Article CONVEX2, MML version 4.99.1005

:: CONVEX2:th 1
theorem
for b1 being non empty RLSStruct
for b2, b3 being convex Element of bool the carrier of b1 holds
b2 /\ b3 is convex(b1);

:: CONVEX2:th 2
theorem
for b1 being non empty RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being FinSequence of the carrier of b1
for b4 being FinSequence of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: for b6 being Element of NAT
              st b6 in (dom b3) /\ dom b4
           holds ex b7 being Element of the carrier of b1 st
              b7 = b3 . b6 & b5 .|. b7 <= b4 . b6}
   holds b2 is convex(b1);

:: CONVEX2:th 3
theorem
for b1 being non empty RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being FinSequence of the carrier of b1
for b4 being FinSequence of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: for b6 being Element of NAT
              st b6 in (dom b3) /\ dom b4
           holds ex b7 being Element of the carrier of b1 st
              b7 = b3 . b6 & b5 .|. b7 < b4 . b6}
   holds b2 is convex(b1);

:: CONVEX2:th 4
theorem
for b1 being non empty RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being FinSequence of the carrier of b1
for b4 being FinSequence of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: for b6 being Element of NAT
              st b6 in (dom b3) /\ dom b4
           holds ex b7 being Element of the carrier of b1 st
              b7 = b3 . b6 & b4 . b6 <= b5 .|. b7}
   holds b2 is convex(b1);

:: CONVEX2:th 5
theorem
for b1 being non empty RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being FinSequence of the carrier of b1
for b4 being FinSequence of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: for b6 being Element of NAT
              st b6 in (dom b3) /\ dom b4
           holds ex b7 being Element of the carrier of b1 st
              b7 = b3 . b6 & b4 . b6 < b5 .|. b7}
   holds b2 is convex(b1);

:: CONVEX2:th 6
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 holds
      for b3 being Element of bool the carrier of b1
      for b4 being Linear_Combination of b3
            st b4 is convex(b1) & b3 c= b2
         holds Sum b4 in b2
   iff
      b2 is convex(b1);

:: CONVEX2:funcnot 1 => CONVEX2:func 1
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;
  func LinComb A2 -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 is Linear_Combination of a2;
end;

:: CONVEX2:def 1
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
for b3 being set holds
      b3 = LinComb b2
   iff
      for b4 being set holds
            b4 in b3
         iff
            b4 is Linear_Combination of b2;

:: CONVEX2:exreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster Relation-like Function-like quasi_total complex-valued ext-real-valued real-valued convex Linear_Combination of a1;
end;

:: CONVEX2:modenot 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  mode Convex_Combination of a1 is convex Linear_Combination of a1;
end;

:: CONVEX2:exreg 2
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be non empty Element of bool the carrier of a1;
  cluster Relation-like Function-like quasi_total complex-valued ext-real-valued real-valued convex Linear_Combination of a2;
end;

:: CONVEX2:modenot 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be non empty Element of bool the carrier of a1;
  mode Convex_Combination of a2 is convex Linear_Combination of a2;
end;

:: CONVEX2:th 7
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 holds
   Convex-Family b2 <> {};

:: CONVEX2:th 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 holds
   b2 c= conv b2;

:: CONVEX2:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being convex Linear_Combination of b1
for b4 being Element of REAL
      st 0 < b4 & b4 < 1
   holds (b4 * b2) + ((1 - b4) * b3) is convex Linear_Combination of b1;

:: CONVEX2:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty Element of bool the carrier of b1
for b3, b4 being convex Linear_Combination of b2
for b5 being Element of REAL
      st 0 < b5 & b5 < 1
   holds (b5 * b3) + ((1 - b5) * b4) is convex Linear_Combination of b2;

:: CONVEX2:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   ex b2 being Linear_Combination of b1 st
      b2 is convex(b1);

:: CONVEX2:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty Element of bool the carrier of b1 holds
   ex b3 being Linear_Combination of b2 st
      b3 is convex(b1);

:: CONVEX2:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1 holds
Up (b2 + b3) = (Up b2) + Up b3;

:: CONVEX2:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1 holds
Up (b2 /\ b3) = (Up b2) /\ Up b3;

:: CONVEX2:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being convex Linear_Combination of b1
for b4, b5 being Element of REAL
      st 0 < b4 * b5
   holds Carrier ((b4 * b2) + (b5 * b3)) = (Carrier (b4 * b2)) \/ Carrier (b5 * b3);

:: CONVEX2:th 16
theorem
for b1, b2 being Relation-like Function-like set
   st b1,b2 are_fiberwise_equipotent
for b3, b4 being set
      st b3 in proj1 b1 & b4 in proj1 b1 & b3 <> b4
   holds ex b5, b6 being set st
      b5 in proj1 b2 & b6 in proj1 b2 & b5 <> b6 & b1 . b3 = b2 . b5 & b1 . b4 = b2 . b6;

:: CONVEX2:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1
for b3 being Element of bool the carrier of b1
      st b3 c= Carrier b2
   holds ex b4 being Linear_Combination of b1 st
      Carrier b4 = b3;