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Theorem List for Metamath Proof Explorer - 25301-25400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
18.7.48  Outside of relationship
 
Syntaxcoutsideof 25301 Declare the syntax for the outside of constant.
 class OutsideOf
 
Definitiondf-outsideof 25302 The outside of relationship. This relationship expresses that  P,  A, and  B fall on a line, but  P is not on the segment  A B. This definition is taken from theorem 6.4 of [Schwabhauser] p. 43, since it requires no dummy variables. (Contributed by Scott Fenton, 17-Oct-2013.)
 |- OutsideOf  =  (  Colinear  \  Btwn  )
 
Theorembroutsideof 25303 Binary relationship form of OutsideOf. Theorem 6.4 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 17-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( POutsideOf
 <. A ,  B >.  <->  ( P 
 Colinear 
 <. A ,  B >.  /\ 
 -.  P  Btwn  <. A ,  B >. ) )
 
Theorembroutsideof2 25304 Alternate form of OutsideOf. Definition 6.1 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 17-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( POutsideOf <. A ,  B >.  <-> 
 ( A  =/=  P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) ) )
 
Theoremoutsidene1 25305 Outsideness implies inequality. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( POutsideOf <. A ,  B >.  ->  A  =/=  P ) )
 
Theoremoutsidene2 25306 Outsideness implies inequality. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( POutsideOf <. A ,  B >.  ->  B  =/=  P ) )
 
Theorembtwnoutside 25307 A principle linking outsideness to betweenness. Theorem 6.2 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) ) 
 ->  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P )  /\  P  Btwn  <. A ,  C >. ) 
 ->  ( P  Btwn  <. B ,  C >. 
 <->  POutsideOf <. A ,  B >. ) ) )
 
Theorembroutsideof3 25308* Characterization of outsideness in terms of relationship to a fourth point. Theorem 6.3 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( POutsideOf <. A ,  B >.  <-> 
 ( A  =/=  P  /\  B  =/=  P  /\  E. c  e.  ( EE
 `  N ) ( c  =/=  P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B ,  c >. ) ) ) )
 
Theoremoutsideofrflx 25309 Reflexitivity of outsideness. Theorem 6.5 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  P  e.  ( EE
 `  N )  /\  A  e.  ( EE `  N ) )  ->  ( A  =/=  P  ->  POutsideOf <. A ,  A >. ) )
 
Theoremoutsideofcom 25310 Commutitivity law for outsideness. Theorem 6.6 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( POutsideOf <. A ,  B >.  <->  POutsideOf
 <. B ,  A >. ) )
 
Theoremoutsideoftr 25311 Transitivity law for outsideness. Theorem 6.7 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) ) 
 ->  ( ( POutsideOf <. A ,  B >.  /\  POutsideOf <. B ,  C >. )  ->  POutsideOf <. A ,  C >. ) )
 
Theoremoutsideofeq 25312 Uniqueness law for OutsideOf. Analog of segconeq 25192. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N ) 
 /\  Y  e.  ( EE `  N ) ) )  ->  ( (
 ( AOutsideOf <. X ,  R >.  /\  <. A ,  X >.Cgr
 <. B ,  C >. ) 
 /\  ( AOutsideOf <. Y ,  R >.  /\  <. A ,  Y >.Cgr <. B ,  C >. ) )  ->  X  =  Y ) )
 
Theoremoutsideofeu 25313* Given a non-degenerate ray, there is a unique point congruent to the segment  B C lying on the ray  A R. Theorem 6.11 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 23-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) ) 
 ->  ( ( R  =/=  A 
 /\  B  =/=  C )  ->  E! x  e.  ( EE `  N ) ( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr <. B ,  C >. ) ) )
 
Theoremoutsidele 25314 Relate OutsideOf to  Seg<_. Theorem 6.13 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( POutsideOf <. A ,  B >.  ->  ( <. P ,  A >.  Seg<_  <. P ,  B >.  <->  A  Btwn  <. P ,  B >. ) ) )
 
Theoremoutsideofcol 25315 Outside of implies colinearity. (Contributed by Scott Fenton, 26-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( POutsideOf
 <. Q ,  R >.  ->  P 
 Colinear 
 <. Q ,  R >. )
 
18.7.49  Lines and Rays
 
Syntaxcline2 25316 Declare the constant for the line function.
 class Line
 
Syntaxcray 25317 Declare the constant for the ray function.
 class Ray
 
Syntaxclines2 25318 Declare the constant for the set of all lines.
 class LinesEE
 
Definitiondf-line2 25319* Define the Line function. This function generates the line passing through the distinct points  a and  b. Adapted from definition 6.14 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 25-Oct-2013.)
 |- Line  =  { <.
 <. a ,  b >. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b
 )  /\  l  =  [ <. a ,  b >. ] `'  Colinear  ) }
 
Definitiondf-ray 25320* Define the Ray function. This function generates the set of all points that lie on the ray starting at  p and passing through  a. Definition 6.8 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 21-Oct-2013.)
 |- Ray  =  { <.
 <. p ,  a >. ,  r >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n )  /\  a  e.  ( EE `  n )  /\  p  =/=  a
 )  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) }
 
Definitiondf-lines2 25321 Define the set of all lines. Definition 6.14, part 2 of [Schwabhauser] p. 45. See ellines 25334 for membership. (Contributed by Scott Fenton, 28-Oct-2013.)
 |- LinesEE  =  ran Line
 
Theoremfunray 25322 Show that the Ray relationship is a function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Fun Ray
 
Theoremfvray 25323* Calculate the value of the Ray function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  ->  ( PRay A )  =  { x  e.  ( EE `  N )  |  POutsideOf <. A ,  x >. } )
 
Theoremfunline 25324 Show that the Line relationship is a function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Fun Line
 
Theoremlinedegen 25325 When Line is applied with the same argument, the result is the empty set. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ALine A )  =  (/)
 
Theoremfvline 25326* Calculate the value of the Line function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  ->  ( ALine B )  =  { x  |  x  Colinear  <. A ,  B >. } )
 
Theoremliness 25327 A line is a subset of the space its two points lie in. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  ->  ( ALine B )  C_  ( EE `  N ) )
 
Theoremfvline2 25328* Alternate definition of a line. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  ->  ( ALine B )  =  { x  e.  ( EE `  N )  |  x  Colinear  <. A ,  B >. } )
 
Theoremlineunray 25329 A line is composed of a point and the two rays emerging from it. Theorem 6.15 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 26-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  ( P  Btwn  <. Q ,  R >.  ->  ( PLine Q )  =  ( (
 ( PRay Q )  u.  { P }
 )  u.  ( PRay R ) ) ) )
 
Theoremlineelsb2 25330 If  S lies on  P Q, then 
P Q  =  P S. Theorem 6.16 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  /\  ( S  e.  ( EE `  N )  /\  P  =/=  S ) )  ->  ( S  e.  ( PLine Q )  ->  ( PLine Q )  =  ( PLine S ) ) )
 
Theoremlinerflx1 25331 Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  ->  P  e.  ( PLine Q ) )
 
Theoremlinecom 25332 Commutativity law for lines. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  ->  ( PLine Q )  =  ( QLine P ) )
 
Theoremlinerflx2 25333 Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  ->  Q  e.  ( PLine Q ) )
 
Theoremellines 25334* Membership in the set of all lines. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( A  e. LinesEE  <->  E. n  e.  NN  E. p  e.  ( EE
 `  n ) E. q  e.  ( EE `  n ) ( p  =/=  q  /\  A  =  ( pLine q ) ) )
 
Theoremlinethru 25335 If  A is a line containing two distinct points  P and  Q, then  A is the line through  P and  Q. Theorem 6.18 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e. LinesEE  /\  ( P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q ) 
 ->  A  =  ( PLine
 Q ) )
 
Theoremhilbert1.1 25336* There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  ->  E. x  e. LinesEE  ( P  e.  x  /\  Q  e.  x ) )
 
Theoremhilbert1.2 25337* There is at most one line through any two distinct points. Hilbert's axiom I.2 for geometry. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by NM, 17-Jun-2017.)
 |-  ( P  =/=  Q  ->  E* x  e. LinesEE ( P  e.  x  /\  Q  e.  x ) )
 
Theoremlinethrueu 25338* There is a unique line going through any two distinct points. Theorem 6.19 of [Schwabhauser] p. 46. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  ->  E! x  e. LinesEE  ( P  e.  x  /\  Q  e.  x ) )
 
Theoremlineintmo 25339* Two distinct lines intersect in at most one point. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e. LinesEE  /\  B  e. LinesEE 
 /\  A  =/=  B )  ->  E* x ( x  e.  A  /\  x  e.  B ) )
 
18.7.50  Bernoulli polynomials and sums of k-th powers
 
Syntaxcbp 25340 Declare the constant for the Bernoulli polynomial operator.
 class BernPoly
 
Definitiondf-bpoly 25341* Define the Bernoulli polynomials. Here we use well-founded recursion to define the Bernoulli polynomials. This agrees with most textbook definitions, although explicit formulae do exist. (Contributed by Scott Fenton, 22-May-2014.)
 |- BernPoly  =  ( m  e.  NN0 ,  x  e.  CC  |->  ( U. { f  |  E. s
 ( f  Fn  s  /\  ( s  C_  NN0  /\  A. e  e.  s  Pred (  <  ,  NN0 ,  e )  C_  s ) 
 /\  A. e  e.  s  ( f `  e
 )  =  ( ( g  e.  _V  |->  [_ ( # `  dom  g
 )  /  n ]_ (
 ( x ^ n )  -  sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `  k )  /  (
 ( n  -  k
 )  +  1 ) ) ) ) ) `
  ( f  |`  Pred
 (  <  ,  NN0 ,  e ) ) ) ) } `  m ) )
 
Theorembpolylem 25342* Lemma for bpolyval 25343. (Contributed by Scott Fenton, 22-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  G  =  ( g  e.  _V  |->  [_ ( # `  dom  g )  /  n ]_ ( ( X ^ n )  -  sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `
  k )  /  ( ( n  -  k )  +  1
 ) ) ) ) )   &    |-  F  =  U. { f  |  E. s
 ( f  Fn  s  /\  ( s  C_  NN0  /\  A. e  e.  s  Pred (  <  ,  NN0 ,  e )  C_  s ) 
 /\  A. e  e.  s  ( f `  e
 )  =  ( G `
  ( f  |`  Pred
 (  <  ,  NN0 ,  e ) ) ) ) }   =>    |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  =  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  (
 ( k BernPoly  X )  /  ( ( N  -  k )  +  1
 ) ) ) ) )
 
Theorembpolyval 25343* The value of the Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.)
 |-  (
 ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  =  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  (
 ( k BernPoly  X )  /  ( ( N  -  k )  +  1
 ) ) ) ) )
 
Theorembpoly0 25344 The value of the Bernoulli polynomials at zero. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( X  e.  CC  ->  ( 0 BernPoly  X )  =  1 )
 
Theorembpoly1 25345 The value of the Bernoulli polynomials at one. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( X  e.  CC  ->  ( 1 BernPoly  X )  =  ( X  -  ( 1 
 /  2 ) ) )
 
Theorembpolycl 25346 Closure law for Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 22-May-2014.)
 |-  (
 ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  e.  CC )
 
Theorembpolysum 25347* A sum for Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 22-May-2014.)
 |-  (
 ( N  e.  NN0  /\  X  e.  CC )  -> 
 sum_ k  e.  (
 0 ... N ) ( ( N  _C  k
 )  x.  ( ( k BernPoly  X )  /  (
 ( N  -  k
 )  +  1 ) ) )  =  ( X ^ N ) )
 
Theorembpolydiflem 25348* Lemma for bpolydif 25349. (Contributed by Scott Fenton, 12-Jun-2014.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ( ph  /\  k  e.  ( 1
 ... ( N  -  1 ) ) ) 
 ->  ( ( k BernPoly  ( X  +  1 )
 )  -  ( k BernPoly  X ) )  =  ( k  x.  ( X ^ ( k  -  1 ) ) ) )   =>    |-  ( ph  ->  (
 ( N BernPoly  ( X  +  1 ) )  -  ( N BernPoly  X ) )  =  ( N  x.  ( X ^ ( N  -  1 ) ) ) )
 
Theorembpolydif 25349 Calculate the difference between successive values of the Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 26-May-2014.)
 |-  (
 ( N  e.  NN  /\  X  e.  CC )  ->  ( ( N BernPoly  ( X  +  1 ) )  -  ( N BernPoly  X ) )  =  ( N  x.  ( X ^
 ( N  -  1
 ) ) ) )
 
Theoremfsumkthpow 25350* A closed-form expression for the sum of  K-th powers. (Contributed by Scott Fenton, 16-May-2014.) (Revised by Mario Carneiro, 16-Jun-2014.)
 |-  (
 ( K  e.  NN0  /\  M  e.  NN0 )  -> 
 sum_ n  e.  (
 0 ... M ) ( n ^ K )  =  ( ( ( ( K  +  1 ) BernPoly  ( M  +  1 ) )  -  (
 ( K  +  1 ) BernPoly  0 ) ) 
 /  ( K  +  1 ) ) )
 
Theorembpoly2 25351 The Bernoulli polynomials at two. (Contributed by Scott Fenton, 8-Jul-2015.)
 |-  ( X  e.  CC  ->  ( 2 BernPoly  X )  =  ( ( ( X ^
 2 )  -  X )  +  ( 1  /  6 ) ) )
 
Theorembpoly3 25352 The Bernoulli polynomials at three. (Contributed by Scott Fenton, 8-Jul-2015.)
 |-  ( X  e.  CC  ->  ( 3 BernPoly  X )  =  ( ( ( X ^
 3 )  -  (
 ( 3  /  2
 )  x.  ( X ^ 2 ) ) )  +  ( ( 1  /  2 )  x.  X ) ) )
 
Theorembpoly4 25353 The Bernoulli polynomials at four. (Contributed by Scott Fenton, 8-Jul-2015.)
 |-  ( X  e.  CC  ->  ( 4 BernPoly  X )  =  ( ( ( ( X ^ 4 )  -  ( 2  x.  ( X ^ 3 ) ) )  +  ( X ^ 2 ) )  -  ( 1  / ; 3 0 ) ) )
 
Theoremfsumcube 25354* Express the sum of cubes in closed terms. (Contributed by Scott Fenton, 16-Jun-2015.)
 |-  ( T  e.  NN0  ->  sum_ k  e.  ( 0 ... T ) ( k ^
 3 )  =  ( ( ( T ^
 2 )  x.  (
 ( T  +  1 ) ^ 2 ) )  /  4 ) )
 
18.7.51  Rank theorems
 
Theoremrankung 25355 The rank of the union of two sets. Closed form of rankun 7618. (Contributed by Scott Fenton, 15-Jul-2015.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( rank `  ( A  u.  B ) )  =  ( ( rank `  A )  u.  ( rank `  B ) ) )
 
Theoremranksng 25356 The rank of a singleton. Closed form of ranksn 7616. (Contributed by Scott Fenton, 15-Jul-2015.)
 |-  ( A  e.  V  ->  (
 rank `  { A }
 )  =  suc  ( rank `  A ) )
 
Theoremrankelg 25357 The membership relation is inherited by the rank function. Closed form of rankel 7601. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  (
 ( B  e.  V  /\  A  e.  B ) 
 ->  ( rank `  A )  e.  ( rank `  B )
 )
 
Theoremrankpwg 25358 The rank of a power set. Closed form of rankpw 7605. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  ( A  e.  V  ->  (
 rank `  ~P A )  =  suc  ( rank `  A ) )
 
Theoremrank0 25359 The rank of the empty set is 
(/) (Contributed by Scott Fenton, 17-Jul-2015.)
 |-  ( rank `  (/) )  =  (/)
 
Theoremrankeq1o 25360 The only set with rank  1o is the singleton of the empty set. (Contributed by Scott Fenton, 17-Jul-2015.)
 |-  (
 ( rank `  A )  =  1o  <->  A  =  { (/)
 } )
 
18.7.52  Hereditarily Finite Sets
 
Syntaxchf 25361 The constant Hf is a class.
 class Hf
 
Definitiondf-hf 25362 Define the hereditarily finite sets. These are the finite sets whose elements are finite, and so forth. (Contributed by Scott Fenton, 9-Jul-2015.)
 |- Hf  =  U. ( R1 " om )
 
Theoremelhf 25363* Membership in the hereditarily finite sets. (Contributed by Scott Fenton, 9-Jul-2015.)
 |-  ( A  e. Hf  <->  E. x  e.  om  A  e.  ( R1 `  x ) )
 
Theoremelhf2 25364 Alternate form of membership in the hereditarily finite sets. (Contributed by Scott Fenton, 13-Jul-2015.)
 |-  A  e.  _V   =>    |-  ( A  e. Hf  <->  ( rank `  A )  e.  om )
 
Theoremelhf2g 25365 Hereditarily finiteness via rank. Closed form of elhf2 25364. (Contributed by Scott Fenton, 15-Jul-2015.)
 |-  ( A  e.  V  ->  ( A  e. Hf  <->  ( rank `  A )  e.  om )
 )
 
Theorem0hf 25366 The empty set is a hereditarily finite set. (Contributed by Scott Fenton, 9-Jul-2015.)
 |-  (/)  e. Hf
 
Theoremhfun 25367 The union of two HF sets is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
 |-  (
 ( A  e. Hf  /\  B  e. Hf  )  ->  ( A  u.  B )  e. Hf  )
 
Theoremhfsn 25368 The singleton of an HF set is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
 |-  ( A  e. Hf  ->  { A }  e. Hf  )
 
Theoremhfadj 25369 Adjoining one HF element to an HF set preserves HF status. (Contributed by Scott Fenton, 15-Jul-2015.)
 |-  (
 ( A  e. Hf  /\  B  e. Hf  )  ->  ( A  u.  { B } )  e. Hf  )
 
Theoremhfelhf 25370 Any member of an HF set is itself an HF set. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  (
 ( A  e.  B  /\  B  e. Hf  )  ->  A  e. Hf  )
 
Theoremhftr 25371 The class of all hereditarily finite sets is transitive. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  Tr Hf
 
Theoremhfext 25372* Extensionality for HF sets depends only on comparison of HF elements. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  (
 ( A  e. Hf  /\  B  e. Hf  )  ->  ( A  =  B  <->  A. x  e. Hf  ( x  e.  A  <->  x  e.  B ) ) )
 
Theoremhfuni 25373 The union of an HF set is itself hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  ( A  e. Hf  ->  U. A  e. Hf  )
 
Theoremhfpw 25374 The power class of an HF set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  ( A  e. Hf  ->  ~P A  e. Hf  )
 
Theoremhfninf 25375  om is not hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  -.  om  e. Hf
 
18.8  Mathbox for Anthony Hart
 
18.8.1  Propositional Calculus
 
Theoremtb-ax1 25376 The first of three axioms in the Tarski-Bernays axiom system. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ps )  ->  ( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
 
Theoremtb-ax2 25377 The second of three axioms in the Tarski-Bernays axiom system. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ph ) )
 
Theoremtb-ax3 25378 The third of three axioms in the Tarski-Bernays axiom system.

This axiom, along with ax-mp 8, tb-ax1 25376, and tb-ax2 25377, can be used to derive any theorem or rule that uses only  ->. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  (
 ( ( ph  ->  ps )  ->  ph )  ->  ph )
 
Theoremtbsyl 25379 The weak syllogism from Tarski-Bernays'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ps  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theoremre1ax2lem 25380 Lemma for re1ax2 25381. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) )
 
Theoremre1ax2 25381 ax-2 6 rederived from the Tarski-Bernays axiom system. Often tb-ax1 25376 is replaced with this theorem to make a "standard" system. This is because this theorem is easier to work with, despite it being longer. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  ( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
 
Theoremnaim1 25382 Constructor theorem for  -/\. (Contributed by Anthony Hart, 1-Sep-2011.)
 |-  (
 ( ph  ->  ps )  ->  ( ( ps  -/\  ch )  ->  ( ph  -/\  ch )
 ) )
 
Theoremnaim2 25383 Constructor theorem for  -/\. (Contributed by Anthony Hart, 1-Sep-2011.)
 |-  (
 ( ph  ->  ps )  ->  ( ( ch  -/\  ps )  ->  ( ch  -/\  ph )
 ) )
 
Theoremnaim1i 25384 Constructor rule for  -/\. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ps  -/\  ch )   =>    |-  ( ph  -/\  ch )
 
Theoremnaim2i 25385 Constructor rule for  -/\. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ch  -/\  ps )   =>    |-  ( ch  -/\  ph )
 
Theoremnaim12i 25386 Constructor rule for  -/\. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   &    |-  ( ps  -/\  th )   =>    |-  ( ph  -/\  ch )
 
Theoremnabi1 25387 Constructor theorem for  -/\. (Contributed by Anthony Hart, 1-Sep-2011.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ph  -/\  ch )  <->  ( ps  -/\  ch )
 ) )
 
Theoremnabi2 25388 Constructor theorem for  -/\. (Contributed by Anthony Hart, 1-Sep-2011.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ch  -/\  ph )  <->  ( ch  -/\  ps )
 ) )
 
Theoremnabi1i 25389 Constructor rule for  -/\. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  ( ph 
 <->  ps )   &    |-  ( ps  -/\  ch )   =>    |-  ( ph  -/\  ch )
 
Theoremnabi2i 25390 Constructor rule for  -/\. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  ( ph 
 <->  ps )   &    |-  ( ch  -/\  ps )   =>    |-  ( ch  -/\  ph )
 
Theoremnabi12i 25391 Constructor rule for  -/\. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  ( ph 
 <->  ps )   &    |-  ( ch  <->  th )   &    |-  ( ps  -/\  th )   =>    |-  ( ph  -/\  ch )
 
Syntaxw3nand 25392 The double nand.
 wff  ( ph  -/\  ps  -/\  ch )
 
Definitiondf-3nand 25393 The double nand. This definition allows us to express the input of three variables only being false if all three are true. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  (
 ( ph  -/\  ps  -/\  ch )  <->  (
 ph  ->  ( ps  ->  -. 
 ch ) ) )
 
Theoremdf3nandALT1 25394 The double nand expressed in terms of pure nand. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  (
 ( ph  -/\  ps  -/\  ch )  <->  (
 ph  -/\  ( ( ps  -/\  ch )  -/\  ( ps  -/\  ch ) ) ) )
 
Theoremdf3nandALT2 25395 The double nand expressed in terms of negation and and. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  (
 ( ph  -/\  ps  -/\  ch )  <->  -.  ( ph  /\  ps  /\ 
 ch ) )
 
Theoremandnand1 25396 Double and in terms of double nand. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  (
 ( ph  /\  ps  /\  ch )  <->  ( ( ph  -/\ 
 ps  -/\  ch )  -/\  ( ph  -/\  ps  -/\  ch )
 ) )
 
Theoremimnand2 25397 An  -> nand relation. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  (
 ( -.  ph  ->  ps )  <->  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps ) ) )
 
18.8.2  Predicate Calculus
 
Theoremquantriv 25398* Any wff can be trivially quantified, so long as the quantifier's set is distinct from said wff.

See also 19.9v 1664. (Contributed by Anthony Hart, 13-Sep-2011.)

 |-  ( A. x ph  <->  ph )
 
Theoremallt 25399 For all sets,  T. is true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  A. x  T.
 
Theoremalnof 25400 For all sets,  F. is not true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  A. x  -.  F.
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