Article RMOD_5, MML version 4.99.1005

:: RMOD_5:attrnot 1 => RMOD_5:attr 1
definition
  let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  let a3 be Element of bool the carrier of a2;
  attr a3 is linearly-independent means
    for b1 being Linear_Combination of a3
          st Sum b1 = 0. a2
       holds Carrier b1 = {};
end;

:: RMOD_5:dfs 1
definiens
  let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  let a3 be Element of bool the carrier of a2;
To prove
     a3 is linearly-independent
it is sufficient to prove
  thus for b1 being Linear_Combination of a3
          st Sum b1 = 0. a2
       holds Carrier b1 = {};

:: RMOD_5:def 1
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of bool the carrier of b2 holds
      b3 is linearly-independent(b1, b2)
   iff
      for b4 being Linear_Combination of b3
            st Sum b4 = 0. b2
         holds Carrier b4 = {};

:: RMOD_5:attrnot 2 => RMOD_5:attr 1
notation
  let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  let a3 be Element of bool the carrier of a2;
  antonym linearly-dependent for linearly-independent;
end;

:: RMOD_5:th 2
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Element of bool the carrier of b2
      st b3 c= b4 & b4 is linearly-independent(b1, b2)
   holds b3 is linearly-independent(b1, b2);

:: RMOD_5:th 3
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of bool the carrier of b2
      st 0. b1 <> 1_ b1 & b3 is linearly-independent(b1, b2)
   holds not 0. b2 in b3;

:: RMOD_5:th 4
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1 holds
   {} the carrier of b2 is linearly-independent(b1, b2);

:: RMOD_5:th 5
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Element of the carrier of b2
      st 0. b1 <> 1_ b1 & {b3,b4} is linearly-independent(b1, b2)
   holds b3 <> 0. b2 & b4 <> 0. b2;

:: RMOD_5:th 6
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b2
      st 0. b1 <> 1_ b1
   holds {b3,0. b2} is linearly-dependent(b1, b2) & {0. b2,b3} is linearly-dependent(b1, b2);

:: RMOD_5:funcnot 1 => RMOD_5:func 1
definition
  let a1 be non empty non degenerated right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed domRing-like doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  let a3 be Element of bool the carrier of a2;
  func Lin A3 -> strict Submodule of a2 means
    the carrier of it = {Sum b1 where b1 is Linear_Combination of a3: TRUE};
end;

:: RMOD_5:def 2
theorem
for b1 being non empty non degenerated right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed domRing-like doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of bool the carrier of b2
for b4 being strict Submodule of b2 holds
      b4 = Lin b3
   iff
      the carrier of b4 = {Sum b5 where b5 is Linear_Combination of b3: TRUE};

:: RMOD_5:th 9
theorem
for b1 being set
for b2 being non empty non degenerated right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed domRing-like doubleLoopStr
for b3 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b2
for b4 being Element of bool the carrier of b3 holds
      b1 in Lin b4
   iff
      ex b5 being Linear_Combination of b4 st
         b1 = Sum b5;

:: RMOD_5:th 10
theorem
for b1 being set
for b2 being non empty non degenerated right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed domRing-like doubleLoopStr
for b3 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b2
for b4 being Element of bool the carrier of b3
      st b1 in b4
   holds b1 in Lin b4;

:: RMOD_5:th 11
theorem
for b1 being non empty non degenerated right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed domRing-like doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1 holds
   Lin {} the carrier of b2 = (0). b2;

:: RMOD_5:th 12
theorem
for b1 being non empty non degenerated right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed domRing-like doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of bool the carrier of b2
      st Lin b3 = (0). b2 & b3 <> {}
   holds b3 = {0. b2};

:: RMOD_5:th 13
theorem
for b1 being non empty non degenerated right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed domRing-like doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of bool the carrier of b2
for b4 being strict Submodule of b2
      st 0. b1 <> 1_ b1 & b3 = the carrier of b4
   holds Lin b3 = b4;

:: RMOD_5:th 14
theorem
for b1 being non empty non degenerated right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed domRing-like doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed strict RightMod-like RightModStr over b1
for b3 being Element of bool the carrier of b2
      st 0. b1 <> 1_ b1 & b3 = the carrier of b2
   holds Lin b3 = b2;

:: RMOD_5:th 15
theorem
for b1 being non empty non degenerated right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed domRing-like doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Element of bool the carrier of b2
      st b3 c= b4
   holds Lin b3 is Submodule of Lin b4;

:: RMOD_5:th 16
theorem
for b1 being non empty non degenerated right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed domRing-like doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed strict RightMod-like RightModStr over b1
for b3, b4 being Element of bool the carrier of b2
      st Lin b3 = b2 & b3 c= b4
   holds Lin b4 = b2;

:: RMOD_5:th 17
theorem
for b1 being non empty non degenerated right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed domRing-like doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Element of bool the carrier of b2 holds
Lin (b3 \/ b4) = (Lin b3) + Lin b4;

:: RMOD_5:th 18
theorem
for b1 being non empty non degenerated right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed domRing-like doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Element of bool the carrier of b2 holds
Lin (b3 /\ b4) is Submodule of (Lin b3) /\ Lin b4;