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Theorem List for Metamath Proof Explorer - 27801-27900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremevthf 27801* A version of evth 18473 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ x F   &    |-  F/_ y F   &    |-  F/_ x X   &    |-  F/_ y X   &    |- 
 F/ x ph   &    |-  F/ y ph   &    |-  X  =  U. J   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  X  =/=  (/) )   =>    |-  ( ph  ->  E. x  e.  X  A. y  e.  X  ( F `  y )  <_  ( F `
  x ) )
 
Theoremcnfex 27802 The class of continuous functions between two topologies is a set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  (
 ( J  e.  Top  /\  K  e.  Top )  ->  ( J  Cn  K )  e.  _V )
 
Theoremfnchoice 27803* For a finite set, a choice function exists, without using the axiom of choice. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  ( A  e.  Fin  ->  E. f
 ( f  Fn  A  /\  A. x  e.  A  ( x  =/=  (/)  ->  (
 f `  x )  e.  x ) ) )
 
Theoremrefsumcn 27804* A finite sum of continuous real functions, from a common topological space, is continuous. The class expression for B normally contains free variables k and x to index it. See fsumcn 18390 for the analogous theorem on continuous complex functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ x ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  ( x  e.  X  |->  B )  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  sum_ k  e.  A  B )  e.  ( J  Cn  K ) )
 
Theoremrfcnpre2 27805 If  F is a continuous function with respect to the standard topology, then the preimage A of the values smaller than a given extended real  B, is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ x B   &    |-  F/_ x F   &    |-  F/ x ph   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  X  =  U. J   &    |-  A  =  { x  e.  X  |  ( F `  x )  <  B }   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  A  e.  J )
 
Theoremcncmpmax 27806* When the hypothesis for the extreme value theorem hold, then the sup of the range of the function belongs to the range, it is real and it an upper bound of the range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  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  ( F `  t ) 
 <_  sup ( ran  F ,  RR ,  <  )
 ) )
 
Theoremrfcnpre3 27807* If F is a continuous function with respect to the standard topology, then the preimage A of the values greater or equal than a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  T  =  U. J   &    |-  A  =  { t  e.  T  |  B  <_  ( F `  t ) }   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  A  e.  ( Clsd `  J )
 )
 
Theoremrfcnpre4 27808* If F is a continuous function with respect to the standard topology, then the preimage A of the values smaller or equal than a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  T  =  U. J   &    |-  A  =  { t  e.  T  |  ( F `
  t )  <_  B }   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  A  e.  ( Clsd `  J )
 )
 
Theoremsumpair 27809* Sum of two distinct complex values. The class expression for  A and  B normally contain free variable  k to index it. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  ( ph  ->  F/_ k D )   &    |-  ( ph  ->  F/_ k E )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  E  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ( ph  /\  k  =  A )  ->  C  =  D )   &    |-  ( ( ph  /\  k  =  B ) 
 ->  C  =  E )   =>    |-  ( ph  ->  sum_ k  e. 
 { A ,  B } C  =  ( D  +  E )
 )
 
Theoremrfcnnnub 27810* Given a real continuous function  F defined on a compact topological space, there is always a natural number that is a strict upper bound of its range. (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  ->  F  e.  C )   =>    |-  ( ph  ->  E. n  e.  NN  A. t  e.  T  ( F `  t )  <  n )
 
Theoremrefsum2cnlem1 27811* This is the core Lemma for refsum2cn 27812: the sum of two continuous real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x F   &    |-  F/_ x G   &    |-  F/ x ph   &    |-  A  =  ( k  e.  { 1 ,  2 }  |->  if ( k  =  1 ,  F ,  G ) )   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  G  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( ( F `  x )  +  ( G `  x ) ) )  e.  ( J  Cn  K ) )
 
Theoremrefsum2cn 27812* The sum of two continuus real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ x F   &    |-  F/_ x G   &    |-  F/ x ph   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  G  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( ( F `  x )  +  ( G `  x ) ) )  e.  ( J  Cn  K ) )
 
18.20.2  Finite multiplication of numbers and finite multiplication of functions
 
Theoremfmul01 27813* Multiplying a finite number of values in [ 0 , 1 ] , gives the final product itself a number in [ 0 , 1 ]. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ i B   &    |- 
 F/ i ph   &    |-  A  =  seq  L (  x.  ,  B )   &    |-  ( ph  ->  L  e.  ZZ )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  L ) )   &    |-  ( ph  ->  K  e.  ( L ... M ) )   &    |-  ( ( ph  /\  i  e.  ( L
 ... M ) ) 
 ->  ( B `  i
 )  e.  RR )   &    |-  (
 ( ph  /\  i  e.  ( L ... M ) )  ->  0  <_  ( B `  i ) )   &    |-  ( ( ph  /\  i  e.  ( L
 ... M ) ) 
 ->  ( B `  i
 )  <_  1 )   =>    |-  ( ph  ->  ( 0  <_  ( A `  K ) 
 /\  ( A `  K )  <_  1 ) )
 
Theoremfmulcl 27814* If ' Y ' is closed under the multiplication of two functions, then Y is closed under the multiplication ( ' X ' ) of a finite number of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  P  =  ( f  e.  Y ,  g  e.  Y  |->  ( t  e.  T  |->  ( ( f `  t )  x.  (
 g `  t )
 ) ) )   &    |-  X  =  (  seq  1 ( P ,  U ) `
  N )   &    |-  ( ph  ->  N  e.  (
 1 ... M ) )   &    |-  ( ph  ->  U :
 ( 1 ... M )
 --> Y )   &    |-  ( ( ph  /\  f  e.  Y  /\  g  e.  Y )  ->  ( t  e.  T  |->  ( ( f `  t )  x.  (
 g `  t )
 ) )  e.  Y )   &    |-  ( ph  ->  T  e.  _V )   =>    |-  ( ph  ->  X  e.  Y )
 
Theoremfmuldfeqlem1 27815* induction step for the proof of fmuldfeq 27816. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ f ph   &    |-  F/ g ph   &    |-  F/_ t Y   &    |-  P  =  ( f  e.  Y ,  g  e.  Y  |->  ( t  e.  T  |->  ( ( f `
  t )  x.  ( g `  t
 ) ) ) )   &    |-  F  =  ( t  e.  T  |->  ( i  e.  ( 1 ... M )  |->  ( ( U `
  i ) `  t ) ) )   &    |-  ( ph  ->  T  e.  _V )   &    |-  ( ph  ->  U : ( 1 ...
 M ) --> Y )   &    |-  ( ( ph  /\  f  e.  Y  /\  g  e.  Y )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  Y )   &    |-  ( ph  ->  N  e.  ( 1 ... M ) )   &    |-  ( ph  ->  ( N  +  1 )  e.  ( 1 ...
 M ) )   &    |-  ( ph  ->  ( (  seq  1 ( P ,  U ) `  N ) `  t )  =  (  seq  1 (  x.  ,  ( F `
  t ) ) `
  N ) )   &    |-  ( ( ph  /\  f  e.  Y )  ->  f : T --> RR )   =>    |-  ( ( ph  /\  t  e.  T ) 
 ->  ( (  seq  1
 ( P ,  U ) `  ( N  +  1 ) ) `  t )  =  (  seq  1 (  x.  ,  ( F `  t ) ) `  ( N  +  1 ) ) )
 
Theoremfmuldfeq 27816* X and Z are two equivalent definitions of the finite product of real functions. Y is a set of real functions from a common domain T, Y is closed under function multiplication and U is a finite sequence of functions in Y. M is the number of functions multiplied together. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ i 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  ->  T  e.  _V )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  U : ( 1 ...
 M ) --> Y )   &    |-  ( ( 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.  T )  ->  ( X `  t )  =  ( Z `  t
 ) )
 
Theoremfmul01lt1lem1 27817* Given a finite multiplication of values betweeen 0 and 1, a value larger than its frist element is larger the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ i B   &    |- 
 F/ i ph   &    |-  A  =  seq  L (  x.  ,  B )   &    |-  ( ph  ->  L  e.  ZZ )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  L ) )   &    |-  ( ( ph  /\  i  e.  ( L
 ... M ) ) 
 ->  ( B `  i
 )  e.  RR )   &    |-  (
 ( ph  /\  i  e.  ( L ... M ) )  ->  0  <_  ( B `  i ) )   &    |-  ( ( ph  /\  i  e.  ( L
 ... M ) ) 
 ->  ( B `  i
 )  <_  1 )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  ( B `  L )  <  E )   =>    |-  ( ph  ->  ( A `  M )  <  E )
 
Theoremfmul01lt1lem2 27818* Given a finite multiplication of values betweeen 0 and 1, a value  E larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ i B   &    |- 
 F/ i ph   &    |-  A  =  seq  L (  x.  ,  B )   &    |-  ( ph  ->  L  e.  ZZ )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  L ) )   &    |-  ( ( ph  /\  i  e.  ( L
 ... M ) ) 
 ->  ( B `  i
 )  e.  RR )   &    |-  (
 ( ph  /\  i  e.  ( L ... M ) )  ->  0  <_  ( B `  i ) )   &    |-  ( ( ph  /\  i  e.  ( L
 ... M ) ) 
 ->  ( B `  i
 )  <_  1 )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  J  e.  ( L ... M ) )   &    |-  ( ph  ->  ( B `  J )  <  E )   =>    |-  ( ph  ->  ( A `  M )  <  E )
 
Theoremfmul01lt1 27819* Given a finite multiplication of values betweeen 0 and 1, a value E larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ i B   &    |- 
 F/ i ph   &    |-  F/_ j A   &    |-  A  =  seq  1 (  x. 
 ,  B )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  B : ( 1 ...
 M ) --> RR )   &    |-  (
 ( ph  /\  i  e.  ( 1 ... M ) )  ->  0  <_  ( B `  i ) )   &    |-  ( ( ph  /\  i  e.  ( 1
 ... M ) ) 
 ->  ( B `  i
 )  <_  1 )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E. j  e.  ( 1 ... M ) ( B `  j )  <  E )   =>    |-  ( ph  ->  ( A `  M )  <  E )
 
Theoremcncfmptss 27820* A continuous complex function restricted to a subset is continuous, using "map to" notation. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F/_ x F   &    |-  ( ph  ->  F  e.  ( A -cn-> B ) )   &    |-  ( ph  ->  C 
 C_  A )   =>    |-  ( ph  ->  ( x  e.  C  |->  ( F `  x ) )  e.  ( C
 -cn-> B ) )
 
Theoremfmptdf 27821* A version of fmptd 5700 using bound-variable hypothesis instead of a distinct variable condition for  ph. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  C )   &    |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( ph  ->  F : A --> C )
 
Theoremrrpsscn 27822 The positive reals are a subset of the complex numbers. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  RR+  C_  CC
 
Theoremmulcncf 27823* The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( X -cn-> CC )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( X -cn-> CC ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A  x.  B ) )  e.  ( X -cn-> CC ) )
 
Theoremmulc1cncfg 27824* A version of mulc1cncf 18425 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
 |-  F/_ x F   &    |- 
 F/ x ph   &    |-  ( ph  ->  F  e.  ( A -cn-> CC ) )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  x.  ( F `
  x ) ) )  e.  ( A
 -cn-> CC ) )
 
Theoreminfrglb 27825* The infimum of a non-empty bounded set of reals is the greatest lower bound. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  (
 ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
 )  /\  B  e.  RR )  ->  ( sup ( A ,  RR ,  `'  <  )  <  B  <->  E. z  e.  A  z  <  B ) )
 
Theoremexpcncf 27826* The power function on complex numbers, for fixed exponent N, is continuous. Similar to expcn 18392. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  ( N  e.  NN0  ->  ( x  e.  CC  |->  ( x ^ N ) )  e.  ( CC -cn-> CC ) )
 
Theoremeluzelcn 27827 A member of a set of upper integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  N  e.  CC )
 
Theoremm1expeven 27828 Exponentiation of negative one to an even power. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  ( N  e.  NN0  ->  ( -u 1 ^ ( 2  x.  N ) )  =  1 )
 
Theoremexpcnfg 27829* If  F is a complex continuous function and N is a fixed number, then F^N is continuous too. A generalization of expcncf 27826. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F/_ x F   &    |-  ( ph  ->  F  e.  ( A -cn-> CC )
 )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( ( F `  x ) ^ N ) )  e.  ( A -cn-> CC ) )
 
18.20.3  Limits
 
Theoremclim1fr1 27830* A class of sequences of fractions that converge to 1 (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F  =  ( n  e.  NN  |->  ( ( ( A  x.  n )  +  B )  /  ( A  x.  n ) ) )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  F  ~~>  1 )
 
Theoremisumneg 27831* Negation of a converging sum. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  sum_ k  e.  Z  A  e.  CC )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  e.  CC )   &    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  sum_ k  e.  Z  -u A  =  -u sum_
 k  e.  Z  A )
 
Theoremclimrec 27832* Limit of the reciprocal of a converging sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  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 ) )
 
Theoremclimmulf 27833* A version of climmul 12122 using bound-variable hypotheses instead of distinct variable conditions. (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  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `  k )  x.  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( A  x.  B ) )
 
Theoremclimexp 27834* The limit of natural powers, is the natural power of the limit. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F/ k ph   &    |-  F/_ k F   &    |-  F/_ k H   &    |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> CC )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  H  e.  V )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `  k ) ^ N ) )   =>    |-  ( ph  ->  H  ~~>  ( A ^ N ) )
 
Theoremcliminf 27835* A bounded monotonic non increasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  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 ,  `'  <  )
 )
 
Theoremclimsuselem1 27836* The subsequence index  I has the expected properties: it belongs to the same upper integers as the original index, and it is always larger or equal than the original index. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  ( I `  M )  e.  Z )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( I `  (
 k  +  1 ) )  e.  ( ZZ>= `  ( ( I `  k )  +  1
 ) ) )   =>    |-  ( ( ph  /\  K  e.  Z ) 
 ->  ( I `  K )  e.  ( ZZ>= `  K ) )
 
Theoremclimsuse 27837* 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 27838* A version of climrec 27832 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 27839* 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 27840* A version of climinf 27835 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 27841* 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 27842 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 ) )
 
18.20.4  Derivatives
 
Theoremdvsinexp 27843* 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 27844 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 ) )
 
18.20.5  Integrals
 
Theoremioovolcl 27845 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 27846 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 27847* 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 27848* 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 27849 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 27850* 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 27851* 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 27852* 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
 ) ) ) )
 
18.20.6  Stone Weierstrass theorem - real version
 
Theoremstoweidlem1 27853 Lemma for stoweid 27915. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90; the key step uses Bernoulli's inequality bernneq 11243. (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 27854* lemma for stoweid 27915: 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 27855* Lemma for stoweid 27915: 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 27856* Lemma for stoweid 27915: 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 27857* 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 27858* Lemma for stoweid 27915: 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 27859* 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 27860* Lemma for stoweid 27915: 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 27861* Lemma for stoweid 27915: 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 27862 Lemma for stoweid 27915. 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 27863* 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 27864* Lemma for stoweid 27915. 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 27865 Lemma for stoweid 27915. 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 27866* 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 27867* 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 27868* Lemma for stoweid 27915. 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 27869* 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 27870* 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 27871* 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 27872* 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 27873* 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 27874* 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 27875* 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 27876* 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 27877* 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 27878* 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 27879* 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 27880* 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 27881* 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 27882* 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 27883* 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 27884* 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 27885* 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 27886* 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 27887* 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 27888* 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 27889* 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 27890* 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 27891* 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 27892* 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 27893* 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 27894* 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 27895* 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 27896* 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 27897* 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 27898* 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 27899* 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 27900* 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 )
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