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Theorem List for Metamath Proof Explorer - 27101-27200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremstoweidlem29 27101* When the hypothesis for the extreme value theorem hold, then the inf of the range of the function belongs to the range, it is real and it a lower bound of the range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |- 
 F/ t ph   &    |-  T  =  U. J   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  T  =/=  (/) )   =>    |-  ( ph  ->  ( sup ( ran  F ,  RR ,  `'  <  )  e.  ran  F  /\  sup ( ran  F ,  RR ,  `'  <  )  e.  RR  /\  A. t  e.  T  sup ( ran 
 F ,  RR ,  `'  <  )  <_  ( F `  t ) ) )
 
Theoremstoweidlem30 27102* This lemma is used to prove the existence of a function p as in Lemma 1 [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Z is used for t0, P is used for p,  ( G `  i ) is used for p(t_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `
  i ) `  t ) ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  G : ( 1 ...
 M ) --> Q )   &    |-  ( ( ph  /\  f  e.  A )  ->  f : T --> RR )   =>    |-  ( ( ph  /\  S  e.  T ) 
 ->  ( P `  S )  =  ( (
 1  /  M )  x.  sum_ i  e.  (
 1 ... M ) ( ( G `  i
 ) `  S )
 ) )
 
Theoremstoweidlem31 27103* This lemma is used to prove that there exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91: assuming that  R is a finite subset of  V,  x indexes a finite set of functions in the subalgebra (of the Stone Weierstrass theorem), such that for all  i ranging in the finite indexing set, 0 ≤ xi ≤ 1, xi < ε / m on V(ti), and xi > 1 - ε / m on  B. Here M is used to represent m in the paper,  E is used to represent ε in the paper, vi is used to represent V(ti). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ h ph   &    |-  F/ t ph   &    |-  F/ w 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 ) ) }   &    |-  G  =  ( w  e.  R  |->  { h  e.  A  |  ( A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 )  /\  A. t  e.  w  ( h `  t )  < 
 ( E  /  M )  /\  A. t  e.  ( T  \  U ) ( 1  -  ( E  /  M ) )  <  ( h `
  t ) ) } )   &    |-  ( ph  ->  R 
 C_  V )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  v : ( 1 ...
 M ) -1-1-onto-> R )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  B 
 C_  ( T  \  U ) )   &    |-  ( ph  ->  V  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  ran 
 G  e.  Fin )   =>    |-  ( ph  ->  E. x ( x : ( 1 ...
 M ) --> Y  /\  A. i  e.  ( 1
 ... M ) (
 A. t  e.  (
 v `  i )
 ( ( x `  i ) `  t
 )  <  ( E  /  M )  /\  A. t  e.  B  (
 1  -  ( E 
 /  M ) )  <  ( ( x `
  i ) `  t ) ) ) )
 
Theoremstoweidlem32 27104* If a set A of real functions from a common domain T is a subalgebra and it contains constants, then it is closed under the sum of a finite number of functions, indexed by G and finally scaled by a real Y. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  P  =  ( t  e.  T  |->  ( Y  x.  sum_ i  e.  ( 1 ... M ) ( ( G `
  i ) `  t ) ) )   &    |-  F  =  ( t  e.  T  |->  sum_ i  e.  (
 1 ... M ) ( ( G `  i
 ) `  t )
 )   &    |-  H  =  ( t  e.  T  |->  Y )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  G : ( 1 ... M ) --> A )   &    |-  ( ( 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  /\  f  e.  A )  ->  f : T --> RR )   =>    |-  ( ph  ->  P  e.  A )
 
Theoremstoweidlem33 27105* If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |-  F/_ t G   &    |-  F/ t ph   &    |-  (
 ( 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  /\  F  e.  A  /\  G  e.  A )  ->  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  e.  A )
 
Theoremstoweidlem34 27106* This lemma proves that for all  t in  T there is a  j as in the proof of [BrosowskiDeutsh] p. 91 (at the bottom of page 91 and at the top of page 92): (j-4/3) * ε < f(t) <= (j-1/3) * ε , g(t) < (j+1/3) * ε, and g(t) > (j-4/3) * ε. Here  E is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |- 
 F/ j ph   &    |-  F/ t ph   &    |-  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 ) } )   &    |-  J  =  ( t  e.  T  |->  { j  e.  ( 1
 ... N )  |  t  e.  ( D `
  j ) }
 )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T  e.  _V )   &    |-  ( ph  ->  F : T --> RR )   &    |-  ( ( ph  /\  t  e.  T ) 
 ->  0  <_  ( F `
  t ) )   &    |-  ( ( ph  /\  t  e.  T )  ->  ( F `  t )  < 
 ( ( N  -  1 )  x.  E ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   &    |-  (
 ( ph  /\  j  e.  ( 0 ... N ) )  ->  ( X `
  j ) : T --> RR )   &    |-  (
 ( ph  /\  j  e.  ( 0 ... N )  /\  t  e.  T )  ->  0  <_  (
 ( X `  j
 ) `  t )
 )   &    |-  ( ( ph  /\  j  e.  ( 0 ... N )  /\  t  e.  T )  ->  ( ( X `
  j ) `  t )  <_  1 )   &    |-  ( ( ph  /\  j  e.  ( 0 ... N )  /\  t  e.  ( D `  j ) ) 
 ->  ( ( X `  j ) `  t
 )  <  ( E  /  N ) )   &    |-  (
 ( ph  /\  j  e.  ( 0 ... N )  /\  t  e.  ( B `  j ) ) 
 ->  ( 1  -  ( E  /  N ) )  <  ( ( X `
  j ) `  t ) )   =>    |-  ( ph  ->  A. t  e.  T  E. j  e.  RR  (
 ( ( ( j  -  ( 4  / 
 3 ) )  x.  E )  <  ( F `  t )  /\  ( F `  t ) 
 <_  ( ( j  -  ( 1  /  3
 ) )  x.  E ) )  /\  ( ( ( t  e.  T  |->  sum_
 i  e.  ( 0
 ... N ) ( E  x.  ( ( X `  i ) `
  t ) ) ) `  t )  <  ( ( j  +  ( 1  / 
 3 ) )  x.  E )  /\  (
 ( j  -  (
 4  /  3 )
 )  x.  E )  <  ( ( t  e.  T  |->  sum_ i  e.  ( 0 ... N ) ( E  x.  ( ( X `  i ) `  t
 ) ) ) `  t ) ) ) )
 
Theoremstoweidlem35 27107* 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. Here  ( q `  i ) is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  F/ w ph   &    |-  F/ h ph   &    |-  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
 ) } }   &    |-  G  =  ( w  e.  X  |->  { h  e.  Q  |  w  =  { t  e.  T  |  0  < 
 ( h `  t
 ) } } )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  X 
 C_  W )   &    |-  ( ph  ->  ( T  \  U )  C_  U. X )   &    |-  ( ph  ->  ( T  \  U )  =/=  (/) )   =>    |-  ( ph  ->  E. m E. q ( m  e. 
 NN  /\  ( q : ( 1 ... m ) --> Q  /\  A. t  e.  ( T 
 \  U ) E. i  e.  ( 1 ... m ) 0  < 
 ( ( q `  i ) `  t
 ) ) ) )
 
Theoremstoweidlem36 27108* This lemma is used to prove the existence of a function pt as in Lemma 1 of [BrosowskiDeutsh] p. 90 (at the beginning of Lemma 1): for all t in T - U, there exists a function p in the subalgebra, such that pt ( t0 ) = 0 , pt ( t ) > 0, and 0 <= pt <= 1. Z is used for t0 , S is used for t e. T - U , h is used for pt . G is used for (ht)^2 and the final h is a normalized version of G ( divided by its norm, see the variable N ). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ h Q   &    |-  F/_ t H   &    |-  F/_ t F   &    |-  F/_ t G   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  T  =  U. J   &    |-  G  =  ( t  e.  T  |->  ( ( F `  t
 )  x.  ( F `
  t ) ) )   &    |-  N  =  sup ( ran  G ,  RR ,  <  )   &    |-  H  =  ( t  e.  T  |->  ( ( G `  t
 )  /  N )
 )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  A 
 C_  ( J  Cn  K ) )   &    |-  (
 ( 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  ->  S  e.  T )   &    |-  ( ph  ->  Z  e.  T )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  ( F `  S )  =/=  ( F `  Z ) )   &    |-  ( ph  ->  ( F `  Z )  =  0 )   =>    |-  ( ph  ->  E. h ( h  e.  Q  /\  0  < 
 ( h `  S ) ) )
 
Theoremstoweidlem37 27109* 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. Z is used for t0, P is used for p,  ( G `  i ) is used for p(t_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `
  i ) `  t ) ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  G : ( 1 ...
 M ) --> Q )   &    |-  ( ( ph  /\  f  e.  A )  ->  f : T --> RR )   &    |-  ( ph  ->  Z  e.  T )   =>    |-  ( ph  ->  ( P `  Z )  =  0 )
 
Theoremstoweidlem38 27110* 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. Z is used for t0, P is used for p,  ( G `  i ) is used for p(t_i). (Contributed by GlaucoSiliprandi, 20-Apr-2017.)
 |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `
  i ) `  t ) ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  G : ( 1 ...
 M ) --> Q )   &    |-  ( ( ph  /\  f  e.  A )  ->  f : T --> RR )   =>    |-  ( ( ph  /\  S  e.  T ) 
 ->  ( 0  <_  ( P `  S )  /\  ( P `  S ) 
 <_  1 ) )
 
Theoremstoweidlem39 27111* This lemma is used to prove that there exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91: assuming that  r is a finite subset of  W,  x indexes a finite set of functions in the subalgebra (of the Stone Weierstrass theorem), such that for all i ranging in the finite indexing set, 0 ≤ xi ≤ 1, xi < ε / m on V(ti), and xi > 1 - ε / m on  B. Here  D is used to represent A in the paper's Lemma 2 (because  A is used for the subalgebra),  M is used to represent m in the paper,  E is used to represent ε, and vi is used to represent V(ti).  W is just a local definition, used to shorten statements. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ h ph   &    |-  F/ t ph   &    |-  F/ w ph   &    |-  U  =  ( T  \  B )   &    |-  Y  =  { h  e.  A  |  A. t  e.  T  ( 0  <_  ( h `  t ) 
 /\  ( h `  t )  <_  1 ) }   &    |-  W  =  { 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  ->  r  e.  ( ~P W  i^i  Fin )
 )   &    |-  ( ph  ->  D  C_ 
 U. r )   &    |-  ( ph  ->  D  =/=  (/) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  B  C_  T )   &    |-  ( ph  ->  W  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   =>    |-  ( ph  ->  E. m  e.  NN  E. v ( v : ( 1
 ... m ) --> W  /\  D  C_  U. ran  v  /\  E. x ( x : ( 1 ... m ) --> Y  /\  A. i  e.  ( 1
 ... m ) (
 A. t  e.  (
 v `  i )
 ( ( x `  i ) `  t
 )  <  ( E  /  m )  /\  A. t  e.  B  (
 1  -  ( E 
 /  m ) )  <  ( ( x `
  i ) `  t ) ) ) ) )
 
Theoremstoweidlem40 27112* This lemma proves that qn is in the subalgebra, as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90. Q is used to represent qn in the paper, N is used to represent n in the paper, and M is used to represent k^n in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t P   &    |- 
 F/ t ph   &    |-  Q  =  ( t  e.  T  |->  ( ( 1  -  (
 ( P `  t
 ) ^ N ) ) ^ M ) )   &    |-  F  =  ( t  e.  T  |->  ( 1  -  ( ( P `  t ) ^ N ) ) )   &    |-  G  =  ( t  e.  T  |->  1 )   &    |-  H  =  ( t  e.  T  |->  ( ( P `  t
 ) ^ N ) )   &    |-  ( ph  ->  P  e.  A )   &    |-  ( ph  ->  P : T --> RR )   &    |-  ( ( 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  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  NN )   =>    |-  ( ph  ->  Q  e.  A )
 
Theoremstoweidlem41 27113* This lemma is used to prove that there exists x as in Lemma 1 of [BrosowskiDeutsh] p. 90: 0 <= x(t) <= 1 for all t in T, x(t) < epsilon for all t in V, x(t) > 1 - epsilon for all t in T \ U. Here we prove the very last step of the proof of Lemma 1: "The result follows from taking x = 1 - qn";. Here  E is used to represent ε in the paper, and  y to represent qn in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  X  =  ( t  e.  T  |->  ( 1  -  ( y `
  t ) ) )   &    |-  F  =  ( t  e.  T  |->  1 )   &    |-  V  C_  T   &    |-  ( ph  ->  y  e.  A )   &    |-  ( ph  ->  y : T --> RR )   &    |-  (
 ( 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  /\  w  e.  RR )  ->  (
 t  e.  T  |->  w )  e.  A )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  A. t  e.  T  ( 0  <_  ( y `  t )  /\  (
 y `  t )  <_  1 ) )   &    |-  ( ph  ->  A. t  e.  V  ( 1  -  E )  <  ( y `  t ) )   &    |-  ( ph  ->  A. t  e.  ( T  \  U ) ( y `  t )  <  E )   =>    |-  ( ph  ->  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 ) ) )
 
Theoremstoweidlem42 27114* This lemma is used to prove that  x built as in Lemma 2 of [BrosowskiDeutsh] p. 91, is such that x > 1 - ε on B. Here  X is used to represent  x in the paper, and E is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ i ph   &    |-  F/ t ph   &    |-  F/_ t Y   &    |-  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  ->  U : ( 1 ...
 M ) --> Y )   &    |-  ( ( ph  /\  i  e.  ( 1 ... M ) )  ->  A. t  e.  B  ( 1  -  ( E  /  M ) )  <  ( ( U `  i ) `
  t ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   &    |-  (
 ( ph  /\  f  e.  Y )  ->  f : T --> RR )   &    |-  (
 ( ph  /\  f  e.  Y  /\  g  e.  Y )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  Y )   &    |-  ( ph  ->  T  e.  _V )   &    |-  ( ph  ->  B 
 C_  T )   =>    |-  ( ph  ->  A. t  e.  B  ( 1  -  E )  <  ( X `  t ) )
 
Theoremstoweidlem43 27115* This lemma is used to prove the existence of a function pt as in Lemma 1 of [BrosowskiDeutsh] p. 90 (at the beginning of Lemma 1): for all t in T - U, there exists a function pt in the subalgebra, such that pt( t0 ) = 0 , pt ( t ) > 0, and 0 <= pt <= 1. Hera Z is used for t0 , S is used for t e. T - U , h is used for pt. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ g ph   &    |-  F/ t ph   &    |-  F/_ h Q   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  T  =  U. J   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  A 
 C_  ( J  Cn  K ) )   &    |-  (
 ( ph  /\  f  e.  A  /\  l  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( l `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  l  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( l `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ( ph  /\  (
 r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) 
 ->  E. g  e.  A  ( g `  r
 )  =/=  ( g `  t ) )   &    |-  ( ph  ->  U  e.  J )   &    |-  ( ph  ->  Z  e.  U )   &    |-  ( ph  ->  S  e.  ( T  \  U ) )   =>    |-  ( ph  ->  E. h ( h  e.  Q  /\  0  < 
 ( h `  S ) ) )
 
Theoremstoweidlem44 27116* 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. Z is used to represent t0 in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ j ph   &    |-  F/ t ph   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `
  i ) `  t ) ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  G : ( 1 ...
 M ) --> Q )   &    |-  ( ph  ->  A. t  e.  ( T  \  U ) E. j  e.  (
 1 ... M ) 0  <  ( ( G `
  j ) `  t ) )   &    |-  T  =  U. J   &    |-  ( 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  ->  Z  e.  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 ) ) )
 
Theoremstoweidlem45 27117* 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 represent  k,  D to represent δ,  E to represent ε, and  P to represent  p. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t P   &    |- 
 F/ t ph   &    |-  V  =  {
 t  e.  T  |  ( P `  t )  <  ( D  / 
 2 ) }   &    |-  Q  =  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N ) ) ^ ( K ^ N ) ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( 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  ->  ( 1  -  E )  <  ( 1  -  ( ( ( K  x.  D )  / 
 2 ) ^ N ) ) )   &    |-  ( ph  ->  ( 1  /  ( ( K  x.  D ) ^ N ) )  <  E )   =>    |-  ( 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 ) )
 
Theoremstoweidlem46 27118* 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 )
 
Theoremstoweidlem47 27119* 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 )
 
Theoremstoweidlem48 27120* 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 )
 
Theoremstoweidlem49 27121* 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 ) )
 
Theoremstoweidlem50 27122* 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 ) )
 
Theoremstoweidlem51 27123* 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 ) ) ) )
 
Theoremstoweidlem52 27124* 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
 ) ) ) )
 
Theoremstoweidlem53 27125* 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 ) ) )
 
Theoremstoweidlem54 27126* 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 ) ) )
 
Theoremstoweidlem55 27127* 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 ) ) )
 
Theoremstoweidlem56 27128* 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
 ) ) ) )
 
Theoremstoweidlem57 27129* 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 ) ) )
 
Theoremstoweidlem58 27130* 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 ) ) )
 
Theoremstoweidlem59 27131* 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 ) ) ) )
 
Theoremstoweidlem60 27132* 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 ) ) ) )
 
Theoremstoweidlem61 27133* 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 ) )
 
Theoremstoweidlem62 27134* 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 )
 
Theoremstoweid 27135* 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 )
 
Theoremstowei 27136* 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 27135: often times it will be better to use stoweid 27135 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
 
18.19.7  Wallis' product for π
 
Theoremwallispilem1 27137*  I is monotone: increasing the exponent, the integral decreases. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  I  =  ( n  e.  NN0  |->  S. ( 0 (,) pi ) ( ( sin `  x ) ^ n )  _d x )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( I `  ( N  +  1 ) )  <_  ( I `  N ) )
 
Theoremwallispilem2 27138* A first set of properties for the sequence  I that will be used in the proof of the Wallis product formula (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  I  =  ( n  e.  NN0  |->  S. ( 0 (,) pi ) ( ( sin `  x ) ^ n )  _d x )   =>    |-  ( ( I `
  0 )  =  pi  /\  ( I `
  1 )  =  2  /\  ( N  e.  ( ZZ>= `  2
 )  ->  ( I `  N )  =  ( ( ( N  -  1 )  /  N )  x.  ( I `  ( N  -  2
 ) ) ) ) )
 
Theoremwallispilem3 27139* I maps to real values (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  I  =  ( n  e.  NN0  |->  S. ( 0 (,) pi ) ( ( sin `  x ) ^ n )  _d x )   =>    |-  ( N  e.  NN0 
 ->  ( I `  N )  e.  RR+ )
 
Theoremwallispilem4 27140*  F maps to explicit expression for the ratio of two consecutive values of  I (Contributed by Glauco Siliprandi, 30-Jun-2017.)
 |-  F  =  ( k  e.  NN  |->  ( ( ( 2  x.  k )  /  ( ( 2  x.  k )  -  1
 ) )  x.  (
 ( 2  x.  k
 )  /  ( (
 2  x.  k )  +  1 ) ) ) )   &    |-  I  =  ( n  e.  NN0  |->  S. (
 0 (,) pi ) ( ( sin `  z
 ) ^ n )  _d z )   &    |-  G  =  ( n  e.  NN  |->  ( ( I `  ( 2  x.  n ) )  /  ( I `  ( ( 2  x.  n )  +  1 ) ) ) )   &    |-  H  =  ( n  e.  NN  |->  ( ( pi  /  2
 )  x.  ( 1 
 /  (  seq  1
 (  x.  ,  F ) `  n ) ) ) )   =>    |-  G  =  H
 
Theoremwallispilem5 27141* The sequence  H converges to 1. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
 |-  F  =  ( k  e.  NN  |->  ( ( ( 2  x.  k )  /  ( ( 2  x.  k )  -  1
 ) )  x.  (
 ( 2  x.  k
 )  /  ( (
 2  x.  k )  +  1 ) ) ) )   &    |-  I  =  ( n  e.  NN0  |->  S. (
 0 (,) pi ) ( ( sin `  x ) ^ n )  _d x )   &    |-  G  =  ( n  e.  NN  |->  ( ( I `  (
 2  x.  n ) )  /  ( I `
  ( ( 2  x.  n )  +  1 ) ) ) )   &    |-  H  =  ( n  e.  NN  |->  ( ( pi  /  2
 )  x.  ( 1 
 /  (  seq  1
 (  x.  ,  F ) `  n ) ) ) )   &    |-  L  =  ( n  e.  NN  |->  ( ( ( 2  x.  n )  +  1 )  /  ( 2  x.  n ) ) )   =>    |-  H  ~~>  1
 
Theoremwallispi 27142* Wallis' formula for π : Wallis' product converges to π / 2 . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F  =  ( k  e.  NN  |->  ( ( ( 2  x.  k )  /  ( ( 2  x.  k )  -  1
 ) )  x.  (
 ( 2  x.  k
 )  /  ( (
 2  x.  k )  +  1 ) ) ) )   &    |-  W  =  ( n  e.  NN  |->  ( 
 seq  1 (  x. 
 ,  F ) `  n ) )   =>    |-  W  ~~>  ( pi  /  2 )
 
Theoremwallispi2lem1 27143 An intermediate step between the first version of the Wallis' formula for π and the second version of Wallis' formula. This second version will then be used to prove Stirling's approximation formula for the factorial. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
 |-  ( N  e.  NN  ->  ( 
 seq  1 (  x. 
 ,  ( k  e. 
 NN  |->  ( ( ( 2  x.  k ) 
 /  ( ( 2  x.  k )  -  1 ) )  x.  ( ( 2  x.  k )  /  (
 ( 2  x.  k
 )  +  1 ) ) ) ) ) `
  N )  =  ( ( 1  /  ( ( 2  x.  N )  +  1 ) )  x.  (  seq  1 (  x.  ,  ( k  e.  NN  |->  ( ( ( 2  x.  k ) ^
 4 )  /  (
 ( ( 2  x.  k )  x.  (
 ( 2  x.  k
 )  -  1 ) ) ^ 2 ) ) ) ) `  N ) ) )
 
Theoremwallispi2lem2 27144 Two expressions are proven to be equal, and this is used to complete the proof of the second version of Wallis' formula for π . (Contributed by Glauco Siliprandi, 30-Jun-2017.)
 |-  ( N  e.  NN  ->  ( 
 seq  1 (  x. 
 ,  ( k  e. 
 NN  |->  ( ( ( 2  x.  k ) ^ 4 )  /  ( ( ( 2  x.  k )  x.  ( ( 2  x.  k )  -  1
 ) ) ^ 2
 ) ) ) ) `
  N )  =  ( ( ( 2 ^ ( 4  x.  N ) )  x.  ( ( ! `  N ) ^ 4
 ) )  /  (
 ( ! `  (
 2  x.  N ) ) ^ 2 ) ) )
 
Theoremwallispi2 27145 An alternative version of Wallis' formula for π ; this second formula uses factorials and it is later used to proof Stirling's approximation formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  V  =  ( n  e.  NN  |->  ( ( ( ( 2 ^ ( 4  x.  n ) )  x.  ( ( ! `
  n ) ^
 4 ) )  /  ( ( ! `  ( 2  x.  n ) ) ^ 2
 ) )  /  (
 ( 2  x.  n )  +  1 )
 ) )   =>    |-  V  ~~>  ( pi  / 
 2 )
 
18.19.8  Stirling's approximation formula for ` n ` factorial
 
Theoremstirlinglem1 27146 A simple limit of fractions is computed. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
 |-  H  =  ( n  e.  NN  |->  ( ( n ^
 2 )  /  ( n  x.  ( ( 2  x.  n )  +  1 ) ) ) )   &    |-  F  =  ( n  e.  NN  |->  ( 1  -  ( 1 
 /  ( ( 2  x.  n )  +  1 ) ) ) )   &    |-  G  =  ( n  e.  NN  |->  ( 1  /  ( ( 2  x.  n )  +  1 ) ) )   &    |-  L  =  ( n  e.  NN  |->  ( 1  /  n ) )   =>    |-  H  ~~>  ( 1  / 
 2 )
 
Theoremstirlinglem2 27147  A maps to positive reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   =>    |-  ( N  e.  NN  ->  ( A `  N )  e.  RR+ )
 
Theoremstirlinglem3 27148 Long but simple algebraic transformations are applied to show that  V, the Wallis formula for π , can be expressed in terms of  A, the Stirling's approximation formula for the factorial, up to a constant factor. This will allow (in a later theorem) to determine the right constant factor to be put into the  A, in order to get the exact Stirling's formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  D  =  ( n  e.  NN  |->  ( A `  ( 2  x.  n ) ) )   &    |-  E  =  ( n  e.  NN  |->  ( ( sqr `  (
 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) )   &    |-  V  =  ( n  e.  NN  |->  ( ( ( ( 2 ^ ( 4  x.  n ) )  x.  ( ( ! `  n ) ^ 4
 ) )  /  (
 ( ! `  (
 2  x.  n ) ) ^ 2 ) )  /  ( ( 2  x.  n )  +  1 ) ) )   =>    |-  V  =  ( n  e.  NN  |->  ( ( ( ( A `  n ) ^ 4
 )  /  ( ( D `  n ) ^
 2 ) )  x.  ( ( n ^
 2 )  /  ( n  x.  ( ( 2  x.  n )  +  1 ) ) ) ) )
 
Theoremstirlinglem4 27149* Algebraic manipulation of  ( ( B n ) - ( B  ( n  +  1 ) ) ). It will be used in other theorems to show that  B is decreasing. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   &    |-  J  =  ( n  e.  NN  |->  ( ( ( ( 1  +  (
 2  x.  n ) )  /  2 )  x.  ( log `  (
 ( n  +  1 )  /  n ) ) )  -  1
 ) )   =>    |-  ( N  e.  NN  ->  ( ( B `  N )  -  ( B `  ( N  +  1 ) ) )  =  ( J `  N ) )
 
Theoremstirlinglem5 27150* If  T is between  0 and  1, then a series (without alternating negative and positive terms) is given that converges to log (1+T)/(1-T) . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  D  =  ( j  e.  NN  |->  ( ( -u 1 ^ ( j  -  1 ) )  x.  ( ( T ^
 j )  /  j
 ) ) )   &    |-  E  =  ( j  e.  NN  |->  ( ( T ^
 j )  /  j
 ) )   &    |-  F  =  ( j  e.  NN  |->  ( ( ( -u 1 ^ ( j  -  1 ) )  x.  ( ( T ^
 j )  /  j
 ) )  +  (
 ( T ^ j
 )  /  j )
 ) )   &    |-  H  =  ( j  e.  NN0  |->  ( 2  x.  ( ( 1 
 /  ( ( 2  x.  j )  +  1 ) )  x.  ( T ^ (
 ( 2  x.  j
 )  +  1 ) ) ) ) )   &    |-  G  =  ( j  e.  NN0  |->  ( ( 2  x.  j )  +  1 ) )   &    |-  ( ph  ->  T  e.  RR+ )   &    |-  ( ph  ->  ( abs `  T )  < 
 1 )   =>    |-  ( ph  ->  seq  0
 (  +  ,  H ) 
 ~~>  ( log `  (
 ( 1  +  T )  /  ( 1  -  T ) ) ) )
 
Theoremstirlinglem6 27151* A series that converges to log (N+1)/N (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  H  =  ( j  e.  NN0  |->  ( 2  x.  (
 ( 1  /  (
 ( 2  x.  j
 )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ ( ( 2  x.  j )  +  1 ) ) ) ) )   =>    |-  ( N  e.  NN  ->  seq  0 (  +  ,  H )  ~~>  ( log `  ( ( N  +  1 )  /  N ) ) )
 
Theoremstirlinglem7 27152* Algebraic manipulation of the formula for J(n) (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  J  =  ( n  e.  NN  |->  ( ( ( ( 1  +  ( 2  x.  n ) ) 
 /  2 )  x.  ( log `  (
 ( n  +  1 )  /  n ) ) )  -  1
 ) )   &    |-  K  =  ( k  e.  NN  |->  ( ( 1  /  (
 ( 2  x.  k
 )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ ( 2  x.  k ) ) ) )   &    |-  H  =  ( k  e.  NN0  |->  ( 2  x.  ( ( 1 
 /  ( ( 2  x.  k )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ (
 ( 2  x.  k
 )  +  1 ) ) ) ) )   =>    |-  ( N  e.  NN  ->  seq  1 (  +  ,  K )  ~~>  ( J `  N ) )
 
Theoremstirlinglem8 27153 If  A converges to  C, then  F converges to C^2 . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F/ n ph   &    |-  F/_ n A   &    |-  F/_ n D   &    |-  D  =  ( n  e.  NN  |->  ( A `
  ( 2  x.  n ) ) )   &    |-  ( ph  ->  A : NN
 --> RR+ )   &    |-  F  =  ( n  e.  NN  |->  ( ( ( A `  n ) ^ 4
 )  /  ( ( D `  n ) ^
 2 ) ) )   &    |-  L  =  ( n  e.  NN  |->  ( ( A `
  n ) ^
 4 ) )   &    |-  M  =  ( n  e.  NN  |->  ( ( D `  n ) ^ 2
 ) )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( D `  n )  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A  ~~>  C )   =>    |-  ( ph  ->  F  ~~>  ( C ^ 2 ) )
 
Theoremstirlinglem9 27154*  ( ( B `  N )  -  ( B `  ( N  +  1
) ) ) is expressed as a limit of a series. This result will be used both to prove that  B is decreasing and to prove that  B is bounded (below). It will follow that  B converges in the reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   &    |-  J  =  ( n  e.  NN  |->  ( ( ( ( 1  +  (
 2  x.  n ) )  /  2 )  x.  ( log `  (
 ( n  +  1 )  /  n ) ) )  -  1
 ) )   &    |-  K  =  ( k  e.  NN  |->  ( ( 1  /  (
 ( 2  x.  k
 )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ ( 2  x.  k ) ) ) )   =>    |-  ( N  e.  NN  ->  seq  1 (  +  ,  K )  ~~>  ( ( B `  N )  -  ( B `  ( N  +  1 ) ) ) )
 
Theoremstirlinglem10 27155* A bound for any B(N)-B(N + 1) that will allow to find a lower bound for the whole  B sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   &    |-  K  =  ( k  e.  NN  |->  ( ( 1 
 /  ( ( 2  x.  k )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ (
 2  x.  k ) ) ) )   &    |-  L  =  ( k  e.  NN  |->  ( ( 1  /  ( ( ( 2  x.  N )  +  1 ) ^ 2
 ) ) ^ k
 ) )   =>    |-  ( N  e.  NN  ->  ( ( B `  N )  -  ( B `  ( N  +  1 ) ) ) 
 <_  ( ( 1  / 
 4 )  x.  (
 1  /  ( N  x.  ( N  +  1 ) ) ) ) )
 
Theoremstirlinglem11 27156*  B is decreasing. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   &    |-  K  =  ( k  e.  NN  |->  ( ( 1 
 /  ( ( 2  x.  k )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ (
 2  x.  k ) ) ) )   =>    |-  ( N  e.  NN  ->  ( B `  ( N  +  1
 ) )  <  ( B `  N ) )
 
Theoremstirlinglem12 27157* The sequence  B is bounded below. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   &    |-  F  =  ( n  e.  NN  |->  ( 1  /  ( n  x.  ( n  +  1 )
 ) ) )   =>    |-  ( N  e.  NN  ->  ( ( B `
  1 )  -  ( 1  /  4
 ) )  <_  ( B `  N ) )
 
Theoremstirlinglem13 27158*  B is decreasing and has a lower bound, then it converges. Since  B is  log A, in another theorem it is proven that  A converges as well. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   =>    |-  E. d  e.  RR  B  ~~>  d
 
Theoremstirlinglem14 27159* The sequence  A converges to a positive real. This proves that the Stirling's formula converges to the factorial, up to a constant. In another theorem, using Wallis' formula for π& , such constant is exactly determined, thus proving the Stirling's formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   =>    |-  E. c  e.  RR+  A  ~~>  c
 
Theoremstirlinglem15 27160* The Stirling's formula is proven using a number of local definitions. The main theorem stirling 27161 will use this final lemma, but it will not expose the local definitions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F/ n ph   &    |-  S  =  ( n  e.  NN0  |->  ( ( sqr `  ( (
 2  x.  pi )  x.  n ) )  x.  ( ( n 
 /  _e ) ^ n ) ) )   &    |-  A  =  ( n  e.  NN  |->  ( ( ! `
  n )  /  ( ( sqr `  (
 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  D  =  ( n  e.  NN  |->  ( A `  ( 2  x.  n ) ) )   &    |-  E  =  ( n  e.  NN  |->  ( ( sqr `  (
 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) )   &    |-  V  =  ( n  e.  NN  |->  ( ( ( ( 2 ^ ( 4  x.  n ) )  x.  ( ( ! `  n ) ^ 4
 ) )  /  (
 ( ! `  (
 2  x.  n ) ) ^ 2 ) )  /  ( ( 2  x.  n )  +  1 ) ) )   &    |-  F  =  ( n  e.  NN  |->  ( ( ( A `  n ) ^ 4
 )  /  ( ( D `  n ) ^
 2 ) ) )   &    |-  H  =  ( n  e.  NN  |->  ( ( n ^ 2 )  /  ( n  x.  (
 ( 2  x.  n )  +  1 )
 ) ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A  ~~>  C )   =>    |-  ( ph  ->  ( n  e.  NN  |->  ( ( ! `  n ) 
 /  ( S `  n ) ) )  ~~>  1 )
 
Theoremstirling 27161 Stirling's approximation formula for 
n factorial. The proof follows two major steps: first it is proven that  S and  n factorial are asymptotically equivalent, up to an unknown constant. Then, using Wallis' formula for π it is proven that the unknown constant is the square root of π and then the exact Stirling's formula is established. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  S  =  ( n  e.  NN0  |->  ( ( sqr `  (
 ( 2  x.  pi )  x.  n ) )  x.  ( ( n 
 /  _e ) ^ n ) ) )   =>    |-  ( n  e.  NN  |->  ( ( ! `  n )  /  ( S `  n ) ) )  ~~>  1
 
Theoremstirlingr 27162 Stirling's approximation formula for 
n factorial: here convergence is expressed with respect to the standard topology on the reals. The main theorem stirling 27161 is proven for convergence in the topology of complex numbers. The variable  R is used to denote convergence with respect to the standard topology on the reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  S  =  ( n  e.  NN0  |->  ( ( sqr `  (
 ( 2  x.  pi )  x.  n ) )  x.  ( ( n 
 /  _e ) ^ n ) ) )   &    |-  R  =  ( ~~> t `  ( topGen `  ran  (,) )
 )   =>    |-  ( n  e.  NN  |->  ( ( ! `  n )  /  ( S `  n ) ) ) R 1
 
18.20  Mathbox for Saveliy Skresanov
 
18.20.1  Ceva's theorem
 
Theoremsigarval 27163* Define the signed area by treating complex numbers as vectors with two components. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  ( Im `  ( ( * `
  A )  x.  B ) ) )
 
Theoremsigarim 27164* Signed area takes value in reals. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  e.  RR )
 
Theoremsigarac 27165* Signed area is anticommutative. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  -u ( B G A ) )
 
Theoremsigaraf 27166* Signed area is additive by the first argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  C ) G B )  =  ( ( A G B )  +  ( C G B ) ) )
 
Theoremsigarmf 27167* Signed area is additive (with respect to subtraction) by the first argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C ) G B )  =  ( ( A G B )  -  ( C G B ) ) )
 
Theoremsigaras 27168* Signed area is additive by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G ( B  +  C ) )  =  ( ( A G B )  +  ( A G C ) ) )
 
Theoremsigarms 27169* Signed area is additive (with respect to subtraction) by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G ( B  -  C ) )  =  ( ( A G B )  -  ( A G C ) ) )
 
Theoremsigarls 27170* Signed area is linear by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  RR )  ->  ( A G ( B  x.  C ) )  =  ( ( A G B )  x.  C ) )
 
Theoremsigarid 27171* Signed area of a flat parallelogram is zero. (Contributed by Saveliy Skresanov, 20-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( A  e.  CC  ->  ( A G A )  =  0 )
 
Theoremsigarexp 27172* Expand the signed area formula by linearity. (Contributed by Saveliy Skresanov, 20-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C ) G ( B  -  C ) )  =  ( ( ( A G B )  -  ( A G C ) )  -  ( C G B ) ) )
 
Theoremsigarperm 27173* Signed area  ( A  -  C ) G ( B  -  C ) acts as a double area of a triangle  A B C. Here we prove that cyclically permuting the vertices doesn't change the area. (Contributed by Saveliy Skresanov, 20-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C ) G ( B  -  C ) )  =  ( ( B  -  A ) G ( C  -  A ) ) )
 
Theoremsigardiv 27174* If signed area between vectors  B  -  A and  C  -  A is zero, then those vectors lie on the same line. (Contributed by Saveliy Skresanov, 22-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   &    |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC ) )   &    |-  ( ph  ->  -.  C  =  A )   &    |-  ( ph  ->  ( ( B  -  A ) G ( C  -  A ) )  =  0 )   =>    |-  ( ph  ->  (
 ( B  -  A )  /  ( C  -  A ) )  e. 
 RR )
 
Theoremsigarimcd 27175* Signed area takes value in complex numbers. Deduction version. (Contributed by Saveliy Skresanov, 23-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   &    |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC )
 )   =>    |-  ( ph  ->  ( A G B )  e. 
 CC )
 
Theoremsigariz 27176* If signed area is zero, the signed area with swapped arguments is also zero. Deduction version. ( Contributed by Saveliy Skresanov, 23-Sep-2017.) (Contributed by Saveliy Skresanov, 24-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   &    |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC )
 )   &    |-  ( ph  ->  ( A G B )  =  0 )   =>    |-  ( ph  ->  ( B G A )  =  0 )
 
Theoremsigarcol 27177* Given three points  A,  B and  C such that  -.  A  =  B, the point  C lies on the line going through  A and  B iff the corresponding signed area is zero. That justifies the usage of signed area as a collinearity indicator. (Contributed by Saveliy Skresanov, 22-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   &    |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC ) )   &    |-  ( ph  ->  -.  A  =  B )   =>    |-  ( ph  ->  (
 ( ( A  -  C ) G ( B  -  C ) )  =  0  <->  E. t  e.  RR  C  =  ( B  +  ( t  x.  ( A  -  B ) ) ) ) )
 
Theoremsharhght 27178* Let  A B C be a triangle, and let  D lie on the line  A B. Then (doubled) areas of triangles  A D C and  C D B relate as lengths of corresponding bases  A D and  D B. (Contributed by Saveliy Skresanov, 23-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   &    |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC ) )   &    |-  ( ph  ->  ( D  e.  CC  /\  ( ( A  -  D ) G ( B  -  D ) )  =  0
 ) )   =>    |-  ( ph  ->  (
 ( ( C  -  A ) G ( D  -  A ) )  x.  ( B  -  D ) )  =  ( ( ( C  -  B ) G ( D  -  B ) )  x.  ( A  -  D ) ) )
 
Theoremsigaradd 27179* Subtracting (double) area of  A D C from  A B C yields the (double) area of  D B C. (Contributed by Saveliy Skresanov, 23-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   &    |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC ) )   &    |-  ( ph  ->  ( D  e.  CC  /\  ( ( A  -  D ) G ( B  -  D ) )  =  0
 ) )   =>    |-  ( ph  ->  (
 ( ( B  -  C ) G ( A  -  C ) )  -  ( ( D  -  C ) G ( A  -  C ) ) )  =  ( ( B  -  C ) G ( D  -  C ) ) )
 
Theoremcevathlem1 27180 Ceva's theorem first lemma. Multiplies three identities and divides by the common factors. (Contributed by Saveliy Skresanov, 24-Sep-2017.)
 |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
 )   &    |-  ( ph  ->  ( D  e.  CC  /\  E  e.  CC  /\  F  e.  CC ) )   &    |-  ( ph  ->  ( G  e.  CC  /\  H  e.  CC  /\  K  e.  CC ) )   &    |-  ( ph  ->  ( A  =/=  0  /\  E  =/=  0  /\  C  =/=  0 ) )   &    |-  ( ph  ->  ( ( A  x.  B )  =  ( C  x.  D )  /\  ( E  x.  F )  =  ( A  x.  G )  /\  ( C  x.  H )  =  ( E  x.  K ) ) )   =>    |-  ( ph  ->  (
 ( B  x.  F )  x.  H )  =  ( ( D  x.  G )  x.  K ) )
 
Theoremcevathlem2 27181* Ceva's theorem second lemma. Relate (doubled) areas of triangles  C A O and 
A B O with of segments  B D and 
D C. (Contributed by Saveliy Skresanov, 24-Sep-2017.)
 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   &    |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC ) )   &    |-  ( ph  ->  ( F  e.  CC  /\  D  e.  CC  /\  E  e.  CC )
 )   &    |-  ( ph  ->  O  e.  CC )   &    |-  ( ph  ->  ( ( ( A  -  O ) G ( D  -  O ) )  =  0  /\  ( ( B  -  O ) G ( E  -  O ) )  =  0  /\  ( ( C  -  O ) G ( F  -  O ) )  =  0 ) )   &    |-  ( ph  ->  ( ( ( A  -  F ) G ( B  -  F ) )  =  0  /\  ( ( B  -  D ) G ( C  -  D ) )  =  0  /\  ( ( C  -  E ) G ( A  -  E ) )  =  0 ) )   &    |-  ( ph  ->  ( ( ( A  -  O ) G ( B  -  O ) )  =/=  0  /\  ( ( B  -  O ) G ( C  -  O ) )  =/=  0  /\  ( ( C  -  O ) G ( A  -  O ) )  =/=  0 ) )   =>    |-  ( ph  ->  (
 ( ( C  -  O ) G ( A  -  O ) )  x.  ( B  -  D ) )  =  ( ( ( A  -  O ) G ( B  -  O ) )  x.  ( D  -  C ) ) )
 
Theoremcevath 27182* Ceva's theorem. Let  A B C be a triangle and let points  F,  D and  E lie on sides  A B,  B C,  C A correspondingly. Suppose that cevians  A D,  B E and  C F intersect at one point  O. Then triangle's sides are partitioned into segments and their lengths satisfy a certain identity. Here we obtain a bit stronger version by using complex numbers themselves instead of their absolute values.

The proof goes by applying cevathlem2 27181 three times and then using cevathlem1 27180 to multiply obtained identities and prove the theorem.

In the theorem statement we are using function  G as a collinearity indicator. For justification of that use, see sigarcol 27177. (Contributed by Saveliy Skresanov, 24-Sep-2017.)

 |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )   &    |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC ) )   &    |-  ( ph  ->  ( F  e.  CC  /\  D  e.  CC  /\  E  e.  CC )
 )   &    |-  ( ph  ->  O  e.  CC )   &    |-  ( ph  ->  ( ( ( A  -  O ) G ( D  -  O ) )  =  0  /\  ( ( B  -  O ) G ( E  -  O ) )  =  0  /\  ( ( C  -  O ) G ( F  -  O ) )  =  0 ) )   &    |-  ( ph  ->  ( ( ( A  -  F ) G ( B  -  F ) )  =  0  /\  ( ( B  -  D ) G ( C  -  D ) )  =  0  /\  ( ( C  -  E ) G ( A  -  E ) )  =  0 ) )   &    |-  ( ph  ->  ( ( ( A  -  O ) G ( B  -  O ) )  =/=  0  /\  ( ( B  -  O ) G ( C  -  O ) )  =/=  0  /\  ( ( C  -  O ) G ( A  -  O ) )  =/=  0 ) )   =>    |-  ( ph  ->  (
 ( ( A  -  F )  x.  ( C  -  E ) )  x.  ( B  -  D ) )  =  ( ( ( F  -  B )  x.  ( E  -  A ) )  x.  ( D  -  C ) ) )
 
18.21  Mathbox for Jarvin Udandy
 
TheoremhirstL-ax3 27183 The third axiom of a system called "L" but proven to be a theorem since set.mm uses a different third axiom. This is named hirst after Holly P. Hirst and Jeffry L. Hirst. Axiom A3 of [Mendelson] p. 35. (Contributed by Jarvin Udandy, 7-Feb-2015.) (Proof modification is discouraged.)
 |-  (
 ( -.  ph  ->  -. 
 ps )  ->  (
 ( -.  ph  ->  ps )  ->  ph ) )
 
Theoremax3h 27184 Recovery of ax-3 7 from hirstL-ax3 27183. (Contributed by Jarvin Udandy, 3-Jul-2015.)
 |-  (
 ( -.  ph  ->  -. 
 ps )  ->  ( ps  ->  ph ) )
 
Theoremaibandbiaiffaiffb 27185 A closed form showing (a implies b and b implies a) same-as (a same-as b) (Contributed by Jarvin Udandy, 3-Sep-2016.)
 |-  (
 ( ( ph