Article RSSPACE, MML version 4.99.1005

:: RSSPACE:funcnot 1 => RSSPACE:func 1
definition
  func the_set_of_RealSequences -> non empty set means
    for b1 being set holds
          b1 in it
       iff
          b1 is Function-like quasi_total Relation of NAT,REAL;
end;

:: RSSPACE:def 1
theorem
for b1 being non empty set holds
      b1 = the_set_of_RealSequences
   iff
      for b2 being set holds
            b2 in b1
         iff
            b2 is Function-like quasi_total Relation of NAT,REAL;

:: RSSPACE:funcnot 2 => RSSPACE:func 2
definition
  let a1 be set;
  assume a1 in the_set_of_RealSequences;
  func seq_id A1 -> Function-like quasi_total Relation of NAT,REAL equals
    a1;
end;

:: RSSPACE:def 2
theorem
for b1 being set
      st b1 in the_set_of_RealSequences
   holds seq_id b1 = b1;

:: RSSPACE:funcnot 3 => RSSPACE:func 3
definition
  let a1 be set;
  assume a1 in REAL;
  func R_id A1 -> Element of REAL equals
    a1;
end;

:: RSSPACE:def 3
theorem
for b1 being set
      st b1 in REAL
   holds R_id b1 = b1;

:: RSSPACE:th 1
theorem
ex b1 being Function-like quasi_total Relation of [:the_set_of_RealSequences,the_set_of_RealSequences:],the_set_of_RealSequences st
   (for b2, b3 being Element of the_set_of_RealSequences holds
    b1 .(b2,b3) = (seq_id b2) + seq_id b3) &
    b1 is commutative(the_set_of_RealSequences) &
    b1 is associative(the_set_of_RealSequences);

:: RSSPACE:th 2
theorem
ex b1 being Function-like quasi_total Relation of [:REAL,the_set_of_RealSequences:],the_set_of_RealSequences st
   for b2, b3 being set
         st b2 in REAL & b3 in the_set_of_RealSequences
      holds b1 .(b2,b3) = (R_id b2) (#) seq_id b3;

:: RSSPACE:funcnot 4 => RSSPACE:func 4
definition
  func l_add -> Function-like quasi_total Relation of [:the_set_of_RealSequences,the_set_of_RealSequences:],the_set_of_RealSequences means
    for b1, b2 being Element of the_set_of_RealSequences holds
    it .(b1,b2) = (seq_id b1) + seq_id b2;
end;

:: RSSPACE:def 4
theorem
for b1 being Function-like quasi_total Relation of [:the_set_of_RealSequences,the_set_of_RealSequences:],the_set_of_RealSequences holds
      b1 = l_add
   iff
      for b2, b3 being Element of the_set_of_RealSequences holds
      b1 .(b2,b3) = (seq_id b2) + seq_id b3;

:: RSSPACE:funcnot 5 => RSSPACE:func 5
definition
  func l_mult -> Function-like quasi_total Relation of [:REAL,the_set_of_RealSequences:],the_set_of_RealSequences means
    for b1, b2 being set
          st b1 in REAL & b2 in the_set_of_RealSequences
       holds it .(b1,b2) = (R_id b1) (#) seq_id b2;
end;

:: RSSPACE:def 5
theorem
for b1 being Function-like quasi_total Relation of [:REAL,the_set_of_RealSequences:],the_set_of_RealSequences holds
      b1 = l_mult
   iff
      for b2, b3 being set
            st b2 in REAL & b3 in the_set_of_RealSequences
         holds b1 .(b2,b3) = (R_id b2) (#) seq_id b3;

:: RSSPACE:funcnot 6 => RSSPACE:func 6
definition
  func Zeroseq -> Element of the_set_of_RealSequences means
    for b1 being Element of NAT holds
       (seq_id it) . b1 = 0;
end;

:: RSSPACE:def 6
theorem
for b1 being Element of the_set_of_RealSequences holds
      b1 = Zeroseq
   iff
      for b2 being Element of NAT holds
         (seq_id b1) . b2 = 0;

:: RSSPACE:th 3
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   seq_id b1 = b1;

:: RSSPACE:th 4
theorem
for b1, b2 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
b1 + b2 = (seq_id b1) + seq_id b2;

:: RSSPACE:th 5
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
   b1 * b2 = b1 (#) seq_id b2;

:: RSSPACE:funcreg 1
registration
  cluster RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) -> strict Abelian;
end;

:: RSSPACE:th 6
theorem
for b1, b2, b3 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
(b1 + b2) + b3 = b1 + (b2 + b3);

:: RSSPACE:th 7
theorem
for b1 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
   b1 + 0. RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) = b1;

:: RSSPACE:th 8
theorem
for b1 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
   ex b2 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) st
      b1 + b2 = 0. RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#);

:: RSSPACE:th 9
theorem
for b1 being Element of REAL
for b2, b3 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
b1 * (b2 + b3) = (b1 * b2) + (b1 * b3);

:: RSSPACE:th 10
theorem
for b1, b2 being Element of REAL
for b3 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
   (b1 + b2) * b3 = (b1 * b3) + (b2 * b3);

:: RSSPACE:th 11
theorem
for b1, b2 being Element of REAL
for b3 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
   (b1 * b2) * b3 = b1 * (b2 * b3);

:: RSSPACE:th 12
theorem
for b1 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
   1 * b1 = b1;

:: RSSPACE:funcnot 7 => RSSPACE:func 7
definition
  func Linear_Space_of_RealSequences -> RLSStruct equals
    RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#);
end;

:: RSSPACE:def 7
theorem
Linear_Space_of_RealSequences = RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#);

:: RSSPACE:funcreg 2
registration
  cluster Linear_Space_of_RealSequences -> non empty strict;
end;

:: RSSPACE:funcreg 3
registration
  cluster Linear_Space_of_RealSequences -> right_complementable Abelian add-associative right_zeroed RealLinearSpace-like;
end;

:: RSSPACE:funcnot 8 => RSSPACE:func 8
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Element of bool the carrier of a1;
  assume a2 is linearly-closed(a1) & a2 is not empty;
  func Add_(A2,A1) -> Function-like quasi_total Relation of [:a2,a2:],a2 equals
    (the addF of a1) || a2;
end;

:: RSSPACE:def 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
      st b2 is linearly-closed(b1) & b2 is not empty
   holds Add_(b2,b1) = (the addF of b1) || b2;

:: RSSPACE:funcnot 9 => RSSPACE:func 9
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Element of bool the carrier of a1;
  assume a2 is linearly-closed(a1) & a2 is not empty;
  func Mult_(A2,A1) -> Function-like quasi_total Relation of [:REAL,a2:],a2 equals
    (the Mult of a1) | [:REAL,a2:];
end;

:: RSSPACE:def 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
      st b2 is linearly-closed(b1) & b2 is not empty
   holds Mult_(b2,b1) = (the Mult of b1) | [:REAL,b2:];

:: RSSPACE:funcnot 10 => RSSPACE:func 10
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Element of bool the carrier of a1;
  assume a2 is linearly-closed(a1) & a2 is not empty;
  func Zero_(A2,A1) -> Element of a2 equals
    0. a1;
end;

:: RSSPACE:def 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
      st b2 is linearly-closed(b1) & b2 is not empty
   holds Zero_(b2,b1) = 0. b1;

:: RSSPACE:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
      st b2 is linearly-closed(b1) & b2 is not empty
   holds RLSStruct(#b2,Zero_(b2,b1),Add_(b2,b1),Mult_(b2,b1)#) is Subspace of b1;

:: RSSPACE:funcnot 11 => RSSPACE:func 11
definition
  func the_set_of_l2RealSequences -> Element of bool the carrier of Linear_Space_of_RealSequences means
    for b1 being set holds
          b1 in it
       iff
          b1 in the_set_of_RealSequences & (seq_id b1) (#) seq_id b1 is summable;
end;

:: RSSPACE:def 11
theorem
for b1 being Element of bool the carrier of Linear_Space_of_RealSequences holds
      b1 = the_set_of_l2RealSequences
   iff
      for b2 being set holds
            b2 in b1
         iff
            b2 in the_set_of_RealSequences & (seq_id b2) (#) seq_id b2 is summable;

:: RSSPACE:funcreg 4
registration
  cluster the_set_of_l2RealSequences -> non empty linearly-closed;
end;

:: RSSPACE:th 15
theorem
RLSStruct(#the_set_of_l2RealSequences,Zero_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Add_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Mult_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences)#) is Subspace of Linear_Space_of_RealSequences;

:: RSSPACE:th 16
theorem
RLSStruct(#the_set_of_l2RealSequences,Zero_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Add_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Mult_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences)#) is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;

:: RSSPACE:th 17
theorem
the carrier of Linear_Space_of_RealSequences = the_set_of_RealSequences &
 (for b1 being set holds
       b1 is Element of the carrier of Linear_Space_of_RealSequences
    iff
       b1 is Function-like quasi_total Relation of NAT,REAL) &
 (for b1 being Element of the carrier of Linear_Space_of_RealSequences holds
    b1 = seq_id b1) &
 (for b1, b2 being Element of the carrier of Linear_Space_of_RealSequences holds
 b1 + b2 = (seq_id b1) + seq_id b2) &
 (for b1 being Element of REAL
 for b2 being Element of the carrier of Linear_Space_of_RealSequences holds
    b1 * b2 = b1 (#) seq_id b2);

:: RSSPACE:th 18
theorem
ex b1 being Function-like quasi_total Relation of [:the_set_of_l2RealSequences,the_set_of_l2RealSequences:],REAL st
   for b2, b3 being set
         st b2 in the_set_of_l2RealSequences & b3 in the_set_of_l2RealSequences
      holds b1 .(b2,b3) = Sum ((seq_id b2) (#) seq_id b3);

:: RSSPACE:funcnot 12 => RSSPACE:func 12
definition
  func l_scalar -> Function-like quasi_total Relation of [:the_set_of_l2RealSequences,the_set_of_l2RealSequences:],REAL means
    for b1, b2 being set
          st b1 in the_set_of_l2RealSequences & b2 in the_set_of_l2RealSequences
       holds it .(b1,b2) = Sum ((seq_id b1) (#) seq_id b2);
end;

:: RSSPACE:def 12
theorem
for b1 being Function-like quasi_total Relation of [:the_set_of_l2RealSequences,the_set_of_l2RealSequences:],REAL holds
      b1 = l_scalar
   iff
      for b2, b3 being set
            st b2 in the_set_of_l2RealSequences & b3 in the_set_of_l2RealSequences
         holds b1 .(b2,b3) = Sum ((seq_id b2) (#) seq_id b3);

:: RSSPACE:funcreg 5
registration
  cluster UNITSTR(#the_set_of_l2RealSequences,Zero_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Add_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Mult_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),l_scalar#) -> non empty strict;
end;

:: RSSPACE:funcnot 13 => RSSPACE:func 13
definition
  func l2_Space -> non empty UNITSTR equals
    UNITSTR(#the_set_of_l2RealSequences,Zero_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Add_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Mult_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),l_scalar#);
end;

:: RSSPACE:def 13
theorem
l2_Space = UNITSTR(#the_set_of_l2RealSequences,Zero_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Add_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Mult_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),l_scalar#);

:: RSSPACE:th 19
theorem
for b1 being RLSStruct
      st RLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#) is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
   holds b1 is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;

:: RSSPACE:th 20
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st for b2 being Element of NAT holds
           b1 . b2 = 0
   holds b1 is summable & Sum b1 = 0;

:: RSSPACE:th 21
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st (for b2 being Element of NAT holds
         0 <= b1 . b2) &
      b1 is summable &
      Sum b1 = 0
for b2 being Element of NAT holds
   b1 . b2 = 0;

:: RSSPACE:funcreg 6
registration
  cluster l2_Space -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like;
end;