HomeHome Metamath Proof Explorer
Theorem List (p. 277 of 325)
< Previous  Next >
Browser slow? Try the
Unicode version.

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

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-22374)
  Hilbert Space Explorer  Hilbert Space Explorer
(22375-23897)
  Users' Mathboxes  Users' Mathboxes
(23898-32447)
 

Theorem List for Metamath Proof Explorer - 27601-27700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremclimsuse 27601* A subsequence  G of a converging sequence  F, converges to the same limit.  I is the strictly increasing and it is used to index the subsequence (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F/ k ph   &    |-  F/_ k F   &    |-  F/_ k G   &    |-  F/_ k I   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  X )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  ( I `  M )  e.  Z )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( I `  ( k  +  1 ) )  e.  ( ZZ>= `  ( ( I `  k )  +  1 ) ) )   &    |-  ( ph  ->  G  e.  Y )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( F `
  ( I `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  A )
 
Theoremclimrecf 27602* A version of climrec 27596 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F/ k ph   &    |-  F/_ k G   &    |-  F/_ k H   &    |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G  ~~>  A )   &    |-  ( ph  ->  A  =/=  0
 )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( G `  k )  e.  ( CC  \  {
 0 } ) )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( 1  /  ( G `  k ) ) )   &    |-  ( ph  ->  H  e.  W )   =>    |-  ( ph  ->  H  ~~>  ( 1  /  A ) )
 
Theoremclimneg 27603* Complex limit of the negative of a sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F/ k ph   &    |-  F/_ k F   &    |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   =>    |-  ( ph  ->  ( k  e.  Z  |->  -u ( F `  k ) )  ~~>  -u A )
 
Theoremcliminff 27604* A version of climinf 27599 using bound-variable hypotheses instead of distinct variable conditions (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F/ k ph   &    |-  F/_ k F   &    |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )   &    |-  ( ph  ->  E. x  e.  RR  A. k  e.  Z  x  <_  ( F `  k
 ) )   =>    |-  ( ph  ->  F  ~~>  sup ( ran  F ,  RR ,  `'  <  )
 )
 
Theoremclimdivf 27605* Limit of the ratio of two converging sequences. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F/ k ph   &    |-  F/_ k F   &    |-  F/_ k G   &    |-  F/_ k H   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  ( ph  ->  B  =/=  0
 )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  ( CC  \  { 0 } )
 )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `  k )  /  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( A  /  B ) )
 
Theoremclimreeq 27606 If  F is a real function, then  F converges to  A with respect to the standard topology on the reals if and only if it converges to  A with respect to the standard topology on complex numbers. In the theorem,  R is defined to be convergence w.r.t. the standard topology on the reals and then  F R A represents the statement " F converges to  A, with respect to the standard topology on the reals". Notice that there is no need for the hypothesis that  A is a real number. (Contributed by Glauco Siliprandi, 2-Jul-2017.)
 |-  R  =  ( ~~> t `  ( topGen `
  ran  (,) ) )   &    |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> RR )   =>    |-  ( ph  ->  ( F R A  <->  F  ~~>  A ) )
 
19.19.4  Derivatives
 
Theoremdvsinexp 27607* The derivative of sin^N . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  ( CC  _D  ( x  e. 
 CC  |->  ( ( sin `  x ) ^ N ) ) )  =  ( x  e.  CC  |->  ( ( N  x.  ( ( sin `  x ) ^ ( N  -  1 ) ) )  x.  ( cos `  x ) ) ) )
 
Theoremdvcosre 27608 The real derivative of the cosine (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  ( RR  _D  ( x  e. 
 RR  |->  ( cos `  x ) ) )  =  ( x  e.  RR  |->  -u ( sin `  x ) )
 
19.19.5  Integrals
 
Theoremioovolcl 27609 An open real interval has finite volume. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( vol `  ( A (,) B ) )  e.  RR )
 
Theoremvolioo 27610 The measure of an open interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( vol `  ( A (,) B ) )  =  ( B  -  A ) )
 
Theoremitgsin0pilem1 27611* Calculation of the integral for sine on the (0,π) interval (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  C  =  ( t  e.  (
 0 [,] pi )  |->  -u ( cos `  t )
 )   =>    |- 
 S. ( 0 (,)
 pi ) ( sin `  x )  _d x  =  2
 
Theoremibliccsinexp 27612* sin^n on a closed interval is integrable. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  N  e.  NN0 )  ->  ( x  e.  ( A [,] B )  |->  ( ( sin `  x ) ^ N ) )  e.  L ^1 )
 
Theoremitgsin0pi 27613 Calculation of the integral for sine on the (0,π) interval (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  S. ( 0 (,) pi ) ( sin `  x )  _d x  =  2
 
Theoremiblioosinexp 27614* sin^n on an open integral is integrable (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  N  e.  NN0 )  ->  ( x  e.  ( A (,) B )  |->  ( ( sin `  x ) ^ N ) )  e.  L ^1 )
 
Theoremitgsinexplem1 27615* Integration by parts is applied to integrate sin^(N+1) (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F  =  ( x  e.  CC  |->  ( ( sin `  x ) ^ N ) )   &    |-  G  =  ( x  e.  CC  |->  -u ( cos `  x ) )   &    |-  H  =  ( x  e.  CC  |->  ( ( N  x.  (
 ( sin `  x ) ^ ( N  -  1 ) ) )  x.  ( cos `  x ) ) )   &    |-  I  =  ( x  e.  CC  |->  ( ( ( sin `  x ) ^ N )  x.  ( sin `  x ) ) )   &    |-  L  =  ( x  e.  CC  |->  ( ( ( N  x.  ( ( sin `  x ) ^ ( N  -  1 ) ) )  x.  ( cos `  x ) )  x.  -u ( cos `  x ) ) )   &    |-  M  =  ( x  e.  CC  |->  ( ( ( cos `  x ) ^ 2
 )  x.  ( ( sin `  x ) ^ ( N  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  S. ( 0 (,) pi ) ( ( ( sin `  x ) ^ N )  x.  ( sin `  x ) )  _d x  =  ( N  x.  S. (
 0 (,) pi ) ( ( ( cos `  x ) ^ 2 )  x.  ( ( sin `  x ) ^ ( N  -  1 ) ) )  _d x ) )
 
Theoremitgsinexp 27616* A recursive formula for the integral of sin^N on the interval (0,π) .

(Contributed by Glauco Siliprandi, 29-Jun-2017.)

 |-  I  =  ( n  e.  NN0  |->  S. ( 0 (,) pi ) ( ( sin `  x ) ^ n )  _d x )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   =>    |-  ( ph  ->  ( I `  N )  =  ( ( ( N  -  1 )  /  N )  x.  ( I `  ( N  -  2
 ) ) ) )
 
19.19.6  Stone Weierstrass theorem - real version
 
Theoremstoweidlem1 27617 Lemma for stoweid 27679. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90; the key step uses Bernoulli's inequality bernneq 11460. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  A 
 <_  1 )   &    |-  ( ph  ->  D 
 <_  A )   =>    |-  ( ph  ->  (
 ( 1  -  ( A ^ N ) ) ^ ( K ^ N ) )  <_  ( 1  /  (
 ( K  x.  D ) ^ N ) ) )
 
Theoremstoweidlem2 27618* lemma for stoweid 27679: here we prove that the subalgebra of continuous functions, which contains constant functions, is closed under scaling. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  ( ( 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  ->  E  e.  RR )   &    |-  ( ph  ->  F  e.  A )   =>    |-  ( ph  ->  (
 t  e.  T  |->  ( E  x.  ( F `
  t ) ) )  e.  A )
 
Theoremstoweidlem3 27619* Lemma for stoweid 27679: if  A is positive and all  M terms of a finite product are larger than  A, then the finite product is larger than A^M. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ i F   &    |- 
 F/ i ph   &    |-  X  =  seq  1 (  x.  ,  F )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 M ) --> RR )   &    |-  (
 ( ph  /\  i  e.  ( 1 ... M ) )  ->  A  <  ( F `  i ) )   &    |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A ^ M )  < 
 ( X `  M ) )
 
Theoremstoweidlem4 27620* Lemma for stoweid 27679: a class variable replaces a set variable, for constant functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  (
 ( ph  /\  x  e. 
 RR )  ->  (
 t  e.  T  |->  x )  e.  A )   =>    |-  ( ( ph  /\  B  e.  RR )  ->  (
 t  e.  T  |->  B )  e.  A )
 
Theoremstoweidlem5 27621* There exists a δ as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: 0 < δ < 1 , p >= δ on  T  \  U. Here  D is used to represent δ in the paper and  Q to represent  T 
\  U in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  D  =  if ( C  <_  ( 1 
 /  2 ) ,  C ,  ( 1 
 /  2 ) )   &    |-  ( ph  ->  P : T
 --> RR )   &    |-  ( ph  ->  Q 
 C_  T )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. t  e.  Q  C  <_  ( P `  t ) )   =>    |-  ( ph  ->  E. d
 ( d  e.  RR+  /\  d  <  1  /\  A. t  e.  Q  d 
 <_  ( P `  t
 ) ) )
 
Theoremstoweidlem6 27622* Lemma for stoweid 27679: two class variables replace two set variables, for multiplication of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t  f  =  F   &    |-  F/ t  g  =  G   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   =>    |-  ( ( ph  /\  F  e.  A  /\  G  e.  A )  ->  ( t  e.  T  |->  ( ( F `  t )  x.  ( G `  t ) ) )  e.  A )
 
Theoremstoweidlem7 27623* This lemma is used to prove that qn as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91, (at the top of page 91), is such that qn < ε on  T  \  U, and qn > 1 - ε on  V. Here it is proven that, for  n large enough, 1-(k*δ/2)^n > 1 - ε , and 1/(k*δ)^n < ε. The variable  A is used to represent (k*δ) in the paper, and  B is used to represent (k*δ/2). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F  =  ( i  e.  NN0  |->  ( ( 1  /  A ) ^ i
 ) )   &    |-  G  =  ( i  e.  NN0  |->  ( B ^ i ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  1  <  A )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  B  <  1 )   &    |-  ( ph  ->  E  e.  RR+ )   =>    |-  ( ph  ->  E. n  e.  NN  ( ( 1  -  E )  < 
 ( 1  -  ( B ^ n ) ) 
 /\  ( 1  /  ( A ^ n ) )  <  E ) )
 
Theoremstoweidlem8 27624* Lemma for stoweid 27679: two class variables replace two set variables, for the sum of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  F/_ t F   &    |-  F/_ t G   =>    |-  ( ( ph  /\  F  e.  A  /\  G  e.  A )  ->  ( t  e.  T  |->  ( ( F `  t )  +  ( G `  t ) ) )  e.  A )
 
Theoremstoweidlem9 27625* Lemma for stoweid 27679: here the Stone Weierstrass theorem is proven for the trivial case, T is the empty set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  ( ph  ->  T  =  (/) )   &    |-  ( ph  ->  (
 t  e.  T  |->  1 )  e.  A )   =>    |-  ( ph  ->  E. g  e.  A  A. t  e.  T  ( abs `  (
 ( g `  t
 )  -  ( F `
  t ) ) )  <  E )
 
Theoremstoweidlem10 27626 Lemma for stoweid 27679. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90, this lemma is an application of Bernoulli's inequality. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  (
 ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  (
 1  -  ( N  x.  A ) ) 
 <_  ( ( 1  -  A ) ^ N ) )
 
Theoremstoweidlem11 27627* This lemma is used to prove that there is a function  g as in the proof of [BrosowskiDeutsh] p. 92, (at the top of page 92): this lemma proves that g(t) < ( j + 1 / 3 ) * ε. Here  E is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  t  e.  T )   &    |-  ( ph  ->  j  e.  ( 1 ...
 N ) )   &    |-  (
 ( ph  /\  i  e.  ( 0 ... N ) )  ->  ( X `
  i ) : T --> RR )   &    |-  (
 ( ph  /\  i  e.  ( 0 ... N ) )  ->  ( ( X `  i ) `
  t )  <_ 
 1 )   &    |-  ( ( ph  /\  i  e.  ( j
 ... N ) ) 
 ->  ( ( X `  i ) `  t
 )  <  ( E  /  N ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  /  3
 ) )   =>    |-  ( ph  ->  (
 ( t  e.  T  |->  sum_
 i  e.  ( 0
 ... N ) ( E  x.  ( ( X `  i ) `
  t ) ) ) `  t )  <  ( ( j  +  ( 1  / 
 3 ) )  x.  E ) )
 
Theoremstoweidlem12 27628* Lemma for stoweid 27679. This Lemma is used by other three Lemmas. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  Q  =  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N ) ) ^ ( K ^ N ) ) )   &    |-  ( ph  ->  P : T --> RR )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  K  e.  NN0 )   =>    |-  ( ( ph  /\  t  e.  T )  ->  ( Q `  t )  =  ( ( 1  -  ( ( P `  t ) ^ N ) ) ^ ( K ^ N ) ) )
 
Theoremstoweidlem13 27629 Lemma for stoweid 27679. This lemma is used to prove the statement abs( f(t) - g(t) ) < 2 epsilon , in [BrosowskiDeutsh] p. 92, the last step of the proof. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  j  e.  RR )   &    |-  ( ph  ->  (
 ( j  -  (
 4  /  3 )
 )  x.  E )  <  X )   &    |-  ( ph  ->  X  <_  (
 ( j  -  (
 1  /  3 )
 )  x.  E ) )   &    |-  ( ph  ->  ( ( j  -  (
 4  /  3 )
 )  x.  E )  <  Y )   &    |-  ( ph  ->  Y  <  (
 ( j  +  (
 1  /  3 )
 )  x.  E ) )   =>    |-  ( ph  ->  ( abs `  ( Y  -  X ) )  < 
 ( 2  x.  E ) )
 
Theoremstoweidlem14 27630* There exists a  k as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90:  k is an integer and 1 < k * δ < 2.  D is used to represent δ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  A  =  { j  e.  NN  |  ( 1  /  D )  <  j }   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  D  <  1 )   =>    |-  ( ph  ->  E. k  e.  NN  ( 1  < 
 ( k  x.  D )  /\  ( ( k  x.  D )  / 
 2 )  <  1
 ) )
 
Theoremstoweidlem15 27631* This lemma is used to prove the existence of a function  p as in Lemma 1 from [BrosowskiDeutsh] p. 90:  p is in the subalgebra, such that 0 ≤ p ≤ 1, p(t_0) = 0, and p > 0 on T - U. Here  ( G `  I ) is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  ( ph  ->  G : ( 1 ... M ) --> Q )   &    |-  ( ( ph  /\  f  e.  A ) 
 ->  f : T --> RR )   =>    |-  (
 ( ( ph  /\  I  e.  ( 1 ... M ) )  /\  S  e.  T )  ->  ( ( ( G `  I
 ) `  S )  e.  RR  /\  0  <_  ( ( G `  I ) `  S )  /\  ( ( G `
  I ) `  S )  <_  1 ) )
 
Theoremstoweidlem16 27632* Lemma for stoweid 27679. The subset  Y of functions in the algebra  A, with values in [ 0 , 1 ], is closed under multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  Y  =  { h  e.  A  |  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) }   &    |-  H  =  ( t  e.  T  |->  ( ( f `  t )  x.  (
 g `  t )
 ) )   &    |-  ( ( 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  /\  f  e.  Y  /\  g  e.  Y )  ->  H  e.  Y )
 
Theoremstoweidlem17 27633* This lemma proves that the function 
g (as defined in [BrosowskiDeutsh] p. 91, at the end of page 91) belongs to the subalgebra. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  X : ( 0 ... N ) --> 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  ->  E  e.  RR )   &    |-  ( ( ph  /\  f  e.  A ) 
 ->  f : T --> RR )   =>    |-  ( ph  ->  ( t  e.  T  |->  sum_ i  e.  (
 0 ... N ) ( E  x.  ( ( X `  i ) `
  t ) ) )  e.  A )
 
Theoremstoweidlem18 27634* This theorem proves Lemma 2 in [BrosowskiDeutsh] p. 92 when A is empty, the trivial case. Here D is used to denote the set A of Lemma 2, because the variable A is used for the subalgebra. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t D   &    |- 
 F/ t ph   &    |-  F  =  ( t  e.  T  |->  1 )   &    |-  T  =  U. J   &    |-  ( ( ph  /\  a  e.  RR )  ->  (
 t  e.  T  |->  a )  e.  A )   &    |-  ( ph  ->  B  e.  ( Clsd `  J )
 )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  D  =  (/) )   =>    |-  ( 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 ) ) )
 
Theoremstoweidlem19 27635* If a set of real functions is closed under multiplication and it contains constants, then it is closed under finite exponentiation. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |- 
 F/ t ph   &    |-  ( ( 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  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 t  e.  T  |->  ( ( F `  t
 ) ^ N ) )  e.  A )
 
Theoremstoweidlem20 27636* If a set A of real functions from a common domain T is closed under the sum of two functions, then it is closed under the sum of a finite number of functions, indexed by G. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  F  =  ( t  e.  T  |->  sum_ i  e.  ( 1 ...
 M ) ( ( G `  i ) `
  t ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( 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 )  ->  f : T --> RR )   =>    |-  ( ph  ->  F  e.  A )
 
Theoremstoweidlem21 27637* Once the Stone Weierstrass theorem has been proven for approximating nonnegative functions, then this lemma is used to extend the result to functions with (possibly) negative values. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t G   &    |-  F/_ t H   &    |-  F/_ t S   &    |-  F/ t ph   &    |-  G  =  ( t  e.  T  |->  ( ( H `  t
 )  +  S ) )   &    |-  ( ph  ->  F : T --> RR )   &    |-  ( ph  ->  S  e.  RR )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ph  ->  A. f  e.  A  f : T --> RR )   &    |-  ( ph  ->  H  e.  A )   &    |-  ( ph  ->  A. t  e.  T  ( abs `  ( ( H `  t )  -  ( ( F `  t )  -  S ) ) )  <  E )   =>    |-  ( ph  ->  E. f  e.  A  A. t  e.  T  ( abs `  (
 ( f `  t
 )  -  ( F `
  t ) ) )  <  E )
 
Theoremstoweidlem22 27638* 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 ph   &    |-  F/_ t F   &    |-  F/_ t G   &    |-  H  =  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )   &    |-  I  =  ( t  e.  T  |->  -u 1 )   &    |-  L  =  ( t  e.  T  |->  ( ( I `  t )  x.  ( G `  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  /\  F  e.  A  /\  G  e.  A )  ->  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  e.  A )
 
Theoremstoweidlem23 27639* This lemma is used to prove the existence of a function pt as in the beginning of Lemma 1 [BrosowskiDeutsh] p. 90: 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. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  F/_ t G   &    |-  H  =  ( t  e.  T  |->  ( ( G `  t )  -  ( G `  Z ) ) )   &    |-  ( ( ph  /\  f  e.  A ) 
 ->  f : T --> RR )   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ph  ->  S  e.  T )   &    |-  ( ph  ->  Z  e.  T )   &    |-  ( ph  ->  G  e.  A )   &    |-  ( ph  ->  ( G `  S )  =/=  ( G `  Z ) )   =>    |-  ( ph  ->  ( H  e.  A  /\  ( H `  S )  =/=  ( H `  Z )  /\  ( H `
  Z )  =  0 ) )
 
Theoremstoweidlem24 27640* This lemma proves that for  n sufficiently large, qn( t ) > ( 1 - epsilon ), for all  t in  V: see Lemma 1 [BrosowskiDeutsh] p. 90, (at the bottom of page 90). 
Q is used to represent qn in the paper,  N to represent  n in the paper,  K to represent  k,  D to represent δ, and  E to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  V  =  { t  e.  T  |  ( P `  t
 )  <  ( D  /  2 ) }   &    |-  Q  =  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N ) ) ^ ( K ^ N ) ) )   &    |-  ( ph  ->  P : T --> RR )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  K  e.  NN0 )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  ( 1  -  E )  <  ( 1  -  ( ( ( K  x.  D )  / 
 2 ) ^ N ) ) )   &    |-  ( ph  ->  A. t  e.  T  ( 0  <_  ( P `  t )  /\  ( P `  t ) 
 <_  1 ) )   =>    |-  ( ( ph  /\  t  e.  V ) 
 ->  ( 1  -  E )  <  ( Q `  t ) )
 
Theoremstoweidlem25 27641* This lemma proves that for n sufficiently large, qn( t ) < ε, for all  t in  T  \  U: see Lemma 1 [BrosowskiDeutsh] p. 91 (at the top of page 91).  Q is used to represent qn in the paper,  N to represent n in the paper,  K to represent k,  D to represent δ,  P to represent p, and  E to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  Q  =  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N ) ) ^ ( K ^ N ) ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( 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  ->  E  e.  RR+ )   &    |-  ( ph  ->  ( 1  /  ( ( K  x.  D ) ^ N ) )  <  E )   =>    |-  ( ( ph  /\  t  e.  ( T 
 \  U ) ) 
 ->  ( Q `  t
 )  <  E )
 
Theoremstoweidlem26 27642* This lemma is used to prove that there is a function  g as in the proof of [BrosowskiDeutsh] p. 92: this lemma proves that g(t) > ( j - 4 / 3 ) * ε. Here  L is used to represnt j in the paper,  D is used to represent A in the paper,  S is used to represent t, and  E is used to represent ε. (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 ) } )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T  e.  _V )   &    |-  ( ph  ->  L  e.  ( 1 ...
 N ) )   &    |-  ( ph  ->  S  e.  (
 ( D `  L )  \  ( D `  ( L  -  1
 ) ) ) )   &    |-  ( ph  ->  F : T
 --> RR )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   &    |-  (
 ( ph  /\  i  e.  ( 0 ... N ) )  ->  ( X `
  i ) : T --> RR )   &    |-  (
 ( ph  /\  i  e.  ( 0 ... N )  /\  t  e.  T )  ->  0  <_  (
 ( X `  i
 ) `  t )
 )   &    |-  ( ( ph  /\  i  e.  ( 0 ... N )  /\  t  e.  ( B `  i ) ) 
 ->  ( 1  -  ( E  /  N ) )  <  ( ( X `
  i ) `  t ) )   =>    |-  ( ph  ->  ( ( L  -  (
 4  /  3 )
 )  x.  E )  <  ( ( t  e.  T  |->  sum_ i  e.  ( 0 ... N ) ( E  x.  ( ( X `  i ) `  t
 ) ) ) `  S ) )
 
Theoremstoweidlem27 27643* 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. Here  ( q `  i ) is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  G  =  ( w  e.  X  |->  { h  e.  Q  |  w  =  { t  e.  T  |  0  < 
 ( h `  t
 ) } } )   &    |-  ( ph  ->  Q  e.  _V )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  Y  Fn  ran  G )   &    |-  ( ph  ->  ran  G  e.  _V )   &    |-  ( ( ph  /\  l  e.  ran  G )  ->  ( Y `  l )  e.  l
 )   &    |-  ( ph  ->  F : ( 1 ...
 M ) -1-1-onto-> ran  G )   &    |-  ( ph  ->  ( T  \  U )  C_  U. X )   &    |- 
 F/ t ph   &    |-  F/ w ph   &    |-  F/_ h Q   =>    |-  ( ph  ->  E. q
 ( M  e.  NN  /\  ( q : ( 1 ... M ) --> Q  /\  A. t  e.  ( T  \  U ) E. i  e.  (
 1 ... M ) 0  <  ( ( q `
  i ) `  t ) ) ) )
 
Theoremstoweidlem28 27644* There exists a δ as in Lemma 1 [BrosowskiDeutsh] p. 90: 0 < delta < 1 and p >= delta on 
T  \  U. Here  d is used to represent δ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t U   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  T  =  U. J   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  P  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  A. t  e.  ( T  \  U ) 0  <  ( P `  t ) )   &    |-  ( ph  ->  U  e.  J )   =>    |-  ( ph  ->  E. d
 ( d  e.  RR+  /\  d  <  1  /\  A. t  e.  ( T 
 \  U ) d 
 <_  ( P `  t
 ) ) )
 
Theoremstoweidlem29 27645* 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 27646* 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 27647* 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 27648* 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 27649* 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 27650* 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 27651* 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 27652* 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 27653* 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 27654* 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 27655* 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 27656* 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 27657* 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 27658* 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 27659* 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 27660* 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 27661* 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 27662* 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 27663* 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 27664* 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 27665* 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 27666* 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 27667* 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 27668* 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 27669* 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 27670* 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 27671* 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 27672* 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 27673* 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 27674* 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 27675* 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 27676* 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 27677* 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 27678* 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 27679* 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 27680* 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 27679: often times it will be better to use stoweid 27679 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
 
19.19.7  Wallis' product for π
 
Theoremwallispilem1 27681*  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  ->