  .^  =  ( u  e. 
 Fin ,  v  e.  Fin  |->  ( ( # `  (
 v  i^i  u )
 )  -  ( # `  ( v  \  u ) ) ) )   =>    |-  ( ( U  e.  Fin  /\  V  e.  Fin )  ->  ( U  .^  V )  =  ( ( # `
  ( V  i^i  U ) )  -  ( # `
  ( V  \  U ) ) ) )
 
21-Apr-2017ballotlemscr 23003 The image of  ( R `  C ) by  ( S `  C ) (Contributed by Thierry Arnoux, 21-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
 ) ) ) ) )   &    |-  E  =  {
 c  e.  O  |  A. i  e.  (
 1 ... ( M  +  N ) ) 0  <  ( ( F `
  c ) `  i ) }   &    |-  N  <  M   &    |-  I  =  ( c  e.  ( O 
 \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N )
 )  |  ( ( F `  c ) `
  k )  =  0 } ,  RR ,  `'  <  ) )   &    |-  S  =  ( c  e.  ( O  \  E )  |->  ( i  e.  ( 1
 ... ( M  +  N ) )  |->  if ( i  <_  ( I `  c ) ,  ( ( ( I `
  c )  +  1 )  -  i
 ) ,  i ) ) )   &    |-  R  =  ( c  e.  ( O 
 \  E )  |->  ( ( S `  c
 ) " c ) )   =>    |-  ( C  e.  ( O  \  E )  ->  ( ( S `  C ) " ( R `  C ) )  =  C )
 
20-Apr-2017stowei 27113 This theorem proves the Stone-Weierstrass theorem for real valued functions: let  J be a compact topology on  T, and  C be the set of real continuous functions on  T. Assume that  A is a subalgebra of  C (closed under addition and multiplication of functions) containing constant functions and discriminating points (if  r and  t are distinct points in  T, then there exists a function  h in  A such that h(r) is distinct from h(t) ). Then, for any continuous function 
F and for any positive real  E, there exists a function  f in the subalgebra  A, such that  f approximates  F up to  E ( E represents the usual ε value). As a classical example, given any a,b reals, the closed interval  T  =  [
a ,  b ] could be taken, along with the subalgebra  A of real polynomials on  T, and then use this theorem to easily prove that real polynomials are dense in the standard metric space of continuous functions on  [ a ,  b ]. The proof and lemmas are written following [BrosowskiDeutsh] p. 89 (through page 92). Some effort is put in avoiding the use of the axiom of choice. The deduction version of this theorem is stoweid 27112: often times it will be better to use stoweid 27112 in other proofs (but this version is probably easier to be read and understood). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  K  =  ( topGen `  ran  (,) )   &    |-  J  e.  Comp   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  A  C_  C   &    |-  ( ( f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( x  e.  RR  ->  ( t  e.  T  |->  x )  e.  A )   &    |-  ( ( r  e.  T  /\  t  e.  T  /\  r  =/=  t )  ->  E. h  e.  A  ( h `  r )  =/=  ( h `  t ) )   &    |-  F  e.  C   &    |-  E  e.  RR+   =>    |-  E. f  e.  A  A. t  e.  T  ( abs `  (
 ( f `  t
 )  -  ( F `
  t ) ) )  <  E
 
20-Apr-2017stoweid 27112 This theorem proves the Stone-Weierstrass theorem for real valued functions: let  J be a compact topology on  T, and  C be the set of real continuous functions on  T. Assume that  A is a subalgebra of  C (closed under addition and multiplication of functions) containing constant functions and discriminating points (if  r and  t are distinct points in  T, then there exists a function  h in  A such that h(r) is distinct from h(t) ). Then, for any continuous function 
F and for any positive real  E, there exists a function  f in the subalgebra  A, such that  f approximates  F up to  E ( E represents the usual ε value). As a classical example, given any a,b reals, the closed interval  T  =  [
a ,  b ] could be taken, along with the subalgebra  A of real polynomials on  T, and then use this theorem to easily prove that real polynomials are dense in the standard metric space of continuous functions on  [ a ,  b ]. The proof and lemmas are written following [BrosowskiDeutsh] p. 89 (through page 92). Some effort is put in avoiding the use of the axiom of choice. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. h  e.  A  ( h `  r )  =/=  ( h `  t ) )   &    |-  ( ph  ->  F  e.  C )   &    |-  ( ph  ->  E  e.  RR+ )   =>    |-  ( ph  ->  E. f  e.  A  A. t  e.  T  ( abs `  (
 ( f `  t
 )  -  ( F `
  t ) ) )  <  E )
 
20-Apr-2017stoweidlem62 27111 This theorem proves the Stone Weierstrass theorem for the non trivial case in which T is nonempty. The proof follows [BrosowskiDeutsh] p. 89 (through page 92). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |- 
 F/ f ph   &    |-  F/ t ph   &    |-  H  =  ( t  e.  T  |->  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  T  =  U. J   &    |-  ( ph  ->  J  e.  Comp )   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  F  e.  C )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  T  =/=  (/) )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   =>    |-  ( ph  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `
  t )  -  ( F `  t ) ) )  <  E )
 
20-Apr-2017stoweidlem61 27110 This lemma proves that there exists a function  g as in the proof in [BrosowskiDeutsh] p. 92:  g is in the subalgebra, and for all  t in  T, abs( f(t) - g(t) ) < 2*ε. Here  F is used to represent f in the paper, and  E is used to represent ε. For this lemma there's the further assumption that the function  F to be approximated is nonnegative (this assumption is removed in a later theorem). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  T  =  U. J   &    |-  ( ph  ->  T  =/=  (/) )   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  F  e.  C )   &    |-  ( ph  ->  A. t  e.  T  0  <_  ( F `  t ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   =>    |-  ( ph  ->  E. g  e.  A  A. t  e.  T  ( abs `  ( ( g `
  t )  -  ( F `  t ) ) )  <  (
 2  x.  E ) )
 
20-Apr-2017stoweidlem60 27109 This lemma proves that there exists a function g as in the proof in [BrosowskiDeutsh] p. 91 (this parte of the proof actually spans through pages 91-92): g is in the subalgebra, and for all  t in  T, there is a  j such that (j-4/3)*ε < f(t) <= (j-1/3)*ε and (j-4/3)*ε < g(t) < (j+1/3)*ε. Here  F is used to represent f in the paper, and  E is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  D  =  ( j  e.  ( 0 ... n )  |->  { t  e.  T  |  ( F `  t
 )  <_  ( (
 j  -  ( 1 
 /  3 ) )  x.  E ) }
 )   &    |-  B  =  ( j  e.  ( 0 ... n )  |->  { t  e.  T  |  ( ( j  +  ( 1 
 /  3 ) )  x.  E )  <_  ( F `  t ) } )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  T  =/=  (/) )   &    |-  ( ph  ->  A 
 C_  C )   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  y  e.  RR )  ->  (
 t  e.  T  |->  y )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  F  e.  C )   &    |-  ( ph  ->  A. t  e.  T  0  <_  ( F `  t ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   =>    |-  ( ph  ->  E. g  e.  A  A. t  e.  T  E. j  e.  RR  ( ( ( ( j  -  (
 4  /  3 )
 )  x.  E )  <  ( F `  t )  /\  ( F `
  t )  <_  ( ( j  -  ( 1  /  3
 ) )  x.  E ) )  /\  ( ( g `  t )  <  ( ( j  +  ( 1  / 
 3 ) )  x.  E )  /\  (
 ( j  -  (
 4  /  3 )
 )  x.  E )  <  ( g `  t ) ) ) )
 
20-Apr-2017stoweidlem59 27108 This lemma proves that there exists a function  x as in the proof in [BrosowskiDeutsh] p. 91, after Lemma 2: xj is in the subalgebra, 0 <= xj <= 1, xj < ε / n on Aj (meaning A in the paper), xj > 1 - \epslon / n on Bj. Here  D is used to represent A in the paper (because A is used for the subalgebra of functions),  E is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  D  =  ( j  e.  ( 0 ... N )  |->  { t  e.  T  |  ( F `  t
 )  <_  ( (
 j  -  ( 1 
 /  3 ) )  x.  E ) }
 )   &    |-  B  =  ( j  e.  ( 0 ...
 N )  |->  { t  e.  T  |  ( ( j  +  ( 1 
 /  3 ) )  x.  E )  <_  ( F `  t ) } )   &    |-  Y  =  {
 y  e.  A  |  A. t  e.  T  ( 0  <_  (
 y `  t )  /\  ( y `  t
 )  <_  1 ) }   &    |-  H  =  ( j  e.  ( 0 ...
 N )  |->  { y  e.  Y  |  ( A. t  e.  ( D `  j ) ( y `
  t )  < 
 ( E  /  N )  /\  A. t  e.  ( B `  j
 ) ( 1  -  ( E  /  N ) )  <  ( y `
  t ) ) } )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  A 
 C_  C )   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  y  e.  RR )  ->  (
 t  e.  T  |->  y )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  F  e.  C )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  E. x ( x : ( 0
 ... N ) --> A  /\  A. j  e.  ( 0
 ... N ) (
 A. t  e.  T  ( 0  <_  (
 ( x `  j
 ) `  t )  /\  ( ( x `  j ) `  t
 )  <_  1 )  /\  A. t  e.  ( D `  j ) ( ( x `  j
 ) `  t )  <  ( E  /  N )  /\  A. t  e.  ( B `  j
 ) ( 1  -  ( E  /  N ) )  <  ( ( x `  j ) `
  t ) ) ) )
 
20-Apr-2017stoweidlem58 27107 This theorem proves Lemma 2 in [BrosowskiDeutsh] p. 91. Here D is used to represent the set A of Lemma 2, because here the variable A is used for the subalgebra of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t D   &    |-  F/_ t U   &    |-  F/ t ph   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  J  e.  Comp
 )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  a  e.  RR )  ->  (
 t  e.  T  |->  a )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  B  e.  ( Clsd `  J ) )   &    |-  ( ph  ->  D  e.  ( Clsd `  J )
 )   &    |-  ( ph  ->  ( B  i^i  D )  =  (/) )   &    |-  U  =  ( T  \  B )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   =>    |-  ( ph  ->  E. x  e.  A  (
 A. t  e.  T  ( 0  <_  ( x `  t )  /\  ( x `  t ) 
 <_  1 )  /\  A. t  e.  D  ( x `  t )  <  E  /\  A. t  e.  B  ( 1  -  E )  <  ( x `
  t ) ) )
 
20-Apr-2017stoweidlem57 27106 There exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. In this theorem, it is proven the non trivial case (the closed set D is nonempty). Here D is used to represent A in the paper, because the variable A is used for the subalgebra of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t D   &    |-  F/_ t U   &    |-  F/ t ph   &    |-  Y  =  { h  e.  A  |  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) }   &    |-  V  =  { w  e.  J  |  A. e  e.  RR+  E. h  e.  A  (
 A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 )  /\  A. t  e.  w  ( h `  t )  < 
 e  /\  A. t  e.  ( T  \  U ) ( 1  -  e )  <  ( h `
  t ) ) }   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  U  =  ( T  \  B )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  A 
 C_  C )   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  a  e.  RR )  ->  (
 t  e.  T  |->  a )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  B  e.  ( Clsd `  J ) )   &    |-  ( ph  ->  D  e.  ( Clsd `  J )
 )   &    |-  ( ph  ->  ( B  i^i  D )  =  (/) )   &    |-  ( ph  ->  D  =/=  (/) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   =>    |-  ( ph  ->  E. x  e.  A  (
 A. t  e.  T  ( 0  <_  ( x `  t )  /\  ( x `  t ) 
 <_  1 )  /\  A. t  e.  D  ( x `  t )  <  E  /\  A. t  e.  B  ( 1  -  E )  <  ( x `
  t ) ) )
 
20-Apr-2017stoweidlem56 27105 This theorem proves Lemma 1 in [BrosowskiDeutsh] p. 90. Here  Z is used to represent t0 in the paper,  v is used to represent  V in the paper, and  e is used to represent ε (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t U   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  y  e.  RR )  ->  (
 t  e.  T  |->  y )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  U  e.  J )   &    |-  ( ph  ->  Z  e.  U )   =>    |-  ( ph  ->  E. v  e.  J  ( ( Z  e.  v  /\  v  C_  U )  /\  A. e  e.  RR+  E. x  e.  A  ( A. t  e.  T  ( 0  <_  ( x `  t ) 
 /\  ( x `  t )  <_  1 ) 
 /\  A. t  e.  v  ( x `  t )  <  e  /\  A. t  e.  ( T  \  U ) ( 1  -  e )  < 
 ( x `  t
 ) ) ) )
 
20-Apr-2017stoweidlem55 27104 This lemma proves the existence of a function p as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Here Z is used to represent t0 in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t U   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  U  e.  J )   &    |-  ( ph  ->  Z  e.  U )   &    |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  W  =  { w  e.  J  |  E. h  e.  Q  w  =  { t  e.  T  |  0  < 
 ( h `  t
 ) } }   =>    |-  ( ph  ->  E. p  e.  A  (
 A. t  e.  T  ( 0  <_  ( p `  t )  /\  ( p `  t ) 
 <_  1 )  /\  ( p `  Z )  =  0  /\  A. t  e.  ( T  \  U ) 0  <  ( p `  t ) ) )
 
20-Apr-2017stoweidlem54 27103 There exists a function  x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. Here  D is used to represent  A in the paper, because here  A is used for the subalgebra of functions.  E is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ i ph   &    |-  F/ t ph   &    |-  F/ y ph   &    |-  F/ w ph   &    |-  T  =  U. J   &    |-  Y  =  { h  e.  A  |  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) }   &    |-  P  =  ( f  e.  Y ,  g  e.  Y  |->  ( t  e.  T  |->  ( ( f `  t )  x.  (
 g `  t )
 ) ) )   &    |-  F  =  ( t  e.  T  |->  ( i  e.  (
 1 ... M )  |->  ( ( y `  i
 ) `  t )
 ) )   &    |-  Z  =  ( t  e.  T  |->  ( 
 seq  1 (  x. 
 ,  ( F `  t ) ) `  M ) )   &    |-  V  =  { w  e.  J  |  A. e  e.  RR+  E. h  e.  A  (
 A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 )  /\  A. t  e.  w  ( h `  t )  < 
 e  /\  A. t  e.  ( T  \  U ) ( 1  -  e )  <  ( h `
  t ) ) }   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  x.  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A )  ->  f : T --> RR )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  W : ( 1 ...
 M ) --> V )   &    |-  ( ph  ->  B  C_  T )   &    |-  ( ph  ->  D  C_ 
 U. ran  W )   &    |-  ( ph  ->  D  C_  T )   &    |-  ( ph  ->  E. y
 ( y : ( 1 ... M ) --> Y  /\  A. i  e.  ( 1 ... M ) ( A. t  e.  ( W `  i
 ) ( ( y `
  i ) `  t )  <  ( E 
 /  M )  /\  A. t  e.  B  ( 1  -  ( E 
 /  M ) )  <  ( ( y `
  i ) `  t ) ) ) )   &    |-  ( ph  ->  T  e.  _V )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  /  3
 ) )   =>    |-  ( ph  ->  E. x  e.  A  ( A. t  e.  T  ( 0  <_  ( x `  t ) 
 /\  ( x `  t )  <_  1 ) 
 /\  A. t  e.  D  ( x `  t )  <  E  /\  A. t  e.  B  (
 1  -  E )  <  ( x `  t ) ) )
 
20-Apr-2017stoweidlem53 27102 This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t U   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  W  =  { w  e.  J  |  E. h  e.  Q  w  =  { t  e.  T  |  0  < 
 ( h `  t
 ) } }   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  J  e.  Comp
 )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  U  e.  J )   &    |-  ( ph  ->  ( T  \  U )  =/=  (/) )   &    |-  ( ph  ->  Z  e.  U )   =>    |-  ( ph  ->  E. p  e.  A  (
 A. t  e.  T  ( 0  <_  ( p `  t )  /\  ( p `  t ) 
 <_  1 )  /\  ( p `  Z )  =  0  /\  A. t  e.  ( T  \  U ) 0  <  ( p `  t ) ) )
 
20-Apr-2017stoweidlem52 27101 There exists a neighborood V as in Lemma 1 of [BrosowskiDeutsh] p. 90. Here Z is used to represent t0 in the paper, and v is used to represent V in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t U   &    |- 
 F/ t ph   &    |-  F/_ t P   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  V  =  { t  e.  T  |  ( P `  t
 )  <  ( D  /  2 ) }   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  a  e.  RR )  ->  (
 t  e.  T  |->  a )  e.  A )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  D  <  1 )   &    |-  ( ph  ->  U  e.  J )   &    |-  ( ph  ->  Z  e.  U )   &    |-  ( ph  ->  P  e.  A )   &    |-  ( ph  ->  A. t  e.  T  ( 0  <_  ( P `  t )  /\  ( P `  t ) 
 <_  1 ) )   &    |-  ( ph  ->  ( P `  Z )  =  0
 )   &    |-  ( ph  ->  A. t  e.  ( T  \  U ) D  <_  ( P `
  t ) )   =>    |-  ( ph  ->  E. v  e.  J  ( ( Z  e.  v  /\  v  C_  U )  /\  A. e  e.  RR+  E. x  e.  A  ( A. t  e.  T  ( 0  <_  ( x `  t ) 
 /\  ( x `  t )  <_  1 ) 
 /\  A. t  e.  v  ( x `  t )  <  e  /\  A. t  e.  ( T  \  U ) ( 1  -  e )  < 
 ( x `  t
 ) ) ) )
 
20-Apr-2017stoweidlem51 27100 There exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. Here  D is used to represent  A in the paper, because here  A is used for the subalgebra of functions.  E is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ i ph   &    |-  F/ t ph   &    |-  F/ w ph   &    |-  F/_ w V   &    |-  Y  =  { h  e.  A  |  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) }   &    |-  P  =  ( f  e.  Y ,  g  e.  Y  |->  ( t  e.  T  |->  ( ( f `  t )  x.  (
 g `  t )
 ) ) )   &    |-  X  =  (  seq  1 ( P ,  U ) `
  M )   &    |-  F  =  ( t  e.  T  |->  ( i  e.  (
 1 ... M )  |->  ( ( U `  i
 ) `  t )
 ) )   &    |-  Z  =  ( t  e.  T  |->  ( 
 seq  1 (  x. 
 ,  ( F `  t ) ) `  M ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  W : ( 1 ...
 M ) --> V )   &    |-  ( ph  ->  U :
 ( 1 ... M )
 --> Y )   &    |-  ( ( ph  /\  w  e.  V ) 
 ->  w  C_  T )   &    |-  ( ph  ->  D  C_  U. ran  W )   &    |-  ( ph  ->  D 
 C_  T )   &    |-  ( ph  ->  B  C_  T )   &    |-  ( ( ph  /\  i  e.  ( 1 ... M ) )  ->  A. t  e.  ( W `  i
 ) ( ( U `
  i ) `  t )  <  ( E 
 /  M ) )   &    |-  ( ( ph  /\  i  e.  ( 1 ... M ) )  ->  A. t  e.  B  ( 1  -  ( E  /  M ) )  <  ( ( U `  i ) `
  t ) )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A )  ->  f : T --> RR )   &    |-  ( ph  ->  T  e.  _V )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   =>    |-  ( ph  ->  E. x ( x  e.  A  /\  ( A. t  e.  T  (
 0  <_  ( x `  t )  /\  ( x `  t )  <_ 
 1 )  /\  A. t  e.  D  ( x `  t )  <  E  /\  A. t  e.  B  ( 1  -  E )  <  ( x `
  t ) ) ) )
 
20-Apr-2017stoweidlem50 27099 This lemma proves that sets U(t) as defined in Lemma 1 of [BrosowskiDeutsh] p. 90, contain a finite subcover of T \ U. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t U   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  W  =  { w  e.  J  |  E. h  e.  Q  w  =  { t  e.  T  |  0  < 
 ( h `  t
 ) } }   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  J  e.  Comp
 )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  U  e.  J )   &    |-  ( ph  ->  Z  e.  U )   =>    |-  ( ph  ->  E. u ( u  e.  Fin  /\  u  C_  W  /\  ( T  \  U ) 
 C_  U. u ) )
 
20-Apr-2017stoweidlem49 27098 There exists a function qn as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91 (at the top of page 91): 0 <= qn <= 1 , qn < ε on  T  \  U, and qn > 1 - ε on  V. Here y is used to represent the final qn in the paper (the one with n large enough),  N represents  n in the paper,  K represents  k,  D represents δ,  E represents ε, and  P represents  p. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t P   &    |- 
 F/ t ph   &    |-  V  =  {
 t  e.  T  |  ( P `  t )  <  ( D  / 
 2 ) }   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  D  <  1 )   &    |-  ( ph  ->  P  e.  A )   &    |-  ( ph  ->  P : T --> RR )   &    |-  ( ph  ->  A. t  e.  T  ( 0  <_  ( P `  t )  /\  ( P `  t )  <_ 
 1 ) )   &    |-  ( ph  ->  A. t  e.  ( T  \  U ) D 
 <_  ( P `  t
 ) )   &    |-  ( ( ph  /\  f  e.  A ) 
 ->  f : T --> RR )   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ph  ->  E  e.  RR+ )   =>    |-  ( ph  ->  E. y  e.  A  ( A. t  e.  T  ( 0  <_  ( y `  t
 )  /\  ( y `  t )  <_  1
 )  /\  A. t  e.  V  ( 1  -  E )  <  ( y `
  t )  /\  A. t  e.  ( T 
 \  U ) ( y `  t )  <  E ) )
 
20-Apr-2017stoweidlem48 27097 This lemma is used to prove that  x built as in Lemma 2 of [BrosowskiDeutsh] p. 91, is such that x < ε on  A. Here  X is used to represent  x in the paper,  E is used to represent ε in the paper, and  D is used to represent  A in the paper (because  A is always used to represent the subalgebra). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ i ph   &    |-  F/ t ph   &    |-  Y  =  { h  e.  A  |  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) }   &    |-  P  =  ( f  e.  Y ,  g  e.  Y  |->  ( t  e.  T  |->  ( ( f `  t )  x.  (
 g `  t )
 ) ) )   &    |-  X  =  (  seq  1 ( P ,  U ) `
  M )   &    |-  F  =  ( t  e.  T  |->  ( i  e.  (
 1 ... M )  |->  ( ( U `  i
 ) `  t )
 ) )   &    |-  Z  =  ( t  e.  T  |->  ( 
 seq  1 (  x. 
 ,  ( F `  t ) ) `  M ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  W : ( 1 ...
 M ) --> V )   &    |-  ( ph  ->  U :
 ( 1 ... M )
 --> Y )   &    |-  ( ph  ->  D 
 C_  U. ran  W )   &    |-  ( ph  ->  D  C_  T )   &    |-  ( ( ph  /\  i  e.  ( 1 ... M ) )  ->  A. t  e.  ( W `  i
 ) ( ( U `
  i ) `  t )  <  E )   &    |-  ( ph  ->  T  e.  _V )   &    |-  ( ( ph  /\  f  e.  A ) 
 ->  f : T --> RR )   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ph  ->  E  e.  RR+ )   =>    |-  ( ph  ->  A. t  e.  D  ( X `  t )  <  E )
 
20-Apr-2017stoweidlem47 27096 Subtracting a constant from a real continuous function gives another continuous function. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |-  F/_ t S   &    |-  F/ t ph   &    |-  T  =  U. J   &    |-  G  =  ( T  X.  { -u S } )   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Top )   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  F  e.  C )   &    |-  ( ph  ->  S  e.  RR )   =>    |-  ( ph  ->  (
 t  e.  T  |->  ( ( F `  t
 )  -  S ) )  e.  C )
 
20-Apr-2017stoweidlem46 27095 This lemma proves that sets U(t) as defined in Lemma 1 of [BrosowskiDeutsh] p. 90, are a cover of T \ U. Using this lemma, in a later theorem we will prove that a finite subcover exists. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t U   &    |-  F/_ h Q   &    |-  F/ q ph   &    |-  F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  W  =  { w  e.  J  |  E. h  e.  Q  w  =  { t  e.  T  |  0  < 
 ( h `  t
 ) } }   &    |-  T  =  U. J   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  A 
 C_  ( J  Cn  K ) )   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. q  e.  A  ( q `  r
 )  =/=  ( q `  t ) )   &    |-  ( ph  ->  U  e.  J )   &    |-  ( ph  ->  Z  e.  U )   &    |-  ( ph  ->  T  e.  _V )   =>    |-  ( ph  ->  ( T  \  U ) 
 C_  U. W )
 
20-Apr-2017stoweidlem45 27094 This lemma proves that, given an appropriate  K (in another theorem we prove such a  K exists), there exists a function qn as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91 ( at the top of page 91): 0 <= qn <= 1 , qn < ε on T \ U, and qn > 1 - ε on  V. We use y to represent the final qn in the paper (the one with n large enough),  N to represent  n in the paper,  K to represe