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Theorem List for Metamath Proof Explorer - 27001-27100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremconss1 27001 Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  (
 ( _V  \  A )  C_  B  <->  ( _V  \  B )  C_  A )
 
Theoremralbidar 27002 More general form of ralbida 2530. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( ph  ->  A. x  e.  A  ph )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. x  e.  A  ch ) )
 
Theoremrexbidar 27003 More general form of rexbida 2531. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( ph  ->  A. x  e.  A  ph )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  A  ch ) )
 
Theoremdropab1 27004 Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( A. x  x  =  y  ->  { <. x ,  z >.  |  ph }  =  { <. y ,  z >.  |  ph } )
 
Theoremdropab2 27005 Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( A. x  x  =  y  ->  { <. z ,  x >.  |  ph }  =  { <. z ,  y >.  |  ph } )
 
Theoremipo0 27006 If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  (  _I  Po  A  <->  A  =  (/) )
 
Theoremifr0 27007 A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  (  _I  Fr  A  <->  A  =  (/) )
 
Theoremordpss 27008 ordelpss 4378 with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( Ord  B  ->  ( A  e.  B  ->  A  C.  B ) )
 
Theoremfvsb 27009* Explicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
 |-  ( E! y  A F y  ->  ( [. ( F `  A )  /  x ]. ph  <->  E. x ( A. y ( A F y 
 <->  y  =  x ) 
 /\  ph ) ) )
 
Theoremfveqsb 27010* Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
 |-  ( x  =  ( F `  A )  ->  ( ph 
 <->  ps ) )   &    |-  F/ x ps   =>    |-  ( E! y  A F y  ->  ( ps 
 <-> 
 E. x ( A. y ( A F y 
 <->  y  =  x ) 
 /\  ph ) ) )
 
TheoremxrltneNEW 27011 'Less than' implies not equal for extended reals. (Contributed by Andrew Salmon, 11-Nov-2011.)
 |-  (
 ( A  e.  RR*  /\  A  <  B ) 
 ->  A  =/=  B )
 
Theoremxpexb 27012 A Cartesian product exists iff its converse does. Corollary 6.9(1) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)
 |-  (
 ( A  X.  B )  e.  _V  <->  ( B  X.  A )  e.  _V )
 
Theoremxpexcnv 27013 A condition where the converse of xpex 4775 holds as well. Corollary 6.9(2) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)
 |-  (
 ( B  =/=  (/)  /\  ( A  X.  B )  e. 
 _V )  ->  A  e.  _V )
 
Theoremtrelpss 27014 An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 4335, ax-reg 7260 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.)
 |-  (
 ( Tr  A  /\  B  e.  A )  ->  B  C.  A )
 
16.18.6  Arithmetic
 
Theoremaddcomgi 27015 Generalization of commutative law for addition. Simplifies proofs dealing with vectors. However, it is dependent on our particular definition of ordered pair. (Contributed by Andrew Salmon, 28-Jan-2012.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  ( A  +  B )  =  ( B  +  A )
 
16.18.7  Geometry
 
Syntaxcplusr 27016 Introduce the operation of vector addition.
 class  + r
 
Syntaxcminusr 27017 Introduce the operation of vector subtraction.
 class  - r
 
Syntaxctimesr 27018 Introduce the operation of scalar multiplication.
 class  . v
 
Syntaxcptdfc 27019  PtDf is a predicate that is crucial for the definition of lines as well as proving a number of important theorems.
 class  PtDf ( A ,  B )
 
Syntaxcrr3c 27020  RR 3 is a class.
 class  RR 3
 
Syntaxcline3 27021  line 3 is a class.
 class  line 3
 
Definitiondf-addr 27022* Define the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  + r  =  ( x  e.  _V ,  y  e. 
 _V  |->  ( v  e. 
 RR  |->  ( ( x `
  v )  +  ( y `  v
 ) ) ) )
 
Definitiondf-subr 27023* Define the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  - r  =  ( x  e.  _V ,  y  e. 
 _V  |->  ( v  e. 
 RR  |->  ( ( x `
  v )  -  ( y `  v
 ) ) ) )
 
Definitiondf-mulv 27024* Define the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  . v  =  ( x  e.  _V ,  y  e. 
 _V  |->  ( v  e. 
 RR  |->  ( x  x.  ( y `  v
 ) ) ) )
 
Theoremaddrval 27025* Value of the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A + r B )  =  (
 v  e.  RR  |->  ( ( A `  v
 )  +  ( B `
  v ) ) ) )
 
Theoremsubrval 27026* Value of the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A - r B )  =  (
 v  e.  RR  |->  ( ( A `  v
 )  -  ( B `
  v ) ) ) )
 
Theoremmulvval 27027* Value of the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A . v B )  =  (
 v  e.  RR  |->  ( A  x.  ( B `
  v ) ) ) )
 
Theoremaddrfv 27028 Vector addition at a value. The operation takes each vector  A and  B and forms a new vector whose values are the sum of each of the values of  A and  B. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  E  /\  B  e.  D  /\  C  e.  RR )  ->  ( ( A + r B ) `  C )  =  ( ( A `  C )  +  ( B `  C ) ) )
 
Theoremsubrfv 27029 Vector subtraction at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  E  /\  B  e.  D  /\  C  e.  RR )  ->  ( ( A - r B ) `  C )  =  ( ( A `  C )  -  ( B `  C ) ) )
 
Theoremmulvfv 27030 Scalar multiplication at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  E  /\  B  e.  D  /\  C  e.  RR )  ->  ( ( A . v B ) `  C )  =  ( A  x.  ( B `  C ) ) )
 
Theoremaddrfn 27031 Vector addition produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A + r B )  Fn  RR )
 
Theoremsubrfn 27032 Vector subtraction produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A - r B )  Fn  RR )
 
Theoremmulvfn 27033 Scalar multiplication producees a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A . v B )  Fn  RR )
 
Theoremaddrcom 27034 Vector addition is commutative. (Contributed by Andrew Salmon, 28-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A + r B )  =  ( B + r A ) )
 
Definitiondf-ptdf 27035* Define the predicate  PtDf, which is a utility definition used to shorten definitions and simplify proofs. (Contributed by Andrew Salmon, 15-Jul-2012.)
 |-  PtDf ( A ,  B )  =  ( x  e. 
 RR  |->  ( ( ( x . v ( B - r A ) ) +v A ) " { 1 ,  2 ,  3 } ) )
 
Definitiondf-rr3 27036 Define the set of all points  RR 3. We define each point  A as a function to allow the use of vector addition and subtraction as well as scalar multiplication in our proofs. (Contributed by Andrew Salmon, 15-Jul-2012.)
 |-  RR 3  =  ( RR  ^m 
 { 1 ,  2 ,  3 } )
 
Definitiondf-line3 27037* Define the set of all lines. A line is an infinite subset of  RR 3 that satisfies a  PtDf property. (Contributed by Andrew Salmon, 15-Jul-2012.)
 |-  line 3  =  { x  e. 
 ~P RR 3  |  ( 2o  ~<_  x  /\  A. y  e.  x  A. z  e.  x  (
 z  =/=  y  ->  ran  PtDf ( y ,  z
 )  =  x ) ) }
 
16.19  Mathbox for Glauco Siliprandi
 
16.19.1  Miscellanea
 
Theoremssrexf 27038 restricted existential quantification follows from a subclass relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  C_  B  ->  ( E. x  e.  A  ph  ->  E. x  e.  B  ph ) )
 
Theoremfnvinran 27039 the function value belongs to its codomain. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  ( ph  ->  F : A --> B )   =>    |-  ( ( ph  /\  C  e.  A )  ->  ( F `  C )  e.  B )
 
Theoremevth2f 27040* A version of evth2 18406 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   &    |-  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 `  x )  <_  ( F `
  y ) )
 
Theoremelunif 27041* A version of eluni 3790 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  e.  U. B 
 <-> 
 E. x ( A  e.  x  /\  x  e.  B ) )
 
Theoremrzalf 27042 A version of rzal 3516 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ x  A  =  (/)   =>    |-  ( A  =  (/)  ->  A. x  e.  A  ph )
 
Theoremfvelrnbf 27043 A version of fvelrnb 5490 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x F   =>    |-  ( F  Fn  A  ->  ( B  e.  ran 
 F 
 <-> 
 E. x  e.  A  ( F `  x )  =  B ) )
 
Theoremrfcnpre1 27044 If F is a continuous function with respect to the standard topology, then the preimage A of the values greater 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  |  B  <  ( F `  x ) }   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  A  e.  J )
 
Theoremubelsupr 27045* If U belongs to A and U is an upper bound, then U is the sup of A. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  (
 ( A  C_  RR  /\  U  e.  A  /\  A. x  e.  A  x  <_  U )  ->  U  =  sup ( A ,  RR ,  <  ) )
 
Theoremfsumcnf 27046* A finite sum of functions to complex numbers from a common topological space is continuous, without disoint var constraint x ph. The class expression for B normally contains free variables k and x to index it. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  K  =  ( TopOpen ` fld )   &    |-  ( 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 ) )
 
Theoremmulltgt0 27047 The product of a negative and a positive number is negative. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  (
 ( ( A  e.  RR  /\  A  <  0
 )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( A  x.  B )  <  0 )
 
Theoremrcla4egf 27048 A version of rcla4ev 2852 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ x ps   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  B  /\  ps )  ->  E. x  e.  B  ph )
 
Theoremrabexgf 27049 A version of rabexg 4124 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ x A   =>    |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )
 
Theoremfcnre 27050 A function continuous with respect to the standard topology, is a real mapping. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  K  =  ( topGen `  ran  (,) )   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  F  e.  C )   =>    |-  ( ph  ->  F : T --> RR )
 
Theoremsumsnd 27051* A sum of a singleton is the term. The deduction version of sumsn 12164. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  ( ph  ->  F/_ k B )   &    |-  F/ k ph   &    |-  ( ( ph  /\  k  =  M ) 
 ->  A  =  B )   &    |-  ( ph  ->  M  e.  V )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  sum_
 k  e.  { M } A  =  B )
 
Theoremevthf 27052* A version of evth 18405 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 27053 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 27054* 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 27055* 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 18322 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 27056 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 27057* 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 27058* 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 27059* 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 27060* 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 27061* Given a real continuous function  F, there is always a natural natural number that is a strict upper bound of it's 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 27062* This is the core Lemma for refsum2cn 27063: the sum of two continuus 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 27063* 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 ) )
 
16.19.2  Finite multiplication of numbers and finite multiplication of functions
 
Theoremfmul01 27064* 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 27065* 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 27066* induction step for the proof of fmuldfeq 27067. (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 27067* 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 27068* 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 27069* 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 27070* 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 )
 
16.19.3  Stone Weierstrass theorem - real version
 
Theoremstoweidlem1 27071 Lemma for stoweid 27133. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90; the key step uses Bernoulli's inequality bernneq 11179. (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 27072* lemma for stoweid 27133: 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 27073* Lemma for stoweid 27133: 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 27074* Lemma for stoweid 27133: 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 27075* 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 27076* Lemma for stoweid 27133: 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 27077* 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 27078* Lemma for stoweid 27133: 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 27079* Lemma for stoweid 27133: 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 27080 Lemma for stoweid 27133. 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 27081* 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 27082* Lemma for stoweid 27133. 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 27083 Lemma for stoweid 27133. 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 27084* 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 27085* 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 27086* Lemma for stoweid 27133. 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 27087* 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 27088* 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 27089* 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 27090* 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 27091* 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 27092* 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 27093* 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 27094* 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 27095* 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 27096* 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 27097* 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 27098* 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 27099* 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 27100* 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 )
 ) )
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