:: Free Modules
:: by Michal Muzalewski
::
:: Received October 18, 1991
:: Copyright (c) 1991-2012 Association of Mizar Users


begin

Lm1:  for R being Ring
for a being Scalar of R st - a = 0. R holds
a = 0. R
proofend;

theorem Th1: :: MOD_3:1
for R being non empty non degenerated right_complementable add-associative right_zeroed doubleLoopStr holds 0. R <> - (1. R)
proofend;

theorem Th2: :: MOD_3:2
for R being Ring
for V being LeftMod of R
for L being Linear_Combination of V
for C being finite Subset of V st Carrier L c= C holds
ex F being FinSequence of the carrier of V st
( F is one-to-one &amp; rng F = C &amp; Sum L = Sum (L (#) F) )
proofend;

theorem Th3: :: MOD_3:3
for R being Ring
for V being LeftMod of R
for L being Linear_Combination of V
for a being Scalar of R holds Sum (a * L) = a * (Sum L)
proofend;

definition
let R be Ring;
let V be LeftMod of R;
let A be Subset of V;
func Lin A ->  strict Subspace of V means :Def1: :: MOD_3:def 1
 the carrier of it =  {  (Sum l) where l is Linear_Combination of A : verum  }  ;
existence
 ex b1 being strict Subspace of V st the carrier of b1 =  {  (Sum l) where l is Linear_Combination of A : verum  }
proofend;
uniqueness
 for b1, b2 being strict Subspace of V st the carrier of b1 =  {  (Sum l) where l is Linear_Combination of A : verum  }  &amp; the carrier of b2 =  {  (Sum l) where l is Linear_Combination of A : verum  }  holds
b1 = b2 by VECTSP_4:29;
end;

:: deftheorem Def1 defines Lin MOD_3:def_1_:_
 for R being Ring
for V being LeftMod of R
for A being Subset of V
for b4 being strict Subspace of V holds
( b4 = Lin A iff the carrier of b4 =  {  (Sum l) where l is Linear_Combination of A : verum  }  );

theorem Th4: :: MOD_3:4
for x being set
for R being Ring
for V being LeftMod of R
for A being Subset of V holds
( x in Lin A iff ex l being Linear_Combination of A st x = Sum l )
proofend;

theorem Th5: :: MOD_3:5
for x being set
for R being Ring
for V being LeftMod of R
for A being Subset of V st x in A holds
x in Lin A
proofend;

theorem Th6: :: MOD_3:6
for R being Ring
for V being LeftMod of R holds Lin ({} the carrier of V) = (0). V
proofend;

theorem :: MOD_3:7
for R being Ring
for V being LeftMod of R
for A being Subset of V holds
( not Lin A = (0). V or A = {} or A = {(0. V)} )
proofend;

theorem Th8: :: MOD_3:8
for R being Ring
for V being LeftMod of R
for A being Subset of V
for W being strict Subspace of V st 0. R <> 1. R &amp; A = the carrier of W holds
Lin A = W
proofend;

theorem :: MOD_3:9
for R being Ring
for V being strict LeftMod of R
for A being Subset of V st 0. R <> 1. R &amp; A = the carrier of V holds
Lin A = V
proofend;

theorem Th10: :: MOD_3:10
for R being Ring
for V being LeftMod of R
for A, B being Subset of V st A c= B holds
Lin A is Subspace of Lin B
proofend;

theorem :: MOD_3:11
for R being Ring
for V being LeftMod of R
for A, B being Subset of V st Lin A = V &amp; A c= B holds
Lin B = V
proofend;

theorem :: MOD_3:12
for R being Ring
for V being LeftMod of R
for A, B being Subset of V holds Lin (A \/ B) = (Lin A) + (Lin B)
proofend;

theorem :: MOD_3:13
for R being Ring
for V being LeftMod of R
for A, B being Subset of V holds Lin (A /\ B) is Subspace of (Lin A) /\ (Lin B)
proofend;

definition
let R be Ring;
let V be LeftMod of R;
let IT be Subset of V;
attrIT is base  means :Def2: :: MOD_3:def 2
( IT is linearly-independent &amp; Lin IT = VectSpStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) );
end;

:: deftheorem Def2 defines base MOD_3:def_2_:_
 for R being Ring
for V being LeftMod of R
for IT being Subset of V holds
( IT is base iff ( IT is linearly-independent &amp; Lin IT = VectSpStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) ) );

definition
let R be Ring;
let IT be LeftMod of R;
attrIT is free  means :Def3: :: MOD_3:def 3
 ex B being Subset of IT st B is base ;
end;

:: deftheorem Def3 defines free MOD_3:def_3_:_
 for R being Ring
for IT being LeftMod of R holds
( IT is free iff ex B being Subset of IT st B is base );

theorem Th14: :: MOD_3:14
for R being Ring
for V being LeftMod of R holds (0). V is free
proofend;

registration
let R be Ring;
cluster non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed free for VectSpStr over R;
existence
 ex b1 being LeftMod of R st
( b1 is strict &amp; b1 is free )
proofend;
end;

Lm2:  for R being Skew-Field
for a being Scalar of R
for V being LeftMod of R
for v being Vector of V st a <> 0. R holds
( (a ") * (a * v) = (1. R) * v &amp; ((a ") * a) * v = (1. R) * v )
proofend;

theorem :: MOD_3:15
for R being Skew-Field
for V being LeftMod of R
for v being Vector of V holds
( {v} is linearly-independent iff v <> 0. V )
proofend;

theorem Th16: :: MOD_3:16
for R being Skew-Field
for V being LeftMod of R
for v1, v2 being Vector of V holds
( v1 <> v2 &amp; {v1,v2} is linearly-independent iff ( v2 <> 0. V &amp; ( for a being Scalar of R holds v1 <> a * v2 ) ) )
proofend;

theorem :: MOD_3:17
for R being Skew-Field
for V being LeftMod of R
for v1, v2 being Vector of V holds
( ( v1 <> v2 &amp; {v1,v2} is linearly-independent ) iff for a, b being Scalar of R st (a * v1) + (b * v2) = 0. V holds
( a = 0. R &amp; b = 0. R ) )
proofend;

theorem Th18: :: MOD_3:18
for R being Skew-Field
for V being LeftMod of R
for A being Subset of V st A is linearly-independent holds
ex B being Subset of V st
( A c= B &amp; B is base )
proofend;

theorem Th19: :: MOD_3:19
for R being Skew-Field
for V being LeftMod of R
for A being Subset of V st Lin A = V holds
ex B being Subset of V st
( B c= A &amp; B is base )
proofend;

Lm3:  for R being Skew-Field
for V being LeftMod of R ex B being Subset of V st B is base
proofend;

theorem :: MOD_3:20
for R being Skew-Field
for V being LeftMod of R holds V is free
proofend;

definition
let R be Skew-Field;
let V be LeftMod of R;
mode Basis of V ->  Subset of V means :Def4: :: MOD_3:def 4
it is base ;
existence
 ex b1 being Subset of V st b1 is base by Lm3;
end;

:: deftheorem Def4 defines Basis MOD_3:def_4_:_
 for R being Skew-Field
for V being LeftMod of R
for b3 being Subset of V holds
( b3 is Basis of V iff b3 is base );

theorem :: MOD_3:21
for R being Skew-Field
for V being LeftMod of R
for A being Subset of V st A is linearly-independent holds
ex I being Basis of V st A c= I
proofend;

theorem :: MOD_3:22
for R being Skew-Field
for V being LeftMod of R
for A being Subset of V st Lin A = V holds
ex I being Basis of V st I c= A
proofend;